The PRIMAS project: Promoting inquirybased learning (IBL) in mathematics and science education across Europe
PRIMAS guide of supporting actions for teachers in promoting inquiry-based learning
PRIMAS is developing a guide for supporting actions for teachers. Such actions are essential to be carried out in order to foster the uptake of inquiry-based learning (IBL) by teachers in schools and in order to create a supportive climate for teachers to do so. This guide introduces key issues in implementing supporting actions and presents a wide range of case studies of actions that have been successfully carried out in PRIMAS. As the project runs until 2013 this guide is a first version of the final guide to be published in 2013.
www.primas-project.eu
The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Project Information Project no. 244390 Project acronym: PRIMAS Start date of project: 01/01/2010 Duration: 48 months Project title:
Promoting inquiry-based learning In mathematics and science education across Europe Dissemination level Thematic Priority: Science in Society Funding scheme: FP7/ CSA/ Capacities
Information about the deliverable Deliverable N° D 5.1 Due date of deliverable: Month 18 Actual submission date: 15/17/2011 Deliverable title:
PRIMAS WP5 – Supporting actions for dissemination among teachers: International guide about supporting actions for teachers (version 1)
Contact Information Coordinator: University of Education Freiburg, Prof. Dr. Katja Maaß Lead partner for this deliverable: Constantine the Philosopher University in Nitra, Slovakia, associated Professor Dr. Sona Ceretkova, PhD. Website: http://www.primas-project.eu/
The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Table of Contents EXECUTIVE SUMMARY ......................................................................................................... 3 Carrying out dissemination activities to support the uptake of IBL in schools across Europe ..................................................................................................................................... 5 About PRIMAS ..................................................................................................................... 5 How will PRIMAS achieves this? .................................................................................................... 5 Inquiry-based learning (IBL) ................................................................................................. 5 What do we mean by inquiry-based learning? ............................................................................... 5 Why inquiry-based learning? .......................................................................................................... 6 Dissemination activities ........................................................................................................ 6 Professional development vs. dissemination activities ................................................................... 6 What do we mean by dissemination activities? .............................................................................. 7 Target groups addressed in this guide ........................................................................................... 7 Guides for other target groups or different activities....................................................................... 8 Planning your dissemination activity .................................................................................... 8 Aims and target groups ................................................................................................................... 8 What contextual factors are important? .......................................................................................... 9 How can we acquire as much attendees as possible? ................................................................... 9 Organisation of the event .............................................................................................................. 10 Dissemination materials ................................................................................................................ 11 Overview of dissemination activities of PRIMAS .............................................................. 12 The case studies .................................................................................................................. 15 Purpose of the PRIMAS case studies ................................................................................ 15 Introduction to the case studies .......................................................................................... 15 Spain: Information about PRIMAS ..................................................................................... 18 Switzerland: Crossed overview on scientific approach in mathematics, physics, chemistry and biology ......................................................................................................................... 22 Germany: Day for head teachers (Schulleitertag) .............................................................. 27 Cyprus: Mathematical Society Annual Meeting and Conference Cyprus ........................... 32 United Kingdom: Professional Mentors’ Update Meeting ................................................... 36 Norway: Introduction of PRIMAS – Teachers’ Problem-solving Beliefs ............................. 38 Germany: Day for Teachers (Lehrertag) ............................................................................ 42 Slovakia: Mathematic Problem-Solving Methods Seminar ................................................. 45 Hungary: Talent Day .......................................................................................................... 49 Romania: PRIMAS piloting Action ...................................................................................... 51 Slovakia: Methods day for teachers ................................................................................... 55 The Netherlands: Ecent conference 2011 .......................................................................... 57 United Kingdom: Introducing Inquiry-based Learning to trainee mathematics teachers in England .............................................................................................................................. 62 Malta: Introducing PRIMAS to heads of schools ................................................................ 68 Annex I – Examples of dissemination materials ............................................................... 70
2 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
EXECUTIVE SUMMARY Too many students in Europe still have little interest in mathematics and science and thus do not choose to follow related careers. Consequently, the needs of Europe’s labour market may not be met – a weak point in our progress to a highly competitive European Union. Implementing innovative ways of teaching such as inquiry based learning in science and mathematics education is a key lever in bringing about change in this situation by raising students’ motivation and interest in these subjects. In order to motivate teachers to implement inquiry-based learning, a variety of activities need to be carried out. First of all, teachers need to be informed of innovative ways of teaching. Second, teachers need to be supported in changing their way of teaching: They need materials and professional development courses. Furthermore, contextual factors, such as assessment practices that do not include inquiry-based learning, need to be addressed. This guide will give readers recommendations and guidelines for the first aspect: On how to reach teachers with so-called dissemination activities. Their purpose is to reach as many teachers as possible and to give them first insight into inquirybased learning. These dissemination activities have to be distinguished from longterm professional development (PD) activities, which are also necessary if teachers are to change their way of teaching. Long-term PD courses, however, are time and cost consuming, and can (initially) reach only a smaller number of teachers. For this reason, activities reaching a wider audience have to be considered. At the same time, core measures in updating teaching practices (such as PD courses and materials) also need to be supported by preceding dissemination activities (successfully motivating teachers to eventually take part in such PD courses or use materials) as well as parallel activities directed towards other key target groups. The dissemination activities we report on here are addressed to teachers and supporting groups such as head teachers, teacher´s networks and associations, school authorities, curriculum & assessment developers, teacher educators, university teachers, researchers. This guide will address the following topics • • • •
What do we mean by inquiry-based learning and why should it be included in math and science lessons? How to distinguish long-term professional development courses from dissemination activities? How to plan dissemination activities: What important issues should be taken into account? What dissemination materials can be used?
3 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
After the general part with recommendations and guidelines, this guide will present selected case studies from 12 European countries which give inspiring insights into the rich variety of possibilities that exist to effectively address and support teachers with dissemination activities. They illustrate not only the breadth of activities available but also point to general issues to be considered and regarded in planning and carrying out dissemination activities and can as well motivate and inspire further activities. Annex I will present some of the dissemination materials, developed and used in the PRIMAS project, to support dissemination activities. Purposefully developed materials such as posters or brochures and their consistent use are an essential part of effective and efficient dissemination strategies. The right supporting materials can effectively address and appeal to specific target groups, create the attention and interest of these as well as communicate key messages and be a first point of contact establishing the potential for more. This guide is based on the work of the EU´s 7th Framework project PRIMAS. In this project, fourteen universities from twelve different countries work together to further promote the uptake of inquiry-based learning (IBL) in mathematics and science.
4 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Carrying out dissemination activities to support the uptake of IBL in schools across Europe About PRIMAS PRIMAS is an international project within the Seventh Framework Program of the European Union. Fourteen universities from twelve different countries are work together to further promote the uptake of inquiry-based learning (IBL) in mathematics and science. How will PRIMAS achieves this? PRIMAS supports teachers to integrate and apply inquiry-based learning pedagogies in their mathematics and science classrooms. During the project’s lifetime (20102013) various resources and support measures will be developed and made available to mainly teachers, parents and students. Among these are: • • • • •
Support for professional development facilitators Professional development courses for teachers Support for teachers Teaching materials Information meetings for students and parents
Furthermore, PRIMAS works in close cooperation with school authorities and other key institutions in national education systems across the partnership. This helps to ensure the best possible conditions for the integration and application of inquirybased learning in classrooms.
Inquiry-based learning (IBL) What do we mean by inquiry-based learning? Inquiry-based learning is an innovative way of learning and teaching. It aims to develop ‘inquiring minds and attitudes’ of young people which will be required in an uncertain future. Fundamentally, learning is based on students adopting an active, questioning approach. Students inquire and pose questions, explore and evaluate, and the problems they address seem ‘real’ to them. Learning is driven by open questions and multiple solution strategies. Teachers are proactive, supporting struggling students and extending those that are succeeding through the use of carefully chosen strategic questions. They value students’ contributions, including their mistakes, and scaffold learning using students’ reasoning and experience. In the classroom there is a shared sense of purpose and ownership.
