GeoGebra - NCETM · GeoGebra represents one such software program that was designed to combine arithmetic, algebra, geometry, and calculus in a single, integrated system (Hohenwarter

NCETM Grants Final Report [G071203]

Zsolt Lavicza, Markus Hohenwarter, Allison Lu 1

Establishing a professional development network with an open-source

dynamic mathematics software

GeoGebra [Ref No: G071203]

Zsolt Lavicza1, Markus Hohenwarter2, and Allison Lu1

University of Cambridge (1), Florida State University (2) [email protected], [email protected], [email protected]

Contributions by Mark Dawes, Keith Jones, Chris Little, and Alison Parish



Table of Contents CONTEXT AND BACKGROUND _____________________________________________________________________ 3

INTRODUCTION _____________________________________________________________________________________ 3 RATIONALE AND AIMS _______________________________________________________________________________ 3 THEORETICAL FRAMEWORK ___________________________________________________________________________ 4

PROJECT ACTIVITIES AND PARTICIPANTS _________________________________________________________ 6 PARTICIPANTS _____________________________________________________________________________________ 6 PROJECT MEETINGS__________________________________________________________________________________ 7 WORKSHOPS _______________________________________________________________________________________ 7 DISSEMINATION ____________________________________________________________________________________ 8 DATA COLLECTION __________________________________________________________________________________ 9 DATA ANALYSIS ____________________________________________________________________________________ 9

RESULTS __________________________________________________________________________________________ 9 COMMUNITY OF PRACTICE FOR PROFESSIONAL DEVELOPMENT_________________________________________________ 9 WORKSHOPS ______________________________________________________________________________________ 11 REFLECTIVE PRACTICE AND RESPONSES FROM PARTICIPANTS ________________________________________________ 14 WORKSHOP PARTICIPANT COMMENTS___________________________________________________________________ 18

Software comments ______________________________________________________________________________ 18 Presenter comments______________________________________________________________________________ 19

FUTURE PROJECTS _______________________________________________________________________________ 20 EVALUATION_____________________________________________________________________________________ 20 SUMMARY _______________________________________________________________________________________ 21 ACKNOWLEDGEMENTS___________________________________________________________________________ 21 REFERENCES_____________________________________________________________________________________ 21 APPENDICES _____________________________________________________________________________________ 23

Study questionnaires _____________________________________________________________________________ 23 NCETM Manchester conference poster_______________________________________________________________ 24 Handout of menus and commands___________________________________________________________________ 25 GeoGebra Commands ____________________________________________________________________________ 26 Further Mathematics Network advertisement __________________________________________________________ 26 BSRLM Working group abstract ____________________________________________________________________ 27 Chris Little: MSOR Connections____________________________________________________________________ 28



Context and Background

Introduction The principal aim of this project was to conduct research on GeoGebra-related professional development materials and approaches. The study was conducted through tight collaboration of educational researchers and mathematics teachers in England utilising frameworks of communities of inquiry (Jaworski, 2006; Brown & Campione, 1996) and collaborative design (Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003). By joint collaboration and design, the aim was to work towards establishing a core group of GeoGebra experts in England who could offer professional development and support for teaches as well as assist in revealing how GeoGebra can fit into the English curriculum. In addition, there is a broader impact of the findings from this project as it not only informs the national mathematics education community about the affordances of new mathematics learning technologies, curriculum design and teacher professional development, but also contributes to a broader international community. Through the international developer and research network of the International GeoGebra Institute (Hohenwarter & Lavicza, 2007) findings of this project should assist in setting up and designing broader projects. In sum, this project contributes to developing self-sustaining support and professional development networks not only in England but also in other parts of the world.

Rationale and aims There is increasing consensus among mathematics educators that technology is becoming an integral part of mathematics teaching and learning, affording new forms of dynamic representations and communication (diSessa, 2007; Heid & Blume, 2008; Kaput, Hegedus, & Lesh, 2007). Indeed, new mathematics learning technologies can provide multiple representational resources and linking mechanisms that support students’ exploration of complex mathematical ideas and structures in a dynamic manner (e.g., Kaput, 1992; Moreno-Armella, Hegedus, & Kaput, 2008). Furthermore, through "thought-revealing activities" (Kelly, Lesh, & Baek, 2008; English et al. 2008), these rich and interconnected representations have the potential to enhance students’ mathematical experiences and foster deep understanding of mathematical concepts (Hiebert & Carpenter, 1992; Hiebert et al., 1997). However, these studies highlight the importance of appropriate professional development to support teachers in designing technology-supported lessons. There is a general agreement among mathematics educators on the powerful pedagogical and cognitive roles of the new dynamic multiple representations (Heid & Blume, 2008). However, for the majority of teachers, solely providing technology is insufficient for the successful integration of these new dynamic tools into their teaching (Cuban, Kilpatrick, & Peck, 2001). It has been suggested that adequate training and collegial support boost teachers' willingness to integrate technology into their teaching and to develop successful technology-assisted teaching practices (Heid & Blume, 2008). Furthermore, mathematics teachers play a vital role in the iterative cycle of curricular planning, development, implementation, and assessment (Clements, 2007). Heid and Blume (2008) recommended a stronger focus on research that “helps teachers, curriculum developers, and researchers to understand how students move between, connect, and reason from multiple representations” (p. 98). The paucity of such research is disproportionate to the increasing popularity of new mathematical learning environments and the growing interest among classroom mathematics teachers in experimenting with new approaches to mathematics education. For teachers to integrate new mathematics learning tools into their daily practice, it is critical that they understand the affordances, constraints, and general pedagogical nature of such new representational resources in relation to the specific mathematical topics of school mathematics (Ruthven & Hennessy, 2002).



