The abstracts are sorted alphabetically by contributor’s first name

# Anne Bjørnestad: Working with tasks enhancing teachers in Secondary Schools practical competence in using GeoGebra in their teaching

Workshop: 45 min

The workshop is a follow-up from Anne’s plenary talk. The participants will be offered the opportunity to work with a few of the tasks Anne’s students have been working with.

# Anne-Mari Jensen:Planning for the population growth in Trondheim

Workshop: 45 min

This workshop is a project for students in upper secondary school. We will use some statistics of the population in Trondheim, pose some problems and try to conclude with some good advices for the people who make the plans for the future in the community.

# Anne-Mari Jensen: Tide water

Workshop: 45 min

This workshop is a project for students in upper secondary school. We will use some data for the tide water for one day, and find a trigonometric function as a mathematical model. In the model there are some parameters. The students have to find them, find the right units and explain what they tell about the tide water at this location the actual day. Next we will find the correct values and compare, and have a discussion about mathematical models.

# Bo Kristensen: GeoGebraBooks as a teaching resource and portfolio platform

Workshop: 90 min

GeoGebraBooks can be used for several purposes, and in this workshop you will be introduced to two different ways of using them in class.

First I will give examples of how you can use GeoGebraBooks to make you own teaching materials. I will show an example of a Danish material aimed at lower secondary in the subject “transformations”.

After this you’ll get the opportunity to make your own material using GeoGebraBooks. You will be introduced to a couple of free and easy-to-use multimedia resources such as screencasts, videos, images, sound files and GeoGebra files. The progression of the workshop will be as follows:

• Creating a GeoGebraBook.
• Creating chapters.
• Using other user’s worksheets and creating your own from scratch.
• Embedding multimedia files in a GeoGebraBook.
• Making screencasts with Screencastify
• Make a sound recording with Clyp.it
• Programming simple applets using GeoGebra

If there is enough time at the end of the workshop, I will show examples of how your pupils can use GeoGebraBooks as their platform for making portfolios in math.

To participate fully in this workshop, you’ll need:

• Laptop with a working mic and earphones.
• Chrome Browser
• An account on GeoGebra.org

# Bo Kristensen & Rikke Teglskov: Get started making animations in GeoGebra

Workshop: 90 min

It’s extremely simple to get started making animations in GeoGebra. A point attached to a line or perimeter and you are ready to go. From here the only limit is your imagination. A lot of pupils find animations motivating and while they are producing animations they are developing skills essential for the 21st century. As a positive side effect, they risk learning a lot of math while doing so.

In this workshop we will briefly argue which learning potentials we see when working with animations. After this introduction the rest of the workshop will be hands-on. We will demonstrate different techniques and categories on the whiteboard but it’s also possible to explore the online material on your own and in your own pace.

When you leave the workshop and get back home you should be able to continue your journey with animations on your own.

# Hannu Mäkiö: GeoGebra as a student's tool in geometry courses

Talk: 25 min

For long Geogebra has been seen as a great tool for inquiry based learning, especially in geometry. The idea of inquiry based learning is stated clearly in the curriculum of finnish upper secondary school, but in practise there are many obstacles to overcome. In my talk I will discuss the ways I have tried to implement inquiry based learning and which challenges I think I am still facing.

# Jan Ronny Birkeland: Practical Problem solving with GeoGebra

Workshop: 45 min

We use bouncing balls, primitive cars and ‘non-stop’ chocolate to identify functions and formulas through GeoGebra. Participants should bring a computer with GeoGebra installed. There will be practical exercises that can be solved with GeoGebra. Participants do not need to be experts in GeoGebra, but it is an advantage to have used it a little.

# Janika Kaljula: Using GeoGebra for getting to know triangles

Talk: 25 min

In Estonia the year 6 students learn a lot of geometry, including triangle, its elements, classification, how to draw them and the properties of an equality. Also finding area of triangle by means of base and height. A website will be introduced where students have tutorials with fill-in questions and instant feedback and also with GeoGebra graphics view to do the tasks. The instructions are built up so that the students would know how to solve problems without a computer as well. The tasks are suitable for independent learning and for demonstration.

# Jarno Koskimäki & Janne Cederberg:GeoGebra and finnish matriculation exam system

Workshop: 90 min

GeoGebra are inside Finnish matriculation exam system. The workshop will introduce you to the system. This thing called Abitti. You will need your own laptop. Because the system is on USB. You need to know how to boot your computer from USB.

Workshop: 90 min

Participants will have full control over this workshop which will be based on their "How do you..." questions. Though any questions are accepted, this workshop is aimed at those who already have a basic understanding of GeoGebra and wish to know more about some advanced topics such as animation, dynamic colour, automatic visibility control, simple scripting and buttons, using GeoGebra groups and exam mode etc.