5 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Why inquiry-based learning? In our dynamic knowledge-based society, students must develop the ability to attain knowledge and competences as well as problem-solving skills. Knowledge of facts alone is not enough in the 21st century. Students should develop competencies to apply their knowledge in realistic problem-solving situations. They also need to develop competencies for self-directed learning and to explore new knowledge areas. IBL can support the development of such competencies. Meaningful and challenging IBL-tasks in maths and science provide: • • •
first-hand experience for students of typical approaches to solving problems opportunities to achieve a deeper understanding of underlying concepts and laws, increased motivation and interest in maths and science as subjects.
Dissemination activities Professional development vs. dissemination activities Implementing inquiry-based learning in the classroom can be a challenge for teachers who are not used to it. Their role in a classroom changes from a teacher who explains everything to a facilitator who supports the students in finding their own solutions and provides help only when needed. Teachers need to have confidence in the competences of their students and need to allow them to go their own ways. But whether or not teachers actually implement innovative methods of teaching does not only depend on the teachers themselves but is also influenced by a number of contextual factors. Among these are assessments which may not include inquirybased learning or an overloaded curriculum which may not easily allow for IBL to be implemented. Further influencing contextual factors are the head teachers, students and parents. A supportive attitude of the head of school seems to be quite important. Additionally, students have their own beliefs and concepts about a subject. They may initially decline changes to these and their reaction may be quite strong and also emotional. Subsequently, the implementation of inquiry-based learning in day-to-day teaching is by no means a straight-forward enterprise. It requires long-term support for teachers as well as effecting important changes in contextual factors. This long-term support can only be guaranteed by long-term professional development courses which are demanding as well as time and cost consuming for both teachers and providers. Before teachers are willing to subscribe to such PD courses, they need to be informed about inquiry-based learning and their interest and an initial understanding of it has to be developed. This is a reason why dissemination activities are needed. A broad and extended dissemination approach will later also help teacher to meet supportive environments for implementing their newly acquired approaches: Their fellow teachers might have heard of it and become interested as well, their head teachers will advocate the approach, parents will probably already
6 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
have an initial understanding and so on. This is a reason why dissemination activities need to address a broad range of target groups and should reach as many persons as possible of these groups. What do we mean by dissemination activities? Dissemination activities support a widespread implementation of inquiry-based learning by • •
giving teachers first insights into the topic and by reaching many teachers (a lot more than you can reach with long term professional development course).
Dissemination activities (or supporting events) could be conferences, one-day-and-off courses, talks or workshops as well as information events. They complement longterm professional development courses and are thus of particular value. Apart from the scope, frame and setting of the activities themselves that need to be carefully selected and planned the right materials, such as informative posters and brochures, are required. They need to be specifically developed for different target groups. Small promotional materials that catch attention can effectively support dissemination activities or raise people’s interested in them. In the following, we will focus on events and activities. Nevertheless we will later come back to the materials and give some illustrating examples, as they play a crucial role in attracting target groups. Target groups addressed in this guide This guide focuses on dissemination actions for teachers that can be carried out in addition to the project´s direct delivery of the classroom materials and PD courses to the teachers involved in the project. Supporting actions for teachers can be split into two areas: 1. They can be directed to teachers and can either be designed to a. help to motivate teachers to participate in professional development courses b. give a wider audience of teachers insight into inquiry-based learning 2. They can be directed towards ‘supportive groups’ for teachers in their uptake of IBL teaching strategies – these groups are in close connection to the school system and therefore play a significant role in supporting teachers to use IBL. Groups within this frame are: head teachers, teacher´s networks and associations, school authorities, curriculum & assessment developers, teacher educators, university teachers, researchers.
7 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Guides for other target groups or different activities Readers who are interested in carrying out dissemination activities for out-of-school groups, which can for example be addressed to parents or to broader communities, might be interested to take a look at our guide for dissemination activities for out-ofschool-target groups. It focuses on the more general activities with a wider scope of outreach to promote a supportive climate for inquiry-based learning. Readers who are interested in running long-term professional development courses may wish to look at our guide for professional development providers. The guides and reports can be obtained from: www.primas-project.eu
Planning your dissemination activity Planning a dissemination activity requires quite some work. In order to make your activity successful, you might like to consider the following aspects. • • • • •
What is the aim? What are the target groups? Which contextual factors are important? What should be the content of the event? How can we acquire as much attendees as possible? How can we set the right frame and create the appropriate setting for the event, making attendees feel appreciated?
Aims and target groups Before starting to plan the event, it is always very helpful to become clear about the focused target group and the aims of the event. Although this sounds easy and trivial, experience shows that this aspect is too often neglected. •
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Is the event directed to teachers? One important issue in relation to this target group is that all of these teachers have their own perspective of effective teaching. Of course, all of these teachers are doing their best to provide good lessons. In these events they should never get the impression that their style of teaching is wrong or bad (because something ‘right’ or ‘better’ is proposed). Inquiry-based learning is just one way of teaching which can supplement their teaching methods. We are thinking of and aiming for an evolution of lessons rather than revolution. Further you could consider the following questions: o Do you want to motivate a wider audience of teachers to use inquirybased learning? o Or do you want to motivate teachers to participate in professional development courses? Is the event directed to ‘supportive groups’ for teachers, such as head teachers, assessment development or teacher educators? Each of these has its own needs which need to be taken into account. If you e. g. address head teachers, you have to be aware that some of them might not teach the subjects of (in our case) maths and science. Further, you have to be clear
8 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
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about their role: How can they, as head teachers, support teachers of their school to include inquiry based learning. You can find examples for this in the case studies of Germany and Malta. Is the event directed to pre-service teachers? Here, it is important to take into account that this target group has but little experience in teaching.
What contextual factors are important? If you want to convince the target group to implement inquiry-based learning, you have to start off from their needs. Again, this sounds trivial but it is not. Starting-off from somebody’s needs means being informed about these needs. However, as school and contextual factors are closely related to policy they may change rather quickly. Important contextual factors in relation to inquiry-based learning in your country might be: • • •
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Is inquiry-based learning included in the curriculum? In how far? Does the curriculum leave enough time for carrying out inquiry-based learning? Are there enough materials for inquiry-based learning? Are related tasks included in the textbooks? Is inquiry-based learning included in the assessments? If not, this can be a real challenge and there is an urgent need for this issue to be addressed in the activities. Interesting case studies might be the ones from Romania and Slovakia. Are students used to this form of teaching? If not, teachers might get negative reactions from students and they will need to be prepared for this (see e.g. the case study of Romania). Will parents appreciate this way of teaching? Again teachers might have to deal with negative reactions and need to be prepared. Are there any reforms or new policies etc. that have an impact on day-to-day teaching? These changes need to be taken into account (for more detailed information please also look at our International analysis of context).
How can we acquire as much attendees as possible? Teachers, as well as head teachers, teacher educators etc. often suffer from an overload of work. Further, if you plan to run a whole day activity, teachers need the permission to attend the event instead of being at school (for more details on the conditions for teachers taking part in professional development activities, please see our International analysis of context). As a consequence, there is a need to carefully reflect on how to attract teachers. They must be convinced that coming to the particular event offers them support for their day-to-day teaching. • • •
Is there a new curriculum of which teachers need support to deal with? Is there a new form of assessment which teachers are not familiar with? Are schools assessed by external evaluators and do they need help in this respect?
9 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Careful reflections will lead to ideas on how to advertise the upcoming event (see for example the case study of Switzerland). The next question arising is how teachers can be best informed about the meeting. A good possibility is the use of existing networks. • • •
Would it be supportive to contact school authorities? Is there a national centre for professional development? Is there a teacher association which would be willing to distribute the information about the event? The Spanish case study gives an interesting example about this.
