1 Introduction

1.1 Previous Related Work

In 2019 the authors, surprised by the remanence of the Computer Algebra System (CAS) DERIVEFootnote 1 more than 10 years after it was discontinued, published an article [19] about how it was still used for teaching and how it was still mentioned in several papers. It included a bibliographic study about the papers (most devoted to educational issues) that still referred to DERIVE.

1.2 First Notes About the Dynamic Geometry System GeoGebra and this Study

Nowadays the Dynamic Geometry System (DGS) GeoGebra has become a very successful piece of software, claiming over 100 million users (!!!) all around the world (many more than any other DGS or CAS). It now addresses not only dynamic geometry but also includes algebraic capabilities.

GeoGebra is a free piece of software that welcomes the contributions and ideas from its users and is spread through a network of the so called GeoGebra Institutes.

We shall give afterwards a brief overview of DGS in general, as well as a summary of the main characteristics and capabilities of GeoGebra. Finally, a bibliographic study of the evolution of the papers mentioning GeoGebra will be presented.

1.3 About the Authors

The first author has taught computational mathematics to students from the School of Education at the Universidad Complutense de Madrid for 36 years within the frame of different subjects about the use of information and communication technologies (ICT) in mathematics teaching. In these subjects he has used different hardware and languages (in the past, mainly, Logo, Derive and The Geometer’s Sketchpad and now, mainly, Scratch, Maple and GeoGebra). He hs also taught computational mathematics to postgraduates at the School of Mathematics along these years. He was beta tester of the DGS The Geometer’s Sketchpad.

The second author has been a teacher of the School of Librarians of the Universidad de Extremadura for 27 years. She is specialized in quantitative studies in Social Sciences and Humanities.

2 About DGS

2.1 The Pioneers

Cabri Géomètre (later renamed Cabri Geometry II [28] and now Cabri II Plus) and The Geometer’s Sketchpad [29, 34], were available in the early ‘90 s. They included the main features of DGS: dynamism and a mouse-based data introduction, allowing to comfortably experiment with plane geometry (Fig. 1).

Fig. 1
figure 1

Circumcircle of a triangle constructed with The Geometer’s Sketchpad v. 4.01. The initial elements of the construction (in this case the vertices of the triangle) can be dragged and dropped with the mouse, consequently altering the whole construction. It is therefore trivial to check that the circumcentre can lie on the outside, the inside or the border of the triangle

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2.2 Some Ulterior DGS

Many other DGS were developed afterwards. We could underline:

  • Cinderella [5, 13], that performs internal computations in \(\mathbb {C}\) (in order to avoid discontinuity problems in animations).

  • Calques 3D [3, 31], devoted to 3D geometry.

  • Geometry Expressions [9, 30], that includes a small internal CAS that allows to directly perform symbolic computations derived from the geometric constructions.

2.3 DGS and Symbolic Computation: Possibilities

Providing the DGS with the possibility to perform symbolic computations opens new and exciting fields like:

  • Automatic Theorem Proving in geometry (ATP) [22]: In a naïve way: the geometric conditions are translated into algebraic conditions, and it is checked whether the (algebraic) thesis condition follows from assuming the (algebraic) hypotheses conditions. Let’s see a trivial example. Theorem: The segment bisectors of the sides of a triangle are concurrent (existence of circumcentre). Proof: Assign general coordinates to the vertices of the triangle, for instance A=(0,0), B=(b1,0), C=(c1,c2) (the reference system is properly chosen). The linear system consisting in the equations of these three lines is compatible, so they are collinear (Fig. 2). The equations systems are not always linear (if circumferences or distances are involved, second degree equations arise and the corresponding equations system are algebraic but not linear). The best known solving methods in such case are Wu’s pseudoremainder method [4, 32, 33] and Gröbner bases method [2, 16] (both even allowing to prove new theorems [23,24,25]).

  • Automatic discovery of theorems in geometry (derived from the previous one, oriented to hypotheses completion) [17].

  • Exact geometric loci finding [1, 23].

  • Applications in physics, for instance to linkages [14].

Fig. 2
figure 2

The perpendicular bisectors of a triangle are collinear (constructed with GeoGebra 6)

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2.4 Incorporating Symbolic Capabilities to DGS

That a DGS can perform symbolic computations can be achieved by different ways:

  • The designers and developers of the DGS can incorporate a CAS to the DGS. That is the case of Geometry Expressions and GeoGebra (Fig. 3).

  • the designers and developers of the DGS can facilitate the communication between the DGS and external CAS (another possibility of Geometry Expressions and GeoGebra).

  • external designers and developers develop a connection between a DGS and a CAS using the output file of the DGS (2D case [18, 20], 3D case [21]).