Nevertheless, the introduction of technology into mathematics teaching has proven to be a much more complex process than initially expected (Cuban, Kirkpatrick, & Peck, 2001). Teaching in general, and teaching with technology, in particular, has been found to be a “wicked problem” (Koehler & Mishra, 2008, p. 10), introducing additional complexity and new layers of structures in the real classroom. To ease the complexity of technology integration in mathematics education, recent research endeavours have resorted to community-based inquiry and curricular design (Jaworski, 2006). These approaches can be used as tools to foster mathematics teachers’ investigation of mathematical content, mathematics teaching, and higher-level reflection on teaching and technology. By engaging classroom teachers as researchers and curriculum designers, research teams are better oriented to address the complexity of mathematics teaching. Collaboratively, teachers and researchers critically explore the multifaceted relationships among the subject matter, teaching, and the use of technologies in a manner that takes into account both the teaching practice and context (Koehler & Mishra, 2008). In this project, we chose to work with a specific software package GeoGebra due to various reasons. At the 11th International Congress on Mathematics Education (ICME-11) held in Monterrey, Mexico in July 2008, not only was the permanence and potential, yet ongoing challenges, of technology use in mathematics education highlighted by several respected speakers (e.g., Artigue, Kilpatrick, Hoyles). But repeated reference was also made in various working group and workshop sessions to the growing demand for widely-accessible, integrated, and quality mathematics software resources that could meet the expanding needs of educators and institutions worldwide, particularly in regions wherein existing commercial software is simply not affordable. The open-source dynamic mathematics environment GeoGebra represents one such software program that was designed to combine arithmetic, algebra, geometry, and calculus in a single, integrated system (Hohenwarter & Preiner, 2007a). GeoGebra is a freely available dynamic mathematics environment and offers dynamically linked multiple representations for mathematical objects (Hohenwarter & Jones, 2007) through its graphical, algebraic, and spreadsheet views for teaching and learning mathematics at all levels. It also allows the creation of interactive on-line resources (Hohenwarter & Preiner, 2007b) which are freely shared by mathematics teachers on a collaborative online platform (www.geogebra.org/wiki). GeoGebra is currently available in 44 languages and used by tens of thousands of teachers in over 190 countries. The large international user and volunteer developer community has formed around GeoGebra, which has led to the recent establishment of an international research and professional development network: the International GeoGebra Institute (Hohenwarter & Lavicza, 2007) within which the findings of this project can attain further impact. Given the open-source nature, web-accessibility, and the growing international user community, GeoGebra is not only an intellectually promising software but also economically sustainable in empowering classroom teachers to support all students in learning significant mathematics (Hohenwarter, Jarvis, & Lavicza, 2009). In sum, the principal aims of the project were to:

• Nurture a community in England around GeoGebra • Find how GeoGebra can be integrated into the curriculum • Develop professional development workshops together with participating teachers, • Professionalise participating teachers by giving workshops for other teachers, involve them in

research, present at conferences, and write papers together

Theoretical framework As was discussed in the previous section, the project was informed by Jaworski’s (2006) theoretical conception of communities of inquiry grounded in decades of her investigation into the complex problem of mathematics teacher education, which extends on Lave and Wenger’s (1991) community of practice and is compatible with Brown and Campione’s (1996) model of Fostering Communities of Learners



(FCL). Jaworski defines a community of inquiry as a community of practice where inquiry is central to all activities, integrating elements of social constructivism and those of sociocultural theory. In a community of inquiry, “participants grow into and contribute to continual reconstitution of the community through critical reflection; inquiry is developed as one of the norms of practice within the community and individual identity develops through reflective inquiry” (p. 202). Inquiry into curriculum design, as seen by Jaworski, is a tool through which inquiry-based learning becomes a way of being a competent teacher as a member of the teaching community and further enables teachers to conduct inquiry-based learning in the classroom. In the domain of mathematics teaching, participants investigate the mathematical content, student learning, the way of mathematics teaching, and even the social identity of a mathematics teacher. In our research project, both participating teachers and teacher educators have been active members in a community of inquiry where they were learners as much as researchers. Jaworski’s framework for such communities of critical inquiry is well aligned with our project goals in addressing the ill-structured problem of teaching mathematics using new technologies in that, firstly, teachers are regarded as active researchers and curriculum designers; and secondly, it brings practice and context into teacher development. We also draw on the literature on design experiments in the development of the project. In a broad sense, design experiments are compatible with several other methods found in mathematics education literature such as developmental research, design research, and teaching experiments. Moreover, the developmental research cycle as explicated by Gravemeijer (1994) was described by Cobb (2000) as analogous to Simon’s (1995b) mathematics teaching cycle, which involves the design of a hypothetical learning trajectory and classroom-based analysis. In essence, design experiments are methodologically aligned with the iterative nature of curricular research (Clements, 2007) and the spiral-like growth of teacher’s technological content knowledge (Koehler & Mishra, 2008). Further, design experiments bring to the fore the teachers’ classroom practice and children's diverse ways of mathematical thinking, through which teachers set out “to discover the kinds of situations that may induce a naive learner first, to embark on a novel way of operating and then, if this new way led to success, to abstract from it an operative conceptual structure” (Von Glaserfeld & Steffe 1991, p. 100) and further construct “living models of students’ mathematics” (Steffe & Thompson, 2000, p. 287). Accordingly, design experiments, as a methodology, have both a pragmatic purpose in creating specific forms of learning and a theoretical orientation in developing domain-specific theories of learning. Therefore, design experiments “result in a greater understanding of a learning ecology—a complex, interacting system involving multiple elements of different types and levels—by designing its elements and by anticipating how these elements function together to support learning” (Cobb et al., 2003, p. 9). Finally, in a design experiment, the teacher-researchers actually teach the students or establish close collaboration with other classroom teachers (Cobb, 2000; Confrey & Lachance, 2000; Steffe, 1991; von Glasersfeld & Steffe, 1991) in order to make first-hand observations, “not only of children’s mathematical activity, but also of how that activity can be influenced” (von Glasersfeld & Steffe, 1991, p. 92), and subsequently to make sense of their mathematical reasoning. However, due to the limited timeframe and the resources of the project we could not engage in true design experiments, but the continuation of the project could draw on this area. Nevertheless, we utilised experiences gained in our participation in previous research projects and utilised materials developed there. Within the U.S. National Science Foundation’s Math and Science Partnership project Standards Mapped Graduate Education and Mentoring at Florida Atlantic University, an extensive set of professional development materials was developed for initial training of mathematics teachers in the use of the open-source software GeoGebra. These materials have been used and evaluated (Preiner, 2008) in the United States and we were able to draw on these resources in our project in the English context.



Project activities and participants In the original NCETM proposal we intended to work with four teachers from across England, we aimed to offer five workshops, and to hold three project meetings. However, due to practical considerations and scheduling problems we had to slightly alter our plans. Nevertheless, we believe that we accomplished more results and carried out wider dissemination of the project than we originally proposed. In this section, we describe the participants of the project and the activities we carried out during the past year.