Another possibility to reach as many teachers or target groups as possible would be to use events to spread information on inquiry-based learning. The cases of Germany, Slovakia and Cyprus contain examples for this. Another possibility is to ‘attach’ the event to a larger conference which attracts the specific target group (this is illustrated in the second German case). Organisation of the event A good organisation of the event is crucial for its success and should start several months before the event. The size of the organisation team should depend on the size of the event. The most important thing during the whole organisation is to keep the needs of the target group in mind. The whole event needs to be appealing to the target group and relevant for their work. The following list shows some important issues which should be considered. Some of the issues are only relevant for big events. •
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Invitations: o How should the invitations be designed to attract the target group? (See also dissemination materials) o In which ways can they be distributed to the target group (e.g. via teacher associations, school authorities)? o At what time should the event be announced? It may be wise to make a preannouncement (‘Save the date’) several months ahead and then another more closely before the event. Registration: o How should participants register for the event? (By online registration tools, via email, no registration)? o Will there be a conference fee or not? Catering: In order to create a nice atmosphere, it is important to offer refreshments and snacks. o Who will provide the refreshments and snacks? o What should be offered? o Should attendees pay for it or is it included in the conference fee? o How will lunch be organised?
10 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
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Breaks: Informal discussions during breaks are vital for these conferences, so breaks should be long enough. Decoration and framework: o How should the venue be decorated to make participants feel welcome? o Should there be exhibitions during the breaks? E.g. from publishers? Schedule of the day: o When should the day start and end? Here, the balance between a too long day and too little input needs to be considered. o The opening session is crucial: It needs to make attendees curious for the day. o How should the closing ceremony be organized, when participants might already be tired? (See e.g. the experiences of Germany and Switzerland) Folder: o Will participants get a package with materials for the day? o What should it include? o Are name badges needed? Information desk o Is an information desk needed? Where should it be situated? Evaluation: Do you want to evaluate the day? How? Rooms and media o Which rooms are needed for the talks and presentations? o What kind of media? o Do you have the keys for the rooms? Summary and follow-up: How do you plan to follow-up on an activity once it has been carried out? What are the results, outcomes and lessons learned from the activity and how can these be further used? What should be communicated to whom?
Dissemination materials Purposeful dissemination materials can very effectively support dissemination activities. For organizing specific event materials, such as posters and flyers (at the right ‘points of display and distribution’), play a crucial role in announcing the event to the target groups. They are crucial in raising initial awareness and attention as well as in transporting the main messages to relevant target groups. Other purposes are to present the project/initiative in a graphically coherent way to ease recognition and association (use of logos, emblems), to create a ‘corporate identity’. Unique and original graphic elements used consistently in every document, presentation, poster, advertisement material and on project webpage can considerably support recognition and remembering of the project or initiative. It should be pointed out again that materials need to be tailored to the target groups (for example teachers might be more interested in different aspects in a brochure or
11 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
an information email than parents) and the use of materials must be appropriate for the purpose at hand. Thus, different materials in different variants (but with the same ‘corporate’ appearance) may be needed. In Annex I, some target group specific materials are presented. You may want to consider the use of the following dissemination materials • • • • • • •
Posters in different sizes One-page flyers Project brochures and leaflets Small promotional materials like: Bookmarks, pocket calendars, rulers, pens with project logo and website/contact information Roll-up or banners Book Website
Overview of dissemination activities of PRIMAS For readers who run professional development projects or aim to spread innovative pedagogical practices, the following report on some facts on the dissemination activities carried out in the PRIMAS project so far, might be interesting. It gives an overview of the extend, type of efforts and activities that can be involved in such an international project. PRIMAS has organized (in 12 participating countries) 91 meetings, events and actions for teachers within the period from January 2010 – June 2011. Most of these events served the purpose to establish contacts and provide first details about implementing IBL in maths and science to different target groups and how the PRIMAS project supports this. Creating interest for the project, establishing networks, raising the interest of teachers (potentially to take part in our long-term PD courses) as well as head teachers, PD providers and relevant networks and associations were the pursued aims and goals of the PRIMAS dissemination activities. A quantitative analysis of the 91 reported events shows following facts: 52 meetings took place in the first year 2010, 39 meetings were organized or attended in the first six months of the year 2011. There was a steady activity over the first 18 months of the project with a peak at the end of the first year (November) and in May 2011. Time frames can be very important, semester schedules and work load peaks have to be regarded when working with teachers and in the education sector in general. According to the documentation, more than 8000 contacts were made within the first 18 months (this excludes website visits and newsletter readership). The intensity of these contacts varies, of course. There were activities ranging from shorter project
12 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
presentations (mainly informative, e.g. at organised forums or conferences) to intensive involvement in pilot testing and training (PRIMAS materials and courses). Teachers are the main target group recorded in the documentation. This is not surprising as teachers are also the key target group in the PRIMAS project. There were exactly 34 events organized particularly for teachers of mathematics and science. 10 of these events involved teachers and their students from upper primary and secondary schools. The scope of the activities is again broad ranging from more information-based activities aiming to raise attention and interest and produce an initial supportive climate (for later project measures or offers) to working with PRIMAS materials and the IBL approach first hand. Other important target groups that have been recorded repeatedly are student teachers, trainees and pre-service teachers (4 events/activities) as well as head teachers who were specifically targeted in three events. There are several other target groups that were addressed by the PRIMAS dissemination activities. These include the groups of academics, researchers and PhD students, teacher educators, advisors and professional mentors, educational materials designers, teacher networks, communities, associations, parents and other key stakeholders. Four activities used internet-based or paper-based publications to address teachers specifically. These dissemination activities were carried out in addition to the PRIMAS website and the PRIMAS newsletter. Therefore, they were more focused and aimed a specific goal (such as the ‘recruitment’ of PRIMAS multipliers). Thirteen of all reported activities were rather large events, such as Science fairs or large international or national conferences for different groups and of different thematic focus (e.g. research vs. professional focus). Usually, the advantage of these events is that messages can be delivered to a vast amount of people from relevant circles. A disadvantage usually is that ‘contacts’ may be more superficial than when using other formats. However, each format has its purposeful setting and a broad and less intensive contact may pave the way for the participation in more intense formats. Contacts have not only been made within the PRIMAS countries but also outside. The representation of PRIMAS at a large European teacher conference (Scientix conference) provided one of these opportunities to reach target groups even outside the PRIMAS countries. PRIMAS also held a launch event in the beginning of the year 2010 (soon after the project started) including a press conference to ‘start off’ with the project. This helped to create initial impetus and mobilised interest for initiatives that followed later. Most of the activities held in the first 18 project months were one-day or half-day meetings/events (65 events) but we also recorded several activities that were ongoing for two or more days. A range of supporting actions and events can be suggested based on this experience:
13 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
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Involvement in larger conferences/events (mostly organised by others) where the own project can be purposefully presented Running smaller events/meetings (self-organized) for a more selected group Invitation to or offering talks to any organization/network/association providing a regular structure Hands-on workshops for teachers and other target groups (intensive involvement) Large-scale and/or broad dissemination ‘campaign’ using effective media strategies (website, newsletter, infomailing, teachers´ webportals) Dissemination actions providing materials (brochures, small posters, sample teaching materials, small promotional materials) at useful points of display and places of distribution Actions for ‘equipping’ interested persons or multipliers with materials and information to pass on to colleagues, etc.
14 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
The case studies Purpose of the PRIMAS case studies After the previous more general introduction about carrying out dissemination activities for teachers this section will illustrate our guidelines with the help of 14 case studies of dissemination activities that have been carried out by partners in the PRIMAS consortium. These examples (mainly directed towards teachers) illustrate in a very concrete way how such events can be organised in a given context. They can be used to develop concrete strategies and ideas how to inform and involve teachers or other target groups. Activities described in these case studies may also reflect existing practices in different countries in relation to the teaching and learning of mathematics and science and teacher education practices. The main target group of the activities covered in the case studies are teachers but several others, such as head teachers or student and pre-service teachers are covered as well. The cases cover a broad range of strategies ranging from electronic media-activities to thematic teacher days on PRIMAS/IBL to introducing teachers to IBL through hands-on use of materials in piloting actions. Most of the cases (11) cover one-day-actions but there are also longer-term activities illustrated.