# Jonas Hall, Thomas Lingefjärd, Malin Christersson, Svetlana Yushmanova & Anders Karlsson: Constructing high quality apps

Talk: 25 min

In Sweden, the National Encyclopedia company has commissioned GeoGebra apps for their new online mathematics textbooks. This has resulted in a big effort to create similar looking apps of high quality, some of which will be shown during this talk. It has also resulted in strengthening the collaboration between individual members of the Swedish GeoGebra Institute and a future "Builder's conference" for top-level construction skills sharing is suggested. Possible topics of this future conference will be shared during the talk.

# Katarzyna Winkowska-Nowak: The report of introducing GeoGebra into Polish curriculum

Talk: 25 min

GeoGebra started to be used by Polish teachers and researchers as soon as it was available on the internet. The first Polish translations of GeoGebra are going past 2007- now is nearly ten years. But the group of active users is still not so big. One of the biggest problem is absence of materials in GeoGebra and about GeoGebra in Polish. Our group connected with Polish GeoGebra Institute in Warsaw worked using government grants to create such materials – one grant for middle school and one grant for high school. The grants run from 20012-2015 and consisted of creating materials in the first phase, testing them in schools and dissemination. These materials are aligned with Polish curriculum and consist of schools program, methodology books for teachers, materials in form of GeoGebra applets, short exercise books for students, and extension for bright students as well as for students with learning problems. In our talk we will describe the project in some details and give examples of work.

# Kin-Keung Poon and Tak-Wah Wong: Dynamic proof of the Wallace-Bolyai-Gerwien Theorem

Talk: 25 min

Wallace-Bolyai-Gerwien Theorem states that any two simple polygons of equal area can be dissected into a finite number of congruent polygonal pieces. It is difficult for high school students and teachers to understand the proof. By using dynamic mathematics software,such as Geogebra, the authors can give the dynamic proof of the theorem so that students and teachers can understand the essential steps of the proof more easily.

# Liene Krievina: Possibilities of intersection design in research exercises of GeoGebra 3D

Talk: 25 min

Approximately 29 % of people have visual perception, but not always students have developed spatial thinking. Mathematics program GeoGebra can facilitate the perception of spatial structure. 3D view is one of the possibilities offered by this program. I suggest sharing my experience about research exercises which are designed to create three-dimensional slits of forms with plane and to examine correlations using possibilities offered by GeoGebra.

# Margrethe Solevåg & Anne Karin Wallace: GeoGebra and 3 Act Math:

Workshop: 45 min

3 Act Math is a concept developed by teachers in North America by using short videos, pictures etc. This method is trying to develop students’ curiosity and mathematical thinking. In our workshop we will give some examples of how we can combine GeoGebra and this pedagogic concept. We will try to present examples which can be used in both lower and upper secondary schools.

# Maria Fahlgren & Mats Brunström:Pros and cons of asking students to do their own GeoGebra constructions

Talk: 25 min

In this talk, we will share some experiences from a design-based research project with four upper secondary teachers (and their classes). Together we have developed and tested a series of student activities with GeoGebra. The topic is functions and graphs, with focus on exponential functions. The target student group is first year students in Swedish upper secondary school.

One design principle was that the students should do all necessary mathematical constructions in GeoGebra by themselves, that is, not using pre-designed applets. We will demonstrate some student responses and discuss pros and cons of this design principle.

# Marius Zakarevičius, Vilja Šileikienė & Rokas Tamošiūnas: Using Geogebra in Lithuania 2012-2016: success and failure

Talk: 25 min

In 2012 Lithuania has started using Geogebra in schools and its popularity has grown. The talk is about what has been done in Lithuania to increase the number of teachers and students to use Geogebra during lessons in 2012 – 2016 school year, how Geogebra was presented to academic community and how it is used in Lithuanian education system.

# Markus Hohenwarter: Learning Management Systems and GeoGebra

Workshop: 45 min

In this workshop, we will try out some of the features of GeoGebra Groups, our learning management system based on the GeoGebra Materials online sharing platform. We will also discuss pros and cons of some other GeoGebra integrations with Moodle, Google Classroom and other learning platforms. Furthermore, I would like to discuss with you the potential of turning the GeoGebra community into a social network for STEM education where you can follow other authors and groups to share materials and ideas.

# Martin Thun Klausen: Danish GeoGebra championship - how and why

Talk: 25 min

In Denmark we will this year be having the first Danish GeoGebra Championship for primary school.