Participants The project team consisted of four research team members (Table 1) and nine participant teachers (Table 2). In our proposal we aimed to work with two research assistant, but Allison Lu was working with us closely and she was able to handle all data collection and analysis activities. Therefore, we opted to work with only one research assistant. Location Institution Name Role Contact Cambridge University of Cambridge Zsolt Lavicza Researcher [email protected]

Tallahassee Florida State University Markus Hohenwarter Researcher [email protected]

Cambridge University of Cambridge Allison Lu Research Assistant [email protected]

Southampton University of Southampton Keith Jones Project

Evaluator [email protected]

Table 1: Project team

Originally we proposed to work with four teachers in this project. However, when we were looking for participants we encountered such enthusiasm that we decided to expand the project. Eventually, we worked with nine participants from various locations in England. Location Institution Name Contact

Woodbridge Formally of Stowmarket High School Alison Parish [email protected]

Birmingham Queen Mary's Grammar School Michael Borcherds [email protected]

Southampton University of Southampton Chris Little [email protected]

Havering Havering Sixth Form College Kathryn Peake [email protected]

Havering Havering Sixth Form College Damian Rogan [email protected]

Comberton Comberton Village College Mark Dawes [email protected]

Cambridge Hills Road Sixth Form College Gary Wing [email protected]

Cambridge Hills Road Sixth Form College Robert Leslie [email protected]

London Further Mathematics Tom Button [email protected]



Network Table 2: Project participants

Participants of the project were teaching at different grade levels in their schools (from Key stage 3 to A-level). In addition, one participant, Tom Button, is working for the Further Mathematics Network (FMN) and he regularly offered ICT-related professional development for teachers within FMN.

Project meetings Initially, we proposed to hold three project meetings and a working group session at the BSRLM conference. Due to the availability of participants we were able to meet twice during the year and organised a working group session at the BSRLM conference in Cambridge. Between meetings we kept close connection with participants via e-mail and we set up an online wiki site where participants could share their ideas and GeoGebra materials. Date Location Number of participants 17 May 2008 Faculty of Education, Cambridge 12 7 Sep 2008 Trumpington House, Cambridge 12 28 Feb 2009 BSRLM conference, Cambridge 6

Table 3: Project meetings

We aim to further organise the collected materials, develop a public website (connected to the NCETM portal), and develop a CD that can be distributed at future professional development activities1.

Workshops After outlining and collaboratively considering the aims of the project, the project participants organised a range of workshops that were held for different audiences and at various locations (Table 4). Workshop participants ranged from primary to A-level teachers, involved pre-service and in-service teachers and heads of mathematics departments. Several workshops were organised within NCETM regional seminars, and workshops were held at schools, universities, and training centres. The individual workshops usually lasted for 2-3 hours, but there were some whole day workshops as well.

Date Location Institution Number of participants

Participant description

23 June 2008 Comberton, Cambridgeshire

Comberton Village College 25 Secondary in-

service teachers

11 July 2008 Havering Havering Sixth Form College 22 Secondary in-

service teachers

2 October 2008 Cambridge NCETM East of England meeting 20 Secondary/ A-level

teachers

26 Nov 2008 On-line Further Mathematics Network 20 Teachers from all

over England

Nov 2008 Cambridge Cambridge A-level mathematics teachers 25 A-level teachers

5 Jan 2009 Cambridge Faculty of Education- PGCE course 15

Secondary mathematics trainee teachers

1 Please find more details later in this report.



21 Jan 2009 Ipswich, Suffolk Belstead House 25 Suffolk Heads of maths

27 Jan 2009 London London Metropolitan University- PGCE course 28

Secondary mathematics trainee teachers

12 Feb 2009 Birmingham University of Birmingham, NCETM Regional meeting

4 In-service teachers

24 Feb 2009 Polesworth Polesworth International Language College 14 Secondary in-

service teachers Future workshops

6-7 March 2009 Sheffield West Midlands Influential Mathematics Teachers Conference

NCETM influential teachers

19-20 March 2009 Cambourne

East of England Influential Mathematics Teachers Conference

NCETM influential teachers

April 2009 Sheffield Sheffield Hallam University Pre-service and in-

service teachers April 2009 East London To be confirmed In-service teachers

May 2009 London NCETM regional meeting In-service teachers

May/June 2009 Cambridge NRICH Project Pre-service and in-service teachers, NRICH staff

Table 4: List of workshops offered by project participants

As shown in Table 4, participants will offer future workshops after the project finishes in March 2009. Within the timeframe of the project new opportunities emerged and the enthusiasm of participants and invitations from different organisations allow the continuation of offering workshops.

Dissemination Table 5 lists the dissemination activities of the project. The dissemination of project activities and results are not constrained to the research staff of the project. Participating teachers have and will actively present results and their own work at different conferences. For example, one of the teachers presented GeoGebra activities for A-level students at the BSRLM conference in London last November. At the BSRLM conference in Cambridge in February 2009 several teachers participated in a geometry working group session and shared their experiences within the project and the materials they developed during the past year.

Date Location Institution/place Type of dissemination

Description of dissemination

15 Nov 2008 London Kings College London BSRLM conference Two Presentations

3 December 2008 Manchester Ramada Manchester

Piccadilly NCETM Conference

Workshop and poster

28 Feb 2008 Cambridge University of Cambridge BSRLM conference

Working group and meeting

25 March 2009 Bristol University of Bristol Conference Two presentations



and poster 06-09 April 2009 Swansea ATM Conference ATM conference Workshop

14-17 April 2009 Cambridge MA conference Robinson College, University of Cambridge

MA Conference Presentation

11-13 July 2009 Linz, Austria RISC, University of Linz CADGME conference Presentation

19-24 July 2009 Thessaloniki, Greece PME conference

Psychology in Mathematics Education conference

Presentation/Poster

Table 5: Dissemination of the project at conferences

Results of the project will be presented by two teachers at the NCETM conference in Bristol in March 2009. In addition, teachers will actively participate at conferences of mathematical associations (ATM, MA). Also, results of the project will be presented at international mathematics education research conferences (e.g. CADGME, PME) during the upcoming year. Beyond disseminating results at conferences we encouraged participants to actively engage in writing about their work in both practitioner and research journals (e.g. ATM, MA, BSRLM, MSOR). Participating teachers also participated in preparing this final report of the project. In addition, results will be reported in research journals. Furthermore, we will prepare a collection of GeoGebra materials and work of teachers and share it on the NCETM portal, GeoGebra Wiki, and prepare a CD with resources that can be distributed at future activities (a sample of writing and proposals can be found in the appendix).