Introduction to the case studies Two case studies which offer a detailed description of one-day meetings that were especially organised for teachers. In Switzerland, project partners organized a day for mathematics and science teachers named ‘Crossed overview on scientific approach in mathematics, physics, chemistry and biology’ and in Germany, a ‘Day for Teachers’ focused on IBL was organized attached to the German conference for mathematics In two case studies the PRIMAS partners describe their experience in organizing activities for head teachers. One is the case study from the University of Malta that organized that day ‘Introducing PRIMAS to Heads of Schools’. The other is from the University of Education Freiburg that organized a ‘Day for head teachers’ that featured IBL and PRIMAS. The common idea was to present lectures and workshops focused on IBL and introduce the PRIMAS project in some detail to head teachers who have an important role in supporting and encouraging teachers to participate in PD courses and applying innovative teaching and learning methods. Both case studies also describe the motivation of the participants to join the activity and the project. Two other case studies focus on conferences where teachers of primary and secondary schools participated. There is the case study from Cyprus on the ‘Cyprus Mathematical Society Annual Meeting and Conference’ that provided space for a PRIMAS project presentation and a half-day workshop on the role of mathematical modelling in IBL. These initiatives for example led to the successful ‘recruitment’ of PRIMAS multipliers for PD courses and to contacts with the Ministry of Education. The second case study is from the Netherlands where the Dutch team used a large
15 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
conference (Ecent conference 2011) to also make a project presentation and a workshop about PRIMAS and its measures. The Romanian case study provides an example of a deep-level dissemination activity that offers extensive insight for the participating teachers into the PRIMAS activities and IBL by providing them with the opportunity to participate in piloting actions for PRIMAS materials and courses. There was a series of piloting events and activities focused on classroom processes, to the role of the teacher and the analysis of difficulties in applying IBL. This case study also illustrates the different successes. Following the piloting actions and getting acquainted with the approach, several teachers could be recruited as multipliers who will receive PD courses to train other teachers in using IBL, the Romanian PRIMAS team subsequently was invited to another summer academy where they will be able to introduce more teachers to IBL. The case study illustrates as well the powerful insights that can be gained from such piloting actions to improve future activities. One day workshops with teachers are described in the case studies from Norway and Slovakia. The Norwegian team organised a Teacher Day in Trondheim which was used as a first meeting to introduce teachers to IBL and offering them to become a PRIMAS multiplier leading PD courses (or PRIMAS teacher) in the future. The Slovakian day presents the ‘Methods day for teachers’ to which the Slovakian PRIMAS team was invited by the organisers. During both activities reported in the Norwegian and Slovak case study, the PRIMAS project’s aims and ideas were extensively presented. Teachers were introduced to IBL principles and examples were discussed and worked on. Some small ‘experiments’ during the workshops showed teachers how beliefs can impact on problem-solving engagement. Such insightful activities raised the interest of the teachers in IBL. Example tasks can be found in both case studies. The UK case study ‘Curriculum development: focusing on mathematical activity’ was focused on trainee teachers. A special seminar was scheduled at the end of their one-year course. The thematic focus was the nature of mathematical activity that pupils can engage in with an emphasis on inquiry pedagogies facilitated by the use of modelling. The other UK case study (‘Professional Mentors’ Update Meeting’) is a training event for professional mentors. The meeting provided the senior teachers with an introduction to PRIMAS, especially the PRIMAS materials and opportunities to reflect on the importance of IBL within subjects. Several schools/ senior teachers formed ties with the project on the basis of these activities and they will be included in the following PD courses. The case study from Hungary describes a ‘Talent Day’ that was held in Budapest that offered the possibility to reach special target groups. The PRIMAS team introduces the project and focused its lecture on working with real-life problems. The Spanish case study describes how internet and (electronic) media can be very successfully used to carry out dissemination activities. Within an extended period, several consecutive measures have been carried out targeting teachers: These 16 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
activities include information about the project PRIMAS for mathematics teachers through SAEM Thales webpage, circulation of a bulletin and coverage in a mathematics education journal Epsilon. The case study shows that electronic media is a powerful means of dissemination. In the next section, all 14 case studies will be presented; they are ordered chronologically.
17 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Spain: Information about PRIMAS Context SAEM Thales is the professional association of mathematics teachers in Andalusia. As an association, they have different means to communicate with teachers: a website, an internal bulletin and a journal. These three channels where used by PRIMAS to disseminate information about IBL, PRIMAS itself, and our Continuing Professional Development programme. SAEM Thales, as the professional association of mathematics teachers in Andalusia, promotes a wide variety of activities related to mathematics education, including professional development for mathematics teachers. The association has three principal means to communicate with mathematics teachers: • • •
Its own webpage (http://thales.cica.es/), which is regularly visited by many teachers. An internal bulletin with information about the activities carried out by the association and distributed to all members via email. A mathematics education journal, Epsilon, which is distributed to many teachers in Andalusia and to other institutions.
Recently, the association has also opened Facebook and Twitter accounts. Activity Date: October 2010 / February 2011 / June 2011 Target Group(s): Primary and secondary school teachers of mathematics Participants: 1400 +300 potentially
In our attempt to draw Andalusian teachers’ attention to PRIMAS, collaboration with SAEM Thales has been crucial, due to the regular communication they have with many teachers and the strong connections the association has with the regional ministry of education and many teachers’ centres. This collaboration was encouraged from the very beginning as we invited the association to participate in our National Consultancy Panel. They accepted our invitation immediately and, since then, a strong collaboration with Mr. Agustín Carrillo de Albornoz, SAEM Thales Secretary, has been established. During the first months of the project, and prior to starting the Continuing Professional Development programme, our main interest was to get in touch with as many teachers as possible. The aim was to inform them about PRIMAS, inquiry-based learning, and the PRIMAS Continuing Professional Development programme, in order to draw their attention to further activities. The main problem is that Andalusia is a large region, with a wide geographical dispersion of teachers. Therefore, we used the information channels of SAEM Thales to send PRIMASrelated information to as many teachers as possible: 18 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
SAEM Thales website: information about the project was published on the SAEM Thales website in October. It included a brief description of the project and a registration form for teachers interested in participating in our CPD programme.
SAEM Thales info bulletin: during February, an info bulletin was distributed to all SAEM Thales members via email. It again included a brief description of PRIMAS and a link to the website.
19 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Epsilon mathematics education journal: in June, together with issue 75 of the Epsilon journal, a PRIMAS flyer was distributed. Analysis It is difficult to measure the real impact of this dissemination activity, although there are some hints that allow us to regard it as a successful strategy: •
•
Visits to the PRIMAS-related information on the homepage: the website gets thousands of visits every month, many of them from regions other than Andalusia and also from Latin American countries. In the first days after posting, about 200 visits to the PRIMAS-related information were counted. Besides this, we got about 30 replies from teachers interested in PRIMAS and about five for possible multipliers. SAEM Thales bulletin: the bulletin was sent to the 1400 members of Thales. The fact that the information in the bulletin is linked to that published on the Thales site might encourage visits to the homepage. However, we do not have data on whether there was an increase in the number of visits. We considered
20 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
•
this method of dissemination as a secondary activity (mainly reinforcing the first one). Epsilon journal: this can be considered as a very successful means of dissemination. Considering the fact that many teachers and schools get the journal, we can now affirm that, with a little effort, thousands of teachers know about PRIMAS and IBL and can get in touch with the national group. Indeed, Thales has 1400 members whom receive the journal, plus 300 extra subscribers. This is highly important in order to make “concept PRIMAS” sound and to attract teachers to the CPD programme.
Conclusions and Recommendations Looking beyond the local sphere, new strategies have to be considered in order to reduce the gap between teachers and the project. Professional associations like SAEM Thales offer an effective and cost-effective way of getting in contact with many teachers. Although this is only an initial contact, a kind of preliminary step, it is important to make the project sound and to draw teachers’ interest towards further activities. This strategy will be replicated for science teachers through their own association, APICE.