Fig. 3
figure 3

GeoGebra 6 “CAS View” showing an exact arithmetic calculation and some symbolic computations

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2.5 About GeoGebra

According to the epigraph “Short History of GeoGebra” of [10]:

“GeoGebra was created by Markus Hohenwarter in 2001/2002 as part of his master’s thesis [11] in mathematics education and computer science at the University of Salzburg in Austria. Supported by a DOC scholarship from the Austrian Academy of Sciences he was able to continue the development of the software as part of his PhD project in mathematics education [12]. During that time, GeoGebra won several international awards, including the European and German educational software awards, and was translated by math instructors and teachers all over the world to more than 25 languages.”

GeoGebra has always been freely available, initially thanks to the support of the Austrian Ministry of Education and later thanks to the American NSF project “Standard Mapped Graduate Education and Mentoring”.

The graphic interface of GeoGebra is similar to those of other DGS (see Figs. 1 and 2). However there is a difference with respect to other DGS like, for instance, The Geometer’s Sketchpad: in The Geometer’s Sketchpad the constructible geometric objects (the selectable “tools”) depend on the already selected geometric objects, while, in GeoGebra the “tools” are firstly chosen and the input geometric objects are selected a posteriori.

2.6 Main Milestones in the Development of GeoGebra

GeoGebra “basic” windows (the Graphical View and the Algebraic View) have a bidirectional connection (Fig. 4) [10]:

  • Changes introduced with the mouse in the Graphical View induce the corresponding changes in the Algebraic View, and,

  • Conversely, the changes introduced through the keyboard in the Algebraic View induce the corresponding changes in the Graphical View.

Fig. 4
figure 4

GeoGebra 6 Algebraic View and Graphical View windows

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Since version 3.2 (2009) [7] it includes a Spreadsheet View, that allows to easily check geometric theorems (Fig. 5).

Fig. 5
figure 5

Checking (not proving) Pythagoras Theorem with GeoGebra 6 and its Spreadsheet View (the area of poly1 is compared with the sum of areas of poly2 and poly3)

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Since version 5.04 (beta v. 2012) [7] it includes 3D capabilities, opening a whole new world of possibilities (Fig. 6).

Fig. 6
figure 6

Arquimedes calculation of the volume of a sphere with GeoGebra (the details can be found in [26])

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The mathematical software GeoGebra has reached an unprecedented success, claiming, as said above, over 100 million users.

We’ll analyse afterwards through a bibliographic study its impact in academic papers.

3 Bibliographic Data from Scopus (as on April 29th 2022)

3.1 General Scopus Data

The search for GeoGebra in the database Scopus [27] in “Title–Abstract–Keywords” (“T–A–K”) finds 832 references. The search for GeoGebra in “All Fields” finds 2264 references. They are distributed as shown in Table 1 and Fig. 7.

The values in “T–A–K” are close to those of a monotonically increasing function.

The values in “All Fields” correspond to a monotonically increasing function if we exclude the 2008 value.

3.2 Scopus Data by Author

According to Scopus, the top authors citing GeoGebra in “T–A–K” are:

  1. 1)

    Kovács, Z. (39)

  2. 2)

    Recio, T. (25)

  3. 3)

    Botana, F. (15)

  4. 4)

    Hohenwarter, M. (11)

Meanwhile, the top authors citing GeoGebra in “All Fields” are:

  1. 1)

    Kovács, Z. (41)

  2. 2)

    Recio, T. (27)

  3. 3)

    Botana, F. (22) ...

  4. 10)

    Hohenwarter, M. (12)

It has to be noticed that the first three authors in the previous lists are working in a remarkable “official” extension of GeoGebra (GeoGebra Discovery), that is able to find and formally proof theorems directly from geometric constructions (using algebraic ATP techniques) [15].

3.3 Scopus Data by Subject Area

The top subject areas where GeoGebra is cited in “T–A–K” in Scopus database are (Fig. 8):

  1. 1)

    Social Sciences (404)

  2. 2)

    Computer Science (299)

  3. 3)

    Mathematics (296)

  4. 4)

    Physics and Astronomy (171)

  5. 5)

    Engineering (85)

Meanwhile, the top subject areas where GeoGebra is cited in “All Fields” in Scopus database are (Fig. 8):

  1. 1)

    Social Sciences (1119)

  2. 2)

    Computer Science (808)

  3. 3)

    Mathematics (687)

  4. 4)

    Physics and Astronomy (424)

  5. 5)

    Engineering (286)

Table 1 Evolution of the cites to GeoGebra in the database Scopus
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Fig. 7
figure 7

Evolution of the cites to GeoGebra in the database Scopus

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Fig. 8
figure 8

Top subject areas where GeoGebra is cited (Scopus database)