Data collection Project meetings were videotaped, and PowerPoint presentations together with GeoGebra files were collected. We also aimed to videotape all workshops. Unfortunately this was not possible logistically; however, we were able to collect video records from five workshops. In addition, we collected works of participants and asked them to fill in a short questionnaire about their experiences in these workshops. Presenters also filled in a short questionnaire about the workshop. In addition, we interviewed several participants and collected presentation materials.

Data analysis Because it would have been too time consuming to transcribe all video recorded materials the research assistant of the project has reviewed video materials and wrote a summary of the events. In addition, she highlighted parts of the videos that were reviewed and transcribed together with the researchers. The data from questionnaires was entered into a spreadsheet file and analysed with SPSS and Excel software. Written responses and interviews were analysed with the help of HyperResearch software. In the data analysis we followed a Grounded Theory approach (Strauss & Corbin, 1998).

Results

Community of practice for professional development Engaging participants in collaborative activities was successful while developing the project aims and planning and preparing workshops together with the project participants. In the fist project meeting we discussed topics that had to be established for the preparation of professional development activities:



• Workshops by Key stages • Workshops by topic • Workshops by the level of GeoGebra knowledge • Format of materials • Pool of problems/materials

The existing expertise of the project team meant that we were aware that in order to develop successful professional development we would need to consider how to organise GeoGebra materials and develop workshops accordingly. It became clear that we needed to aim for a range of teaching levels and develop workshops according to Key stages (fitting different pedagogical approaches). Within these stages we would need to develop materials for different teaching topics. In addition, beyond the presentations we needed to develop handouts and tutoring materials. In the meeting we reviewed the already available teaching materials from the U.S. and agreed that they would need to be altered considerably for the English audience. The U.S. materials were too detailed and participants suggested that English teachers require more openness and creativity so levelling down the details could improve the acceptance by teachers. Furthermore, the developed materials should not be prescribed, but rather customisable. To achieve these aims we decided to develop a pool of materials, in line with the philosophy of the project, from which participants can draw for their own workshop. Therefore, we established an online wiki site where participants could share materials and comment on their use. We also discussed the necessity of producing materials for people with different GeoGebra knowledge, but we assumed that most participants will have little experience with GeoGebra and thus we concentrated on introductory workshops. Based on a consensus we agreed to develop workshop materials that put emphasis on the ease of use of GeoGebra in classrooms. Because most teachers have limited experience in working with computers in their teaching we wanted to reassure teachers that using GeoGebra in their classroom would be easy and they can create mathematical models with the software on the fly. We also emphasized the importance of sharing experiences with each other. The second important aim of each workshop would be to show teachers how and where GeoGebra can fit into the aims of the national curriculum as well as to discuss pedagogical approaches for the use of dynamic mathematics materials. These pedagogical approaches will be discussed later in this section. Another important discussion emerged on the types of workshops we could produce. According to participants it is difficult to organise workshops for teachers, because of their tight schedule and cost of release. Also, dealing with logistics and scheduling could cause difficulties. In addition, the offered workshops should raise teachers' attention, because they would only devote their free time for clearly worthwhile activities. Therefore, we needed to be creative in offering workshops. Teachers came up with the following ways for offering workshops:

• Local face-to-face • On-line courses (e.g. using Elluminate) • Teacher conferences

As mentioned at the beginning of the report, most workshops were organised at local schools or universities and some of them were part of the NCETM regular monthly seminars. In addition, within the Further Mathematics Network an on-line GeoGebra course was offered with great success. We also discussed the organisations and institutions where professional development activities could be held. Based on the following list, participants and the project team agreed to make contacts to organise some initial workshops before the next project meeting.



• Local schools and LEAs • STEM network • NCETM coordinators • Trainee teachers • NRich • National Strategy network • Further Mathematics Network

Furthermore, we decided to set up an e-mail list and online wiki site for more effective communication and sharing of materials. After the meeting, notes were written up and shared among participants. Between the first and second meetings two workshops were given by participants. The materials for these workshops were developed from a pool of shared materials on the wiki and according to the discussions in the first meeting. In the project meeting in September, participants who offered the workshops shared their experiences and we discussed the successes and difficulties emerged in these workshops. In addition, one of these workshops was videotaped and summarised the important issues emerged in these workshop and reflected on feedbacks from teachers. According to these discussions we outlined plans for further workshop contents (ideas outlined below). In addition, we discussed workshop organisation and dissemination plans. We kept working on our project wiki and we will share these materials later on the NCETM portal. We discussed the importance to develop a local support network together with a pool of problems in England. Thus, we will create local wiki sites on the GeoGebra website and have local moderators who will review and organise materials on these web pages, but it would require substantial resources in the future. However, the project supplied important information and materials for the development of such websites. We also discussed the conferences and journals in which we will report our study and the materials developed from it (see Table 5)

Workshops The developed workshops include an introduction to GeoGebra including hands on activities in which some basic problems from geometry and algebra were given. Several presenters used a slightly modified version of the GeoGebra Quickstart document from the GeoGebra website: http://www.geogebra.org/help/geogebraquickstart_en.pdf. However, it was also important to discuss how GeoGebra fits into the national curriculum of England. In workshops participants received a summary table from the national curriculum and jointly discussed opportunities for GeoGebra use in their teaching. More importantly, participants discussed the importance of pedagogical approaches in GeoGebra environments. Below, we give an example of the pedagogical framework that was presented at our workshops. Workshop participants received an A3 sheet with the figure below which they used in group discussions about different ways how GeoGebra could be used in their teaching.



Figure 1: Pedagogical approaches with GeoGebra

Examples of these three categories were presented at the workshop. Figure 2a, b shows an example of a teacher demonstration to engage students in discussing the dynamic construction. Firstly, the teacher can ask questions about the objects on the screen and what students expect would happen if some parts of the configuration are moved or changed. Then either the teacher or some students change the construction to check their predictions.

Figure 2a, b: Example of teachers’ demonstrations: "Enlarging by a Factor"

According to discussions within our project, such demonstrations allow teachers who have little experience using technology in classroom teaching to experiment with technologies with small risk. In addition, this kind of use requires less change in the classroom environment and needs fewer resources than organising classes in computer labs. When teachers become more comfortable with computers they can provide pre-developed worksheets for students with tasks and students can experiment with dynamic objects. For example in Figure 3a, b students explore triangles inscribed into circles and try to formulate conjectures for further generalisation.