21 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Switzerland: Crossed overview on scientific mathematics, physics, chemistry and biology
approach
in
Context In Geneva, a new “plan d’études”, valid for all French speaking Switzerland (PER: Plan d’études Romand), will be introduced in September 2011, for all compulsory education (ages 4 to 15) With regards to mathematics and sciences this new PER is divided into five major branches, one being mathematics and the natural sciences. This is a novelty since mathematics and the sciences have traditionally been taught as different school subjects. On the other hand, this novelty is not completely radical since even if they appear in the same branch, mathematics and sciences are still taught separately. However, placement in one branch aims at making the two subjects closer. In particular, in the text of the plan d’études, a common introduction has been written that presents what a scientific approach should entail, referring both to mathematics and the sciences. This new situation required some discussion between science teachers. This was also a good opportunity to introduce the PRIMAS project to a large audience, not only superficially but in relation to a major real-life issue in Geneva. In coordination with the Presidents of Group1 (PG) in mathematics, we set up a oneday training course that any teacher could voluntarily attend. It had been announced in a formal offer at the end of school year 2009/2010. It took place on 18 November, 2010, at a school in Geneva. Around 50 teachers, mostly of lower secondary education, but also of upper secondary, in mathematics physics/chemistry and biology2 signed up for it. Event Date: 28 November 2010 Target Group(s): Secondary school teachers Participants: 50
After several discussions between the PG and PRIMAS team, we set up the following schema for the training course (see annexe for a precise programme of the day). We started with a joint presentation between the PG and PRIMAS team, in order to summarise what the aims of the PER are, and how PRIMAS and, more generally, promoting IBL in the teaching of mathematics and sciences could help better implementation of the new ideas in the PER.
1
At the level of lower secondary education (Cycle d’Orientation, CO) in Geneva, there are elected representative teachers of each school subject: two in mathematics, two in physics/chemistry and one in biology. 2 It would be difficult to give an overview of the division between disciplines since, due to a lack of mathematics teachers, many physics and biology teachers actually also teach mathematics.
22 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Then, three specialists presented, over half-anhour, what a scientific approach could represent in their field. A condition we set them was not to limit their talk to teaching but actually to the larger issue of the epistemology of each scientific subject. More than an hour of discussion was planned after the three presentations. In the afternoon the participants were divided into small groups (of four or five) containing at least one teacher of each subject and one PG or PRIMAS team person. They were given four questions and asked to choose two in each group and tackle them from the standpoint of their own background and culture. Here are four questions - each group should choose two to work on: You must tackle the questions from your own field of expertise, which may be complementary. The goal is not to arrive at a definitive answer, but to employ your diverse opinions and scientific backgrounds to tackle these questions and to come to an agreement within the group. In each group, for each chosen question, you will have to give an account of the debate in the group (use transparencies or a powerpoint presentation for five minutes). 1) What is the best way to heat up one litre of water: resistance, a micro-wave oven, electrical hotplates, a kettle, etc.? 2) How can one determine the average yearly temperature of the Earth? 3) It is commonly said that forests, especially the Amazon, are the lungs of the Earth. What does this mean? Is this arguable? 4) Two identical glasses are filled with the same quantity of wine and water respectively. With a spoon, one takes some wine from the first glass and pours it into the glass of water and mixes it with the spoon. Then, with the same spoon, one takes exactly the same quantity as before from the glass containing the mixture of wine and water and pours it into the glass of wine,
23 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
then mixes it. Which has the most? The wine in the glass of water or the water in the glass of wine? The explicit idea (see how we formulate the question in the annexe) was not that they reach a definitive answer but rather that they put forward their similarities and differences in opinion in a way that made it possible to tackle the questions despite their different scientific backgrounds. At the end of the day each group presented, on one transparency sheet, the results of their discussions. The presentations were made grouped by questions. The main ideas that guided our organisational decisions were: ‐ We wanted teachers to be fed with some information about the scientific approach in each subject by specialists, with some opportunity to respond (morning). ‐ It was a great opportunity to have teachers of different disciplines gathered for a training course; we had to give them the opportunity to work together. We wanted to disseminate PRIMSAS ideas without losing focus of the aim of the day. It was in this spirit that we set up the afternoon programme. At the end of the day we presented PRIMAS in more detail, gave a flyer to each participant and advertised events to come. Analysis The morning was quite interesting. The three presentations contrasted quite significantly, but were universally appreciated. The timing was just right and the discussion was rich, with many participants asking questions and discussing issues. Rapidly some distinction between scientific approaches appeared, mostly between mathematics and the other sciences, but also between biology and physics, especially concerning the role of experiments. We used the entry of a glossary given in the PER defining words like hypotheses, experiment, etc., in order to structure the debate. The group-work in the afternoon was usually very interesting even if, as is usual, some groups worked together better than others. Here again, the mathematics way of thinking appeared very specific compared to other sciences. However, mathematics teachers were quite puzzled by questions like the one about forests being the lungs of the Earth and realised that
24 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
this could actually lead to some scientific debate in which they could be involved. The presentations at the end of the day were a bit too long. This could have been predicted and is actually always a pitfall in this kind of event. We have to think of a new way of closing a session, when a long time has already been spent in group work! Conclusions and recommendations Overall, we have several reasons to be quite satisfied with what happened during this day. First of all, we had quite a few teachers (even if 50 does not seem a lot it is when one takes into account that Geneva is quite small and that attendance was voluntary). Secondly, the format of the day, and the fact that we had taken a long time to prepare, produced consistent and productive results. We only have to revise the way we organised the end of the day. Teachers were universally happy, and the fact that they got to meet teachers from other subjects and exchange ideas with them seemed like the main source of satisfaction. Finally, the consistency of PRIMAS objectives with the theme of the day was genuine and it was a great opportunity to present our project in a situation that made sense in a local curriculum context.
25 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Programme of the training course Crossed overview on the scientific approach in mathematics, physics, chemistry and biology (CO-00121) – 18th November 2010 8h30 -9h00 Presentation of the day. Context PER and PRIMAS (Bernd Hatlanek, Christian Colongo and Jean-Luc Dorier) 9h00 – 9h25 Scientific approach in biology Christian Orange 9h25 – 9h50 Scientific approach in physics Guy Robardet 9h50 – 10h15 Coffee break 10h15 – 10h40 Scientific approach in mathematics Pierre-Alain Cherix 10h40 – 12h00 Discussion 12h00 – 13h30 Lunch 13h30 – 15h00 Group work
15h00 – 16h00 Presentation of the groups’ work 16h00 – 16h30 Conclusion
26 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Germany: Day for head teachers (Schulleitertag) Context The following case study describes the Day for head teachers (12/10/2010) at the University of Education in Freiburg, Germany. Head teachers have been identified as a key factor concerning the implementation of change in classrooms (such as changing from transmission teaching to inquiry-based learning). They can either hinder or support teachers’ intentions to integrate inquirybased learning into their lessons. For example, they can support teachers by providing some spare time in the schedule for cooperation with colleagues, by giving them appropriate time in maths or science classes (such as longer lessons - 90 min instead of 45 min per lesson) or by encouraging teachers to participate in Continuous Professional Development. For this reason, it seemed highly relevant to introduce head teachers to inquiry-based learning and its principles at the beginning of the project. In order to reach as many head teachers as possible, the PRIMAS project took advantage of the annual Day of head teachers, which is organised regularly by the University of Education and the regional school authority. Normally, this event tackles more general topics such as assessment, full-time school or heterogeneity of students. In 2010 we were successful in convincing the organisers of the relevance of inquiry-based learning in mathematics and science; thus, the whole day was devoted to this topic. Event Date: 03 December 2010 Target Group(s): Head teachers Participants: 100
The programme was carefully designed to give head teachers an appealing and informative introduction which would motivate them to encourage teachers at their school to implement inquiry-based learning in their mathematics and science lessons. At the beginning, two keynotes where given to introduce head teachers to inquirybased learning – one from the perspective of mathematics (including an introduction to the project itself), and one from the perspective of science. The first one was given by Katja Maaß, the coordinator, the second one by a partner from the Netherlands, Elwin Savelsberg. After this, acknowledging the particular role of school principals, a discussion was held about how they can promote the integration of inquirybased learning at school. This proved to be a highly motivating and important part of the day as 27 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
head teachers eagerly took part in the discussion and developed a lot of ideas on how they could support their teachers. After the lunch break, three parallel workshops were offered – all highlighting the idea of connecting subjects or different age groups in order to point out the relevance of inquiry-based learning across all boundaries. Workshop 1: Task and lesson culture of inquiry based learning – overcoming subject borders Workshop 2: Inquiry based-learning and problem solving as a continuous educational principle Workshop 3: Researching and learning in authentic problem situations – connecting physics and chemistry At the end of the day the closing ceremony gave room for reflection including particular needs heads saw in relation to the professional development of teachers. The whole day was arranged in a way that the heads felt very welcome. We took care of refreshments for coffee breaks, a nice lunch, careful display and a good atmosphere as a whole. Refreshments offered at the end of the day offered more opportunities for discussions. Analysis of the day Sixty percent of the participating head teachers were principals of lower secondary level schools, 30 percent were from higher secondary schools and 10 percent were principals of elementary schools. Most of them (>80%) had teaching experience of more than 15 years; 78 percent of the participants taught maths and science. The whole day was highly appreciated by the principals. They appreciated the good balance between workshops and lectures as well as the discussions. In relation to the workshops, head teachers acknowledged, in particular, that they were interesting, well-prepared, well-structured and that the workshop leaders were very encouraging. They also approved of the materials and the pedagogies used. All participants acknowledged that inquiry-based learning and the PRIMAS project offer opportunities to improve the teaching culture in Germany. They were all willing and encouraged to recommend inquiry-based learning in their schools to get multipliers and teachers joining the PRIMAS project.