Danish GeoGebra competition is a class competition for pupils in primary school (0. to 3. grade) and the purpose of the competition is to inspire pupils and teachers to work creatively and aesthetically in mathematics on a open tasks, with many solutions and involving use of GeoGebra.

I will present our background and ideer and also present the outcome and the classes work. Also I will present how the first years students of the teacheruducation was involved in their own GeoGebra competition.

# Mats Brunström & Maria Fahlgren:Task design with focus on exploration, explanation, and generalization using GeoGebra

Talk: 25 min

In the talk, we will elaborate on how to turn a traditional proof task into a “proving activity” that has a more open approach, consisting of a sequence of tasks, and which utilizes the affordances provided by GeoGebra.

We will introduce a general model and illustrate it by using an illuminating example. The purpose of the model is to foster students’ capability to explore, conjecture, verify, explain, and make generalizations with a particular focus on the last element.

The talk is based on a research paper.

# Mikko Rahikka:Using GeoGebra with Google Apps for Education

Talk: 25 min

I have been using GeoGebra in my math and physics courses with GAFE and especially Classroom for two years. I will show the way I use it.

# Nanna Filt Christensen: Moving in sync with Geogebra

Workshop: 45 min

In continuation of the talk “Moving in sync with Geogebra” this workshop will offer a closer look at the concept of “student animations”. The participants will be challenged with some children movements, where they have to devise the right Geogebra animation to simulate the reality. They will be paired up to create both a little dance move and a so called “ghost trail” for each other.

# Rikke Teglskov: CAS-project. A project on developing teacher skills in using IT in their classroom. Primary and lower secondary

Talk: 25 min

This session will be a looking on the use of CAS (Computer Algebra System) in primary and lower secondary schools in Denmark. I will give quick overview of the demands for using CAS in the Danish curriculum, and I will give a presentation on a teacher network and the experiences we have had from a project that has run in Denmark from 2014-2016 with making math teachers work with using ICT in their teaching. How do you get teachers to share and collaborate on planning and developing lesson plans, teaching materials etc.?

The upcoming school year we have a project on how to use CAS in math lessons more widespread. CAS is the least used digital tool in Danish schools even though it is compulsory to use from year 4. I will present the ideas behind the CAS-project, and we will work with some of the material and ideas from the project.

# Sirje Pihlap & Liis Mardi: Worksheets with feedback

Workshop: 45 min

Creating GeoGebra worksheets that give feedback to students using check box and worded feedback in different kind of tasks.

# Sirje Sild & Sirje Pihlap: Notice mathematics around you - fireworks for the birthday of Estonia

Talk: 25 min

An overview of the best works of the student competition and analysis of students' answers to two questions 1) what did you enjoy about doing this task and 2) what did you learn.

# Sirje Sild, Sirje Pihlap & Kristi Kreutzberg: Notice mathematics around you - art

Workshop: 45 min

You will learn how to make patterns using animated graphs of functions. Making these patters helps students to understand the relationship between the formula of the function and its graph.

# Svetlana Yushmanova & Anders Karlsson: Swedish National Exam - done in a coffee break with GeoGebra

Workshop: 90 min

Swedish National Exams in Mathematics consist of 2 parts, in one of which students are allowed to use calculators and/or computers. These tests are often considered to be stressful; "So many questions, so little time". We believe the reason to be a poor choice of digital tool (i.e. calculators), or perhaps a poor training in how to use the tool chosen.

We will present a dozen or so of actual exam questions, to which we have decent GeoGebra solutions, and challenge the participants to find even better ones. (A small prize will be awarded to the most elegant solutions.)

A well working strategy for "showing your workings" will also be presented and discussed.

The questions are from upper secondary Swedish exams, but the mathematical content should apply to all countries. All presented material will be in English, as well as the workshop itself.

Teaser (graded as one of the most difficult questions in year 1):

It holds for a function f where f(x)=kx+m that

• f(x+2)− f(x)=3
• f (4) = 2m

Find the function f .

# Thomas Lingefjärd: Teaching mathematics to students from different countries

Talk: 25 min

Vygotsky placed emphasis on the role of language in cognitive development. For Vygotsky, cognitive development results from an internalization of language. In fact it is probably quite impossible to learn anything without a language. Immigrant students who arrives to schools in the Nordic countries and elsewhere have to understand and internalize mathematical objects and mathematical concepts in a different language than their mother tongue.

The learning of mathematical objects and our development of internal concepts are intimately connected. We learn from early years by noticing external representations of mathematical objects and then we use them to construct our own internal representations of the same mathematical objects. It can be from concrete, but imprecise, objects such as toys or cartons, or from repeating the number sequence orally, such as “1, 2, 3...” given by parents or other adults. We need the ability to mirror as well as the ability to use our imagination.