Figure 3a, b: Examples of students interacting with teacher created files

Teacher demonstration

Pupils interact with teacher-created files

Pupils create their own files

Pedagogy



In Figure 4, students are presented with a symmetric face where they can play with its shape to make conjectures about the properties of symmetries. In addition, when they are becoming familiar with the construction they can create new objects on the face (such as hair, spots) that bear the same dynamic properties as the already existing face.

Figure 4: Example of students interacting with teacher created file

Figure 5 shows a sequence of similar activities. First, students are experimenting with the images in the dynamic worksheet. For instance, the stick figures can fight or kiss (Figure 5a), but the example could be extended to defining points on objects and explore the changes of these coordinates in the picture. In this way, not only students become familiar with the software and the properties of the dynamic environment, but also teachers learn how to deal with issues emerging in such situations.

Figure 5a, b: Examples of students interacting with teacher created files

Finally, teachers can ask students to create their own files to help them develop conjectures, theories, and proofs (Figure 6a, b).



Figure 6a, b: Examples of students creating their own files

The sequence of these activities was discussed by the participants of the project. We agreed that this could be a good way to present GeoGebra for teachers with limited technology integration experience. This development of activities is very similar to a study carried out by Laborde (2001). She followed two teachers for two years and tried to understand how novice teachers integrate technology into their teaching. Laborde (2001) found that teachers gradually grant control of technology and the learning environment to students. In other words, they mostly use demonstration at first and then slowly allow the individual use of computers by students in their classroom. However, technology integration is a more complex process than just considering teachers’ actual activities. Ruthven (2007) indentified five aspects, 1) working environment; 2) resource system; 3) activity format; 4) curriculum script; 5) time economy; that influence teachers decisions in using technology for teaching. Ruthven (2007) explained the interaction of these aspects in teaching decisions and activities: “Each of these modifications of an established activity format calls for the establishment of new classroom norms for participation, and of classroom routines to support smooth functioning.” (p. 10). In future studies it would be advisable to consider not only the development of teachers’ activities, but also the factors that influence teachers’ decisions in classrooms. In sum, teachers engaged in discussion and came up with activities and ideas that could be useful for further improvement of professional development activities. These ideas and activities also corresponded with previous research. Furthermore, through the project meetings and e-mail/wiki exchanges participants formed a community that has clearly contributed to the success of the project. It also became apparent that the professional development activities highly depended on the accessibility, sharing, and exchange of resources and ideas among participants. Thus, developing on-line, easily accessible and searchable local resources is crucial for the future.

Reflective practice and responses from participants Sharing experiences from workshops in the second project meeting was an important part of the project. In this section, we will outline the results from the survey distributed among workshop participants2. In some workshops, surveys were filled in on-line and in other ones (with no Internet access) completed on paper. We received 78 responses, because not all participants filled in the questionnaire. Also, there were some missing items in the data, but the number of responses (n) is indicated at each result in parentheses. Close to three quarters (74%) of the participants attended the workshop voluntarily, but some participants were required to take part in workshops. Figure 7 shows that most participants heard about the workshop from advertisements in schools, LEA or from the university (in this case trainee teachers), but almost one fifth of teachers were referred to attend the workshop by their colleagues.

2 Images of surveys can be found in the Appendix.



17%

27%

31%

26%

0

0

0

0

0

0

0

0

ColleagueLEASchoolUniversity

Figure 7: How did participants hear about the workshops (n=78)

Figure 8 summarises the distribution of student age groups that participant teachers taught. It can be seen that most participants taught students from ages 11-16, but there were several teachers teaching A-level and further mathematics courses as well.

Years taught

0% 0% 4% 0% 4%

60%65%

71%

80% 80%84%

51%45%

11%

0%

10%

20%

30%

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6 7 8 9 10 11 12 13 14 15 16 17 18 19

Years taught

Figure 8: Age of students taught by workshop participants (n=55)

Figure 9 highlights teachers’ perception of their knowledge of computers, mathematical content, and GeoGebra. It can be seen in the figure that the majority of teachers had a limited knowledge of GeoGebra preceding the workshop. It is interesting to observe that almost one third of the participating teachers declared poor knowledge of computers and mathematical content knowledge. We examined the correlation matrix and cross-tabulation tables with other variables to uncover patterns for this



phenomenon, but the data did not provide clear explanation. There is a high correlation between computer knowledge and mathematical content knowledge (.69) and between content knowledge and prior GeoGebra knowledge (.72), but we could not draw further conclusions from the data.

0%

10%

20%

30%

40%

50%

60%

70%

Poor Fair Good Excellent

General computer useMathematical content knowledgeGeoGebra kowledge

Figure 9: Participants assessment about their own knowledge (n=57)

Figure 10 highlights the list of different technology applications workshop participants use in their teaching. Only 7% of participants used GeoGebra in their teaching, but 33% claimed the use of other DGS packages. It is interesting to see that after text processing and presentation tools, spreadsheets are the most widely used packages to teach mathematics.

78%73%

67%

7%

33%

24%

9%

.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

Word Presentation Spreadsheet GeoGebra DGS Calculator CAS

Figure 10: Types of technologies used by participants (n=57)



On the evaluation of workshops almost all participants at least agreed (Figure 11) that the workshop was interesting, the activities were appropriate, and the time was used effectively.

.0

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Strongly disagree Disagree Neutral Agree Strongly agree

Interesting workshopActivities appropriateTime was used effectively

Figure 11: Participants’ evaluation of the workshops (n=58)

Similarly, Figure 12 shows that the majority of participants rated the presenters as experts of GeoGebra and enthusiastic about the material taught. In addition, they agreed that GeoGebra is an easy to use software package and it might be useful for their own teaching in the future.

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Strongly disagree Disagree Neutral Agree Strongly agree

Presenter GGB expertPresenter enthusiasticGGB easy to useGGB useful in own teaching

Figure: Participants’ evaluation of presenters (n=58)



Overall, workshop participants appreciated the content and the delivery of workshops. Many of them became enthusiastic about experimenting with the software in their own teaching. However, it would be interesting to follow up these teachers and find whether or not they incorporated GeoGebra into their teaching.

Workshop participant comments In this section we will highlight some comments from workshop participants.

Software comments Overall, participants appreciated the easy to use nature of GeoGebra:

The software is so easy to use. Congratulations on its development and success so far. Very easy to start using. Seems a lot more intuitive than the others!