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Due to the characteristics and purpose of GeoGebra, we believe that most “Social Sciences” papers will correspond to educational papers This is confirmed if we check the journals where the papers in this area have been published. For instance, the most recent publications in this area indexed in Scopus are published in the journals:

  • International Journal of STEM Education

  • Thinking Skills and Creativity

  • Journal of Mathematical Behavior

  • Computer Applications in Engineering Education

  • i-com (Human-Computer Interaction)

  • Creativity Studies

  • International Journal of Instruction

  • Nurse Education Today (mentioned in a reference, nothing to do)

  • International Journal of Science and Mathematics Education

  • Education Sciences

3.4 Scopus Data by Country

The top 10 countries when looking for GeoGebra cites in “T–A–K” (Scopus database) are (Fig. 9):

  1. 1)

    Indonesia (127)

  2. 2)

    Spain (78)

  3. 3)

    Turkey (71)

  4. 4)

    Austria (60)

  5. 5)

    Czech Republic (50)

  6. 6)

    Brazil (43)

  7. 7)

    United States (42)

  8. 8)

    Slovakia (34)

  9. 9)

    Italy (28)

  10. 10)

    Malaysia (26)

It is surprising to us that the US occupies the 7th place, the UK the 18th, Germany the 24th and China the 26th.

Meanwhile, the top 14 countries when looking in “All Fields” instead are (Fig. 9):

  1. 1)

    Indonesia (395)

  2. 2)

    Spain (175)

  3. 3)

    United States (171) (7th in “T–A–K”)

  4. 4)

    Turkey (151)

  5. 5)

    Brazil (98)

  6. 6)

    China (96) (26th in “T–A–K”)

  7. 7)

    Czech Republic (81)

  8. 8)

    Austria (78)

  9. 9)

    Malaysia (78)

  10. 10)

    Italy (69)

  11. 11)

    Slovakia (67)

  12. 12)

    Germany (62) (24th in “T–A–K”)

  13. 13)

    Israel (59)

  14. 14)

    United Kingdom (58) (18th in “T–A–K”)

The position changes of the US, China and Germany are remarkable and worth a deeper study. Can they be related, for instance, to cultural issues?

The distribution by countries can also be visualized in the maps of Figs. 10 and 11.

Fig. 9
figure 9

Top countries when looking for GeoGebra cites in “T–A–K” and in “All Fields” in Scopus database

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Fig. 10
figure 10

Top countries when looking for GeoGebra cites in “T–A–K” in Scopus database

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Fig. 11
figure 11

Top countries when looking for GeoGebra cites in “All Fields” in Scopus database

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4 Bibliographic Data from Web of Science (as on April 29th 2022)

The search for GeoGebra in the database Web of Science [6] in “Title” finds 330 references. The search for GeoGebra in “Topic” finds 800 references. They are distributed as shown in Table 2 and Fig. 12. Both lists of values show an increasing general tendency, although with more oscillations than when using Scopus as data source.

5 Bibliographic Data from Google Scholar (as on April 29th 2022)

The general search for GeoGebra in the database Google Scholar [8] results in an impressive \(\sim \) 73,800 references. The advanced search in “Title” finds 9820 references. They are distributed as shown in Table 3 and Fig. 13.

Table 2 Evolution of the cites to GeoGebra in the database Web of Science
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Fig. 12
figure 12

Evolution of the cites to GeoGebra in the database Web of Science

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Table 3 Evolution of the cites to GeoGebra in the database Scopus
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Fig. 13
figure 13

Evolution of the cites to GeoGebra in the database Google Scholar

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The values for the general case correspond to a monotonically increasing function from 2006 onwards. The values for the search in “Title” also show an increasing tendency, with a slight maximum in 2018, and are more or less stabilized since 2017.

The dates in this source are less accurate than those in Scopus and WoS (e.g., many documents are dated before GeoGebra existed due to dating errors!).

6 Conclusions

The available DGS are great tools for exploring geometry. The huge number of papers using GeoGebra and their constant growth confirms this fact and, moreover, the success of this particular piece of software.

The three bibliographic sources used (Scopus, Web of Science and Google Scholar) provide data with similar tendencies (constant growth). In the three sources consulted, we perceive a slight decrease in the number of citations in 2020, coinciding with the pandemic.

It is noticeable that very many papers are published in educational journals.

We guess that the success is due to the good policy of this software:

  • It is free,

  • Training has been provided but the GeoGebra Institutes,

  • It is multilingual,

  • It development has been opened to the contribution and suggestions of the users community.

There are open questions:

  • Which are the reasons for the changes in the positions of the US, the UK, Germany and China if ordering countries by publications indexed in Scopus mentioning GeoGebra in “T–A–K” or in “All Fields”?

  • Which are the reasons for the growth of references in Google Scholar in the general search and the stabilization of references in the search in the title?