Imagination and the ability to construct and explore internal representations, sometimes in an intimate interplay with external representations, forms a fundamental base for learning mathematics. See Lingefjärd and Ghosh (2016) for a more throughout discussion about the interplay between internal and external discussion.

My video clips are also based on the learning theory of three mathematical worlds by Tall (2004). Tall (2004) suggests a possible categorisation of cognitive growth into three worlds of mathematics or three distinct but interacting developments. Three worlds of mathematics are founded on the assumption that the learning of mathematical concepts is individual and develop at different levels, through perception, through symbols or through axioms.

The first world is the conceptual – embodied world, the world we meet through perception, the visual and spatial mathematical world. Most of us have a concept image of what a circle is. A circle is round, it may be large or small and it may be red or blue. We have not learned this through educational efforts; instead we have learned this through the physical world and through observations. The first mathematical world consists of objects we have discovered and observed in the real world, knowledge we have gained through our senses. It also contains mental conceptions of non-existing objects such as point with no size and lines with no thickness.

The second world is the proceptual – symbolic world. In this world we find symbols and actions that we have to perform when we, for example, are dealing with manipulations in algebra. Central in this world is the concept of procept which consist of the first part of process and the end of the word concept. Tall & Gray (1994) introduced the concept procept to describe a central part of the learning of mathematical concepts. It is important to learn how to apprehend mathematical symbols both as concepts and as parts of a process at the same time.

An elementary procept is the amalgam of three components: a process which produces a mathematical object, and a symbol which is used to represent either process or object. (Tall & Gray 1994. p. 12)

According to Tall & Gray (1994), the expression 2 + 3 may be perceived as a process (addition) or as a concept (sum). When the individual is in this symbolic world may he/she use and reflect over the mathematical symbolic language and its function, meaning and application.

The third mathematical world is the formal mathematical world. Is this world are axioms, theorems and proofs in focus. Based on given assumptions regarding the proportion and relation between mathematical objects are axiom based structures built and used as foundations for mathematical theorems. Mathematical thinking is thereby based on perception developing subtly in sophistication through the mental world of conceptual embodiment, operation developing through actions that become mathematical operations in a world of operational symbolism and increasingly subtle use of verbal reason that leads to formal aspects of embodiment and symbolism and eventually to a world of axiomatic formalism. The development takes account of the individual's previous experience which may operate successfully in one context yet remain supportive or become problematic in another, giving rise to emotional reactions to mathematics, leading to a spectrum of success and failure over the longer term (Tall, 2004).

Tall & Vinner (1981) and Tall & Gray (1984) claims that theories about cognitive development of mathematical knowledge is in many ways quite comparable with the historical development of mathematics as an axiomatic science. My video clips, in turn, can be seen as a vehicle that helps students to move from one mathematical world to another.

References

Gray, E. M. & Tall, D. O. (1994). Duality, ambiguity, and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 25, 116 - 140.

Lingefjärd, T & Ghosh, J 2016. Learning mathematics as an interplay between internal and external representations. Far East Journal of Mathematical Education, 16 (3), 271-297.

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169.

Tall, D. (2004).Thinking Through Three Worlds of Mathematics. Proceedings of the 28th Conference of PME, Bergen, Norway, 4, s. 281-288.

# Tor Espen Kristensen: 3D

Workshop: 45 min

GeoGebra has many nice tool that makes it possible to construct geometrical objects in space. In this workshop we will look at different strategies made possible by GeoGebra to solve some geometric problems in 3D.

# Tor Espen Kristensen: Simulations with GeoGebra

Workshop: 90 min

We will use the worksheet and lists in GeoGebra to create simulations of various random experiments.

Examples of such attempts include:

• rolling dices
• draw balls from a box
• binomial trials
• random normally distributed trials

We will use buttons with scripts to make it all work and look nice.

# Valgarð Már Jakobsson: Is the focus on the software or the math? Teaching statistics with GeoGebra

Talk: 25 min

In my talk I will discuss my teaching of statistical analysis the past 7 years in an Icelandic high school. My freedom of selecting teaching materials and assignments has allowed me to experiment with GeoGebra for teaching and exploring statistical concepts from mean, mode, and standard deviation to the normal distribution and testing hypotheses. In my talk I will show some tasks I have been using in my classroom.

# Øistein Gjøvik: The power of a tangent

Workshop: 45 min

In this workshop, we will work with a classic exercise, now on GeoGebra. Originally, I met this exercise in a workshop with John Mason some ten years ago. I will lead us through the activity and show how I have used it in teacher education.