However, they recommended new features and improvements in the software:

A "clear all" button on the screen so you don't have to open a new window each time you want a blank document Able to Rotate shapes without having a centre of rotation. I think that it would be useful to use a simpler version in primary schools. Would like to see compasses, protractors embedded into the project.

In addition, they valued the availability of on-line resources:

GeoGebra Wiki is excellent. But, participants wanted to have resources more specifically suited to the English curriculum:

Readily available resources linked to specific national curriculum objectives, ideally including suggestions for ideas around key points or questions that can be built around the application. Have a UK school scheme of work that links files to topics.

In addition to resources, pedagogical approaches should be offered:

Would be good to have information that considers how to deliver lessons involving the use of GeoGebra. Style of delivery, ways of pulling out the learning.

In this way several teachers could integrate GeoGebra into their teaching more easily and benefit students:

Seeing how GeoGebra can be used to enhance topics I already taught.



Be able to use GeoGebra easily and see its visual benefits for my students. Participants valued the experience of learning the software:

Learning how to use the software and having access to readymade resources. But, there were a number of comments that they enjoyed working with and learning from colleagues as well as with presenters at the workshops:

Learning from other people who are also learning. Sharing ideas for teaching maths using GeoGebra. Working with others to understand how the software works and use and create resources. Hands on with tutors to help out.

Some improvements were recommended that should be considered for designing future workshops. These were concerned with the use of the software:

For a non specialist I think an introductory day course on the actual mechanics of using the software would be useful.

Connection with the curriculum:

Run through of the wide range of applications of GeoGebra mapped to curriculum topics. Designing lesson plans and pedagogical approaches:

Help develop techniques to create efficient lesson plans quickly using the software. See some example on how to turn GeoGebra into full blown lessons.

Specific instructions for creating objects:

More activities/instructions of how to create files, objects and images. Lessons for specific topics or materials:

Workshops specific to Sixth Form teachers. And having more time:

MORE TIME - although this is impossible Certainly, it is not possible to have more time at a workshop, but sequences of workshops could be designed in the future.

Presenter comments Six presenter questionnaires were completed. Most participants rated the success of the workshop as good (one very good) and the facilities appropriate for purposes of workshops. Both teachers’ computer and mathematical content knowledge were rated as good by presenters. In most workshops participants used there own laptops, but at two workshops laptops were provided to participants. In most cases GeoGebra was not installed on computers, but it was downloaded from the Internet or installed from CDs provided



by presenters. However, on some computers the Java version was not up-to-date which caused some problems with installation. The computer skills of workshop participants were variable and it caused difficulty for presenters:

One thing to work on is the difficulty to meet the needs when there is a huge range of participants' IT skills, and previous knowledge concerning software use in teaching maths.

One presenter highlighted conceptual difficulties in a workshop:

It was interesting that most of the groups, in the focused discussion on pedagogy, did not understand what "pedagogy" is!

While another recommended improving resources for presenters: Trouble with people not having mice (2 pens, some trackpads) - need to budget for presenters to have mice & memory sticks next time!

But, most presenters were delighted by the positive feedback from workshop participants: Most comments were pleased with the experiences in the workshop and the appreciation of participants.

In the interview participants mentioned that they would be pleased to continue working in future projects and offer workshops with GeoGebra. However, they acknowledged that technology integration could be more challenging in England than in some other countries, because of the curriculum and available resources:

The number of teachers who use GeoGebra in Norway is more than England and it is a small country. They’ve got textbooks written with the use of GeoGebra. The existing teachers need to be taken to teacher training.

Future projects This project provided the groundwork for extensions for developing professional development networks in England. The limitation of the project did not allow the development of full extent of materials and pedagogy. It would be interesting to follow up workshop participants and continue working with teachers participating in the project in order to provide future support. A possible extension would be to incorporate a theoretical framework from Ruthven (2007) considering the aspects of teachers working life. In addition, keeping the community of practice idea and employing design experiments with several cycles could further contribute to the value of further projects. As was discussed earlier, in design experiments teachers together with researchers iteratively design and evaluate curricular materials over a longer period of time (Cobb et al., 2003).

Evaluation The project was constantly monitored by the project evaluator Dr Keith Jones from the University of Southampton. Dr Jones participated in project meetings and led the BSRLM working group on the project. In addition, all documents were shared with Dr Jones and he reviewed the final report. In addition, as was described in the previous section, workshops were evaluated by a short questionnaire by participants.



Summary We believe that we reached most objectives of the project we proposed. With a group of nine enthusiastic participants we initiated a core group which can begin developing support and professional development to other teachers in England. We also investigated ways in which GeoGebra could be integrated into the English curriculum and begun developing and collecting materials that can be used in mathematics teaching. In addition, we laid down the principles for establishing a local searchable GeoGebra site and on-line support structure. Finally, but not last, participants became interested in research and sharing their experience with other teachers in conferences and through various publications. Thus, we believe that with this project we contributed to the professional development of teachers in England and supported the aims of NCETM.

Acknowledgements We would like to acknowledge the great continuous support and assistance of Peter Hall and Sandy Cowling from NCETM. We especially thank Dr Keith Jones about his invaluable advices and help in organising the BSRLM working group. We are very thankful to Grace Lloyd and Nicola Hern for their assistance in administrative and organisational issues. Finally, the project would not have been possible without the enthusiasm and generosity of all our project participants.

References Clements, D. H. (2007). Curriculum research: Toward a framework for "research-based curricula".

Journal for Research in Mathematics Education, 38, 35-70. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational

research. Educational Researcher, 32(1), 9-13. Confrey, J., & Lachance, A. (2000). Transformative teaching experiments through conjecture-driven

research design. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 231-265). Mahwah, NJ: Lawrence Erlbaum Associates.

Cuban, Kirkpatrick, & Peck (2001). High access and low use of technologies in high school classrooms: Explaining an apparent paradox. American Educational Research Journal, 38, 813-834.

diSessa, A. A. (2007). Systemics of learning for a revised pedagogical agenda. In R. A. Lesh, E. Hamilton & J. J. Kaput (Eds.), Foundations for the future in mathematics education (pp. 245-261). Mahwah, NJ: Lawrence Erlbaum Associates.

English, L. D., Bussi, M. B., Jones, G. A. & Lesh, R. A. Sriraman, B., & Tirosh, D. (Eds.) (2008). Handbook of international research in mathematics education (2nd ed.). New York: Routledge.

Gravemeijer, K. (1994). Educational development and developmental research in mathematics education. Journal for Research in Mathematics Education, 25, 443-471.