28 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
There was a common understanding among head teachers that inquiry-based learning will raise students’ problem solving competencies, which is highly relevant for their future. They did not see any problems concerning the students’ parents (that they might prefer more traditional ways of teaching, for example) nor in relation to students’ understanding of the content of the subjects. The only concerns raised were time restriction and the curriculum (especially external assessment). Participants also pinpointed that they were motivated to join the PRIMAS project. Due to this event, the attention of school heads has been attracted, which is a highly relevant step towards a widespread implementation of inquiry-based learning. Motivated teachers will be convinced by their head teachers. In addition to this, principals have the opportunity to arrange framework conditions such as timetables, classroom equipment and financial support to facilitate the efforts of their colleagues. The whole event has supported us enormously in attracting schools towards becoming PRIMAS schools.
29 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
The programme The beginning The Day for head teachers started at nine o’clock in the morning. After a wake-up coffee the Deputy Chair of the Department for School Development of the school authority welcomed the guests and presented the day’s schedule. She also announced the following key speakers: Keynote 1: Katja Maaß – University of Education Freiburg, Germany “The PRIMAS project” Her talk tackled the following aspects: 1. 2. 3. 4. 5. 6.
Why do we need maths and science? Why inquiry-based learning? What is inquiry-based learning? Inquiry-based learning in day-to-day teaching Why is IBL the main topic at the Day for head teachers? About PRIMAS
Keynote 2: Elwin Savelsbergh – Freudenthal Institute Utrecht, Netherlands “Problem solving and inquiry-based learning in mathematics and science.” His talk tackled the following aspects: 1. What is inquiry-based learning from the perspective of science? 2. Examples for inquiry-based learning 3. The approach of Martin Wagenschein and examples for related lessons 4. Students’ learning inquiry-based in research communities 5. Inquiry-based learning and mathematical modelling Discussion: How can school administrators support PRIMAS? This discussion offered school heads the opportunity to reflect upon possibilities to support teachers who want to implement inquiry-based learning. Upcoming questions were discussed and answered by the PRIMAS team. It also offered the PRIMAS team insight into the obstacles heads of schools have to deal with. Many heads fear that including inquiry-based learning requires a complete change of teaching. The idea of introducing aspects of inquiry-based learning step-by-step seemed to be a solution for them. It was also necessary to point out that inquirybased learning is only one piece of the teaching “jigsaw”.
30 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Workshops After lunch, three parallel workshops on inquiry-based learning took place: Workshop 1: Task and lesson culture of inquiry-based learning – overcoming subject borders Mr. Holzäpfel´s and Ms. Bröll´s workshop worked on interdisciplinary tasks where aspects of maths, physics, chemistry and biology were analysed. Accordingly, the workshop team used topics like caries, behavior of sticklebacks and traffic flow. They presented and tested several teaching techniques e.g. concept mapping, think-pairshare and the placemat method. Workshop 2: Inquiry-based learning and problem solving as a continuous educational principle from grade 1 to grade 13. In this workshop led by Andreas Eichler and Stefanie Schuler, teachers were introduced to inquiry-based learning as a continuous principle from the beginning of school till the very end. Examples for inquiry-based learning covered all age groups of students from 6 to 18. General principles of inquiry-based learning were highlighted as well as the necessity to start with inquiry-based learning very early as to allow students to develop the needed competencies in dealing with open and complex situations. Workshop 3: Researching and learning in authentic problem situations (Physics/Chemistry) The workshop led by Ms. Von Gehlen, Ms. Mikelskis-Seifert and Mr. Claus used tasks of the project “Girls for Science”. The aim of the workshop was to get teachers to experience inquiry-based learning in science, in authentic situations. Participants could chose between several intriguing experiments and hands-on activities. Closing ceremony At the end of the day head teachers were given the opportunity to discuss important aspects of professional development initiatives concerning inquiry-based learning – from their own perspective. Further on, the event was evaluated by heads. After the official programme, everybody was invited to continue the discussions in small groups in a casual atmosphere. Evaluation
31 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Cyprus: Mathematical Society Annual Meeting and Conference Cyprus Context The Cyprus Mathematical Society (www.cms.org.cy) is the largest association and society in mathematics and mathematics education in Cyprus. Among its activities are the organisation of competitions in mathematics in Cyprus and internationally, the organisation of an annual meeting and a conference in mathematics and mathematics education, and the organisation of professional development courses for mathematics teachers. The annual meeting and conference is the major event of the CMS in which not only mathematics teachers but also policy makers, inspectors, and head teachers participate. The conference and workshops focus not only on professional development issues but also on theory and current research in mathematics and mathematics education. As a result, most of the participants have an interest in mathematics research and the majority of them also hold postgraduate degrees in mathematics and mathematics education. Conference presentations are published in conference proceedings and a proceedings book is also distributed to all schools in Cyprus. The annual meeting and conference has been identified as an event that can contribute to PRIMAS objectives and forthcoming activities, as many of the participants are committed to and enthusiastic in being involved with research projects; therefore, on behalf of the PRIMAS team there was an intention to involve teachers in PRIMAS’s subsequent activities and possibly involve some of them as multipliers. Activity Date: 05/06 February 2011 Target Group(s): Primary and secondary school teachers of mathematics Participants: 32
The annual meeting and conference took place on the 5th and 6th of February, 2011, and it was co-organised by the Cyprus Mathematical Society and the Mathematics Teachers Association. Conference started with a plen ary talk entitled “Mathematical Modelling and Applications”. The next plenary talk focused on the new curriculum in mathematics in Cyprus. The two plenary talks were followed by a number of parallel sessions and a roundtable discussion on the innovations integrated in the new mathematics curriculum. Dr Mousoulides participated in the roundtable, addressing issues related to mathematical modelling and IBL, and how interdisciplinary teaching was addressed in the new curriculum.