Heid, M. K., & Blume, G. W. (2008). Algebra and function development. In M. K. Heid & G. W. Blume (Eds.), Research on technology and the teaching and learning of mathematics: Research Syntheses (Vol. 1, pp. 55-108). Charlotte, NC: Information Age Publishing.

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York: Macmillan.

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.



Hohenwarter, M., Jarvis, D., & Lavicza, Z. (2009, in press). Linking Geometry, Algebra, and Mathematics Teachers: GeoGebra Software and the Establishment of the International GeoGebra Institute. The International Journal for Technology in Mathematics Education (IJTME).

Hohenwarter, M., & Jones, K. (2007). Ways of linking geometry and algebra: the case of GeoGebra. In D. Küchemann (Ed.), Proceedings of the British Society for Research into Learning Mathematics. 27(3):126-131, University of Northampton, UK: BSRLM

Hohenwarter, M., & Lavicza, Z. (2007). Mathematics teacher development with ICT: towards an International GeoGebra Institute. In D. Küchemann (Ed.), Proceedings of the British Society for Research into Learning Mathematics. 27(3):49-54. University of Northampton, UK: BSRLM

Hohenwarter, M., & Preiner, J. (2007a). Dynamic mathematics with GeoGebra. Journal of Online Mathematics and its Applications. ID 1448, vol. 7, March 2007

Hohenwarter, M., & Preiner, J. (2007b). Creating mathlets with open source tools. Journal of Online Mathematics and its Applications. ID 1574, vol. 7, August 2007

Jaworski, B. (2006). Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education, 9, 187-211.

Kaput, J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515-556). New York: Macmillan.

Kaput, J., Hegedus, S., & Lesh, R. A. (2007). Technology becoming infrastructural in mathematics education. In R. A. Lesh, E. Hamilton & J. J. Kaput (Eds.), Foundations for the future in mathematics education (pp. 173-191). Mahwah, NJ: Lawrence Erlbaum Associates.

Kelly, A. E., Lesh, R. A. (Eds.). (2008). Handbook of Research Design in Mathematics and Science Education. Mahwah, NJ: Lawrence Erlbaum Associates.

Laborde, C. (2001). Integration of technology in the design of geometry tasks with Cabri-geometry. International Journal of Computers for Mathematical Learning, 6, 283-317.

Moreno-Armella, L., Hegedus, S. J., & Kaput, J. J. (2008). From static to dynamic mathematics: Historical and representational perspectives. Educational Studies in Mathematics, 68, 99-111.

Preiner, J. (2008). Introducing Dynamic Mathematics Software to Mathematics Teachers: the Case of GeoGebra. Doctoral dissertation in Mathematics Education. Faculty of Natural Sciences, University of Salzburg, Austria

Ruthven, K. (2007). Teachers, technologies and the structures of schooling. Proceedings of the Conference of the European Society for Research in Mathematics Education, Cyprus. Retrieved from the World Wide Web: http://ermeweb.free.fr/CERME%205/Plenaries/Plenary3.pdf

Ruthven, K., & Hennessy, S. (2002). A practitioner model of the use of computer-based tools and resources to support mathematics teaching and learning. Educational Studies in Mathematics 49(1), 47-88

Steffe, L. P. (1991). The constructivist teaching experiment: Illustrations and implications. In E. von Glaserfeld (Ed.), Radical constructivism in mathematics education (pp. 177-194). Dordrecht, Netherlands: Klumer Academic Publishers.

Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of Research Design in Mathematics and Science Education (pp. 267-306). Mahwah, NJ: Lawrence Erlbaum Associates.



Appendices

Study questionnaires



NCETM Manchester conference poster



Handout of menus and commands Mode Descriptions - Using the default toolbar groupings

Move New point Polygon Line through two points

Rotate around point Intersect two objects Regular polygon Segment between two points

Midpoint or center

Segment with given length from point

Ray through two points

Vector between two points

Vector from point

Perpendicular line Circle with center through point Move drawing pad

Parallel line Circle with center and radius Zoom in

Line Bisector Circle through three points Zoom out

Angular Bisector Semicircle Show/hide object

Tangents Circular arc with center through two points Show/hide label

Polar or diameter line Circumcircular arc through three points Copy visual style

Locus Circular sector with center through two points Delete object

Circumcircular sector through three points

Conic through five points

Angle Mirror object at line Slider

Angle with given size Mirror object at point Check box to show and hide

objects



Distance or length Rotate object around point by angle Insert text

Area Translate object by vector Insert image

Slope Dilate object from point by factor Relation between two objects GeoGebra Commands

AffineRatio Angle AngularBisector Arc Area Asymptote Axes Center Centroid Circle CircularArc CircularSector CircumcircularArc CircumcircularSector Circumference Conic Corner CrossRatio Curvature CurvatureVector Curve Delete Derivative Diameter Dilate Direction Directrix Distance Div Eccentricity Element Ellipse Extremum FirstAxis FirstAxisLength Focus Function If InflectionPoint Integral Intersect Iteration IterationList Length Line LineBisector Locus LowerSum Max Midpoint Min Mirror Mod Name OsculatingCircle Parabola Parameter Perimeter Perpendicular PerpendicularVector Point Polar Polygon Polynomial Radius Ray Relation Root Rotate SecondAxis SecondAxisLength Sector Segment Semicircle Sequence Slope Tangent TaylorPolynomial Translate UnitPerpendicularVector UnitVector Uppersum Vector Vertex

Further Mathematics Network advertisement



BSRLM Working group abstract

British Society for Research into Learning Mathematics Day Conference

28 February 2009 Faculty of Education, University of Cambridge

Geometry Working Group Session:

Establishing a professional development network with an open-source dynamic mathematics software – GeoGebra

By Keith Jones, Zsolt Lavicza, Markus Hohenwarter, Allison Lu, Mark Dawes, Alison Parish, Michael Borcherds

Abstract: This working group session provides an opportunity to work with participants in an NCETM-funded project that is working on establishing a professional development network with open-source mathematical software – GeoGebra – in England. In addition, we would like to invite to this session colleagues who are interested in professional development issues in technology-equipped classroom environments. The GeoGebra project has involved nine experienced teachers reviewing, modifying, and implementing existing teaching materials in their classrooms and developing ways of providing professional development and support for other teachers across England. The participating teachers not only contributed to the development of professional development materials but also led workshops for other teachers. Project meetings and workshops were videotaped and participants were interviewed during the past year. Participating teachers were also involved in the analysis of data and the refinement



of workshop materials and teaching approaches. The working group session provides the opportunity to find out some results of the project and to discuss potential future work around this topic. We hope that the working group further contributes to nurturing a community of teachers and researchers in England who are interested in developing and using open-source technology in schools and in teacher education.

Chris Little: MSOR Connections

Differentiation in three easy, Geogrebra-style, lessons Chris Little Why is calculus difficult? It required a conjunction of geometric and algebraic thinking to invent or discover its concepts. What makes Geogebra exciting as a tool for calculus is that it offers an opportunity to represent differentiation as both an algebraic and geometric process simultaneously. My experience of working with teachers is that the software is sufficiently straightforward and user-friendly that it doesn’t take a lot of practice, or insider-knowledge about how it works, to give the teacher, or even the learner, the tools to discover the hidden treasures of calculus, and to develop insights which do not require the towering genius of its originators or discoverers, Newton and Leibnitz. Take differentiation. First, we have the geometric image of a point moving round a curve, with its tangent varying in gradient (see Lesson 1)

Lesson 1 1. Input f(x) = x^2 2. Create a point on f(x). Relabel (right click) the point as P. 3. Construct tangent to the curve at P. 4. Measure the slope (gradient) of the tangent (m). 5. Move P, and record the slope of the tangent at points with x-coordinates −2, −1, 0, 1, 2, … 6. Predict the gradient at the point with x-coordinate 4. Verify your prediction. 7. Predict the gradient at the point with x-coordinate 0.5. Verify your prediction. 8. Predict the gradient at the point with x-coordinate −2.5. Verify your prediction. 9. In your own words, write a rule for calculating the gradient of f(x) = x2. 10. Repeat with f(x) = x^3, x^4, 2x^2, 2x^2 + 1, etc. Second, we have the geometric image of a gradient function, traced by the value of the derivative as the point travels round the curve (see Lesson 2)



Lesson 2 1. Input Q = (x(P), m) [x(P) is the x-coordinate of the point P]. 2. Move P and observe what happens to the point Q. 3. Right click Q and turn trace on. 4. Move P, and observe the trace of Q. Find the equation of this trace line. 5. Undo trace (button top right) 6. Construct locus of Q (locus, then click on Q, then click on P) Find the equation of this locus line. 7. Input f’(x) = 2x. Verify that this is the same line as the locus line. Change colour (right click) to red. 8. What is the gradient of the tangent to f(x) = x2 at (10, 100)? 9. What is the point on f(x) = x2 where the gradient is 12? 10. What can you say about the gradient at x = a and x = −a? 11. Repeat for f(x) = x^3, x^4, …, x^n… Third, we have the algebraic process of differentiation, mirrored by its geometric image of the limit of the secant line as a tangent (see Lesson 3). All this could of course be done using other dynamic geometry packages, and Marcus Hohenwarter would no doubt acknowledge the ground-breaking work of Cabri and Geometer’s Sketchpad in pioneering computer interfaces which make all this possible! However, I believe that Geogebra represents a significant advance, through its ease of use, uncluttered screen, and friendly on-screen help, which enables the user to focus on the maths rather than the mechanics of running the program. Moreover, the closer match between the algebraic notation required for first principles differentiation, and the input of algebraic expressions in Geogebra, might provide the learner with a vital, multiple – embodied, link between the geometric image (on the right) and algebraic process (on the left) which will help the process of encapsulation (Dubinsky, 1991, Sfard, 1991) necessary to make sense of differentiation. DUBINSKY, E. (1991) Reflective Abstraction in Advanced Mathematical Thinking. IN TALL, D. (Ed.)

Advanced Mathematical Thinking. Dordrecht, Kluwer. SFARD, A. (1991) On the Dual Nature of Mathematical Conceptions: Reflections on Processes and

Objects as Different Sides of the Same Coin. Educational Studies in Mathematics, 22, 1-36.

f’(x) is called the gradient function of derivative of f(x). So the derivative of f(x) = x2 is f’(x) = 2x.



Lesson 3 1. Input f(x) = x^2 and create a point on the curve, labelling it P. 3. Input a = x(P). Remember that a is the x-coordinate of P 3. Construct the tangent to the curve at P. Colour it red. 4. Input h=1. Right click and show object. In properties – slider – change min to 0, max to 1 and interval to 0.01. We are now going to create a point Q on the curve with x-coordinate x(P) + h. 5. Input Q = (a+h, f(a+h)) (be careful with the brackets!) 6. Change the value of h using the slider. Observe what happens to Q. 7. Create the line through PQ. 8. Measure the gradient (slope) of PQ (m). 9. Move the point P to (1, 1). Now gradually change the value of h until it approaches zero. Observe what happens to the chord PQ, and to the value of m. To get a closer look, you can zoom into P using the move menu. What happens to m when h = 0? Why? The gradient of the chord approaches (or tends to) the gradient of the tangent as h approaches (tends to) zero. 10. Input S = (a,m). 11. Construct the locus of S as P varies on the curve. Colour it red. This locus is the gradient of PQ plotted as a function of a, the x-coordinate of P. 12. What happens to this locus as h approaches zero? As h equals zero? We have so far used the slope function to find the gradient of PQ. Can we use algebra to do this? In the diagram (right) , PR = h, and QR = y-coord of Q − y-coord of P = f(a+ h) − f(a)

So gradient of chord PQ = QR f( ) f( )PR

a h ah

+ −=

13. Input m’=(f(a+h) − f(a))/h. Check that m = m’ as you change P or h. Now for f(x) = x2, f(a) = a2 and f(a+h) = (a+h)2

So the gradient of the chord PQ = m’ = 2 2f( ) f( ) ( )a h a a h a

h h+ − + −

=

14 Expand (a + h)2, and hence show that m’ = 2a + h (provided h is not zero!) 15 Input g(x) = 2x + h. Check that line is the locus of S. What happens to g(x) when h = 0?

We say that the derivative f ‘(x) is the limit of f( ) f( )a h a

h+ −

as h tends to zero, or 0

f( ) f( )limh

a h ah→

+ −⎡ ⎤⎢ ⎥⎣ ⎦

16 Repeat with f(x) = x^3, 2 x^2, etc.

P(a, f(a))

Q(a +h, f(a+h))

R

Documents

GeoGebra - NCETM · GeoGebra represents one such software program that was designed to combine arithmetic, algebra, geometry, and calculus in a single, integrated system (Hohenwarter