32 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Dr Mousoulides’s presentation, entitled “The Role of Mathematical Modeling in Inquiry-based Learning in Mathematics” attempted to address issues like “the role of modelling and problem-based learning in mathematics curricula,” “using modelling as a means to promote student communication skills,” and “interdisciplinary IBL materials for mathematical learning”. During the presentation PRIMAS objectives were presented and PRIMAS was addressed as one means to target, among other research questions, the ones raised above. In the discussion that followed a number of questions posed by teachers further underlined the importance of providing students with opportunities to enhance their problem-solving skills. More experienced teachers tackled the issue of time and a debate that followed focused on finding ways to overcome time problems. During the second day of the annual meeting and conference a three-hour workshop took place. The workshop led teachers to discover the importance of 21st-century learning and the shift in paradigms that mathematics education is currently going through. Through PRIMAS materials, it provided teachers with hands-on experiences of how to plan inquiry-based classes that meet the requirements of the new curriculum and deliver on content requirements, develop student skills, and begin the path for differentiation. The workshop worked well for teachers at the beginning and middle of integrating inquiry-based learning into their curriculum. The workshop was organised into the following strands: • • •
What is IBL and how can IBL be integrated in mathematics teaching and learning. Modelling and IBL. PRIMAS and IBL in Mathematics and Science.
The workshop targeted the participants’ mathematics knowledge, as well as enhanced their teaching strategies through inquiry-based, cooperative learning, integration of modelling skills into mathematics, and the use of technology. As previous research showed, inquiry-based learning is a very effective method of enhancing student learning. Various PRIMAS activities aligned with the new mathematics curriculum were presented to teachers to show how inquiry-based learning, complemented with modelling and technology, can work together to increase students’ performance and improve attitudes towards mathematics.
33 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
During the last part of the workshop participants working in pairs or groups of three worked with PRIMAS materials, reflecting in possible ways to integrate existing or modified materials into their teaching. A whole group discussion followed, in which they discussed their findings and any differences of opinions they might have had. Analysis Participants were very engaged in both the presentation and the workshop that followed. A number of teachers participated in the discussion after the presentation and raised important issues related to mathematical modelling, IBL, and interdisciplinary teaching. Furthermore, four of them asked to participate as multipliers in the PRIMAS project, while an inspector from the Ministry of Education also expressed her willingness to be involved in the PRIMAS project. Teachers left the workshop with practical experience of how to plan and structure their classes to create engaging and visible learning, using IBL in mathematics. Although short in time (three hours), the workshop can be considered successful. Through activities with PRIMAS materials and reflective discussions, the workshop addressed some core elements from modelling and IBL in mathematics and provided to teachers with little experience an in-depth overview of the role IBL can play as an integral part of mathematics teaching. Discussions with teachers that followed further underlined their positive attitude towards the workshop and PRIMAS in general and five more teachers applied to participate in the PRIMAS project as multipliers. Some of them also volunteered to work as facilitators during the next meeting that will take place as part of teacher professional development in the new curricula. In this meeting, more than 150 teachers are expected to participate, and Dr Mousoulides will link new curricula to IBL and mathematical modelling and present related PRIMAS 34 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
materials. Conclusions and Recommendations The presentation and workshop can be considered successful in informing and engaging teachers on IBL and PRIMAS. However, the rather small number of participants underlines the necessity to run the workshop for more teachers. The workshop, with a special orientation towards science teachers or an orientation to mathematics and science teachers, will also be replicated in collaboration with the physics teachers’ and the mathematics teachers’ associations. The results from the presentation and the workshop that followed also stressed the significance of informing target groups other than teachers. Policy makers and inspectors at the Ministry of Education, as well as teacher trainers providing professional development courses, have also been identified as important groups that could facilitate the successful integration of IBL and PRIMAS materials in mathematics and science classes in Cyprus. To this end, meetings with policy makers at the Ministry of Education and a workshop for teacher trainers have been planned and will take place in the next months.
35 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
United Kingdom: Professional Mentors’ Update Meeting Context: Initial teacher training in England is a shared responsibility between universities and schools. Among several patterns of training, the most common for secondary schools is the Postgraduate Certificate in Education, a year-long course leading to the award of Qualified Teacher Status. During the year, trainee teachers spend two-thirds of their time in schools, and one-third in the university. Tutors from the university place trainees in appropriate schools and visit them in schools to monitor their progress. Subject teachers acting as mentors are trained in the university, and in each school a “Professional Mentor” is responsible for organising the work of the subject mentors and overseeing the progress of trainees in different subjects. The professional mentor is always a senior teacher in the school, often an assistant or deputy head teacher, and often with responsibility for the CPD of teachers newly appointed to the school, as well as trainees. Activity Date: 02 November 2011 Target Group(s): Professional mentors – senior teachers with responsibility for teacher training in their respective schools Participants: 30
The network meeting is a twice-yearly event at the university - a training and updating event for these senior teachers. Typically there are presentations from several professional mentors about innovative aspects of their own practice, followed by discussion. At this particular event, several schools introduced work that they had been doing in ICT, and the PRIMAS project was introduced as a university-led activity with school involvement. Andy introduced inquiry-based learning and the PRIMAS project to the group, making links to previous related projects in maths and science in which local schools had participated, and to opportunities in the 11-14 curriculum for a greater focus on process in maths and science. Some of the results of previous projects were highlighted, showing the adaptations of resources by participating teachers and trainees working together on rich-context problems. Participants were invited to engage in a short problem-solving activity in science, and literature about the project was distributed, including bookmarks. The website attracted considerable interest, with professional mentors commenting on the need for such resources in subjects other than maths, and science as well. After the event professional mentors took details of the project back to their schools. Subsequently, three schools asked to be included in the next stage of the project focusing on interdisciplinary enquiry in maths and science, which took place in June 2011.
36 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
The backing of senior staff is extremely important if teachers are to be able to implement such activity in their classrooms. Without such support and approval they can neither get the time to prepare together nor do they have support if parents, or others, question the value of the activities as distinct from ‘normal’ lessons. This dissemination activity helped to create the conditions in schools in which teachers feel supported and encouraged to take some risks and change their practice. We are now considering how to provide literature which schools can distribute to parents at open evenings etc. to generate further support for inquiry-based learning (link to WP6).
37 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Norway: Introduction of PRIMAS – Teachers’ Problem-solving Beliefs Context Teachers often have particular beliefs concerning problem solving and these tend to negatively impact on their choice of problems for their students in class. Furthermore, their willingness to engage with problem solving is shaped by their beliefs, “the set of understandings about mathematics that establish the psychological context within which individuals do mathematics” (Schoenfeld, 1985, p.5). According to Schoenfeld (1985), one’s “beliefs about mathematics can determine how one chooses to approach a problem […] and how long and how hard one will work on it.” (p.45) This also applies to students, and students often believe that once they understand the content, they can solve the problems associated with it in ‘five minutes’. Students with this belief may give up on problems that do not fit their expectations, which in turn makes it difficult for teachers to teach problem solving. Based on these assumptions, we wanted to design a small experiment, to show teachers how their beliefs may interact with their problem-solving engagement. At a later stage we also wanted to help them discuss and defend IBL with, and in front of, their peers. The group of teachers consisted of 25 mathematics and science teachers, most of them teaching at lower secondary level. They were invited for a day to the university college in order to work with teacher trainers and find out whether they wanted to participate in the PRIMAS programme (as multipliers). Activity Date: 14 February 2011 Target Group(s): Potential multipliers Participants: 25
The case consists of working on beliefs concerning mathematics and science teaching- the ‘Blonde Hair Problem’ (amended from Philippou & Chiristou, 1995, p.132). For this we selected a small experiment ‘Blonde Hair Problem’ to show teachers how beliefs can impact on problem-solving engagement. In particular we aimed at four learning goals: • Recognise that some problems they perceive to be ‘unsolvable’ can actually be solvable and within their ‘capabilities’. • Realise that problem solving may require perseverance. • See that the formulation of a mathematical problem can include more than just clearly identifiable mathematical referents (numbers or formulas); and
38 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
• Appreciate that problem solving can be a satisfying and enjoyable activity. We base our rationale for choosing this approach on the literatur e, in particular Nespor (1987) who claims that “belief systems are composed mainly of episodically stored material” and that “episodic memory” is “organised in terms of personal experiences, episodes or events” (p.320). Thus, we aimed to engineer a ‘dramatic’ and positive episodic memory event, which would be engineered by means of purposeful implementation of a carefully-designed problem (Stylianides & Stylianides, 2011). The problem we used for our small intervention was the following:
The Blonde Hair Problem (BHP) fulfilled our theoretical framework criteria (and learning goals). We conducted the implementation in the following way:
39 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
•
The instructor presents the BHP without commenting on whether the problem is solvable or not. Teachers are asked to think about it individually, and write down (a) their initial reactions to the BHP (individually); (b) whether this problem differs in any ways from the problems they know, and how- the aim here is to help students to become aware of their current beliefs around problem solving.
•
The instructor collects the teachers’ responses to (a) and (b), and at this point reassures them that the problem is solvable. Teachers are asked to think carefully about the information that each hint may give them. Teachers are also encouraged to continue their work (they can work together now).
•
Small group work is encouraged until all groups come up with a solution.- this is important for goals 1,2 and 3.
•
Whole group discussion on the solution of the BHP.
•
Students are then asked to write down their experiences of working on and solving the BHP.
Solution of the BHP The first hint (line 5), together with the fact that the ages are natural numbers (line 15), reveals that there are eight possibilities for the children’s ages. With regard to the second hint (lines 8-9), although we are not told the number of Pythagoras’s address, we can infer that Pythagoras knows this number since he is trying to see how this hint could be useful to him (line 10). Eventually, Pythagoras concludes that he is still unable to figure out the children’s ages (line 10-11). This suggests that there are at least 2 possibilities in which the sum of the ages is equal to the number of Pythagoras’ address; the only two possibilities in which this happens are 1,6,6, and 2,2,9. In the final hint (line 12), the hair colour is a distraction; the relevant information is that there is an oldest child. So the ages of Hypatia’s children are 2, 2 and 9 (given that the ages can only be natural numbers, for a child to be recognised as older than another, their ages should correspond to different natural numbers. (amended from Stylianides & Stylianides, 2011) Analysis Teachers wrote down their initial reactions which ranged from “laughing, funny, surprising, unexpected ending” to “the BHP is not a mathematical task” to “Don’t know what to do” and “at first I did not understand the 40 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
task”. Most wanted more information to be able to solve the task, and found it “hard to consider what is necessary”. When they started working together on the task, many found the answer and mentioned ‘with relief’ that “although it is an open task, you can work on it systematically”. As we have not coded all of the teachers’ responses, and with respect to the goals, we cannot provide any quantitative or more in-depth analysis. Conclusions and Recommendations The overall impression was, and this was supported by the questionnaire data, that our Norwegian teachers were open and positive towards problem-solving tasks and teaching, and they wanted support to be able to teach in these ways. Our plan is to analyse this data more systematically, and perhaps link it to the questionnaire data on beliefs. We think that this would give interesting insights, as 96% of the teachers participating in this day ‘want [their] pupils to be motivated by IBL’ tasks and teaching; and 92% of the baseline data in Norway claimed that they ‘want to use more IBL in [their] teaching’. References Schoenfeld, A. (1985) Mathematical Problem Solving, Orlando: Academic Press. Philippou, G. & Chiristou, K. (1995) Teaching mathematics. Athens: Dardanos. Stylianides, G.J. & Stylianides, A.J. (2011) An intervention on students’ problem solving beliefs, Presentation at CERME 2011 in Polen.
41 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Germany: Day for Teachers (Lehrertag) Context The following case study describes the Day for Teachers (22.02.2011) at the University of Education in Freiburg, Germany. The Day for Teachers was integrated into the annual conference of the GDM that took place in Freiburg, Germany, from 21.02.2011 to 25.02.2011. The Society of Didactics of Mathematics (Gesellschaft für Didaktik der Mathematik, GDM) is a scientific society pursuing the goal of fostering mathematics education, particularly in German-speaking countries. The connection to the GDM conference offered teachers who participated in the Day for Teachers the exciting opportunity to meet and discuss with experts of mathematics education – an ideal bridge between theory and practice. The main topic of the day was: “The challenge of competence-orientated teaching in mathematics” and thus inquiry-based learning. The day offered a huge variety of interesting workshops and talks for teachers of both primary and secondary level. Additionally, publishers of educational literature presented their latest releases in an exhibition, and networks like ISTRON demonstrated their range of work - a valuable supplement to the workshops and talks, and in which teachers took an eager interest during breaks. Event Date: 22 February 2011 Target Group(s): Teachers Participants: 300 approx.
The Day for Teachers started at 8.30am with two parallel p lenary talks, one for primary and one for secondary education. Bernd Wollring, a top-level lecturer gave an interesting introduction to: “Geometry in primary school: competence-oriented and inquirybased learning by using Tangram”. Regina Bruder gave a talk on “Pedagogies to foster the problem-solving competences of individual students”. Both Ms. Bruder and Mr. Wollring introduced the topic of the GDM conference in a more than adequate manner.
42 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
After these two plenaries, workshops relevant to day-to-day teaching were offered. There were four categories of workshop, each of which comprised three workshops. There was one workshop category especially for primary school teachers and three for secondary school teachers (one for each of the types of German secondary school: Hauptschule, Realschule and Gymnasium. In total there were 12 interactive workshops, for example: 1. Gerhard Metzger – Land surveying: a possibility to explore, and inquiry applications in mathematics. 2. Reinhard Schmitt-Hartmann – Folding paper: Inquiry in mathematics lesson 3. Michael Marxer – Ways to a mathematical understanding of mathematical terms and term-structures via action orientation and inquiry-based tasks 4. Gabi Heintze: Learning cooperatively in mathematics lessons 5. Renate Diehl: Assessment as a basis to support individual students in problem solving. Closing After the official part of the conference, participants were invited to a café where they had the chance to share ideas in a casual atmosphere. Teachers and education experts were able to interact very easily. Conclusions and Recommendations The Day for Teachers was a very important event in which approximately 300 teachers participated. Feedback given by teachers showed that teachers appreciated the special atmosphere of a teachers’ day connected to a big conference and the opportunity to meet maths educators, as well as the workshops and talks on offer. Furthermore, the exhibition was also very much appreciated: teachers were informed about latest book releases and got the opportunity to meet in the exhibition room. To make sure that an event like this is successful, it is necessary to have everything well-planned and organised in advance. Accordingly, a large organisation team took care of everything. In particular, only workshops and talks which were considered to be of relevance to day-today teaching were offered.
43 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
One particular issue are the questions: when and how teachers decide which workshop they visit, as with workshops in particular there is a limited number of participants. There are two main options: either teachers choose via the website when registering, or they decide on the morning of the event. As one’s opinion might change between registering and visiting an event, we offered teachers the possibility to choose on the very morning of the event. In the morning, everybody was given the opportunity to obtain tickets for his or her preferred workshops. These tickets were affixed to a pin board in the entrance area of the conference. Those interested took one of the provided tickets from the pin board in order to join a workshop. When no tickets were left, the other participants knew that the workshop was already fully booked. Thus, it ensured that no workshop was oversubscribed.
44 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Slovakia: Mathematic Problem-Solving Methods Seminar Context The case study describes a training event where an elementary mathematical model ”In the Hands of Time“ was taught. The teaching unit was included in the subject called Methods of Solving Mathematical Problems and was realised in March 2011 at the Department of Mathematics, FNS UKF in Nitra. The participants were two groups of students who study in the first year of Master’s study in the programme of teaching academic subjects: mathematics in combination with another subject. The following list specifies students’ second subjects and the number of students studying them: Informatics (4), Geography (4), Civics (4), English (3), Biology (3), Physics (3), History (1), Psychology (1), Physical Education (1). Each group of students attended three 90–minute seminars. During the first seminar, a motivating introduction to the topic of time was presented during the final, 45minute-long part of the seminar. During the third, and final, seminar, the first 45 minutes of the seminar were devoted to the problems about time, during which the students evaluated the topic and the teaching methods that were used. The second and third seminars were recorded on video. In the course of these seminars, an interactive board with an applet of hours was used to model the situations. The applet is available at: http://www.wmnet.org.uk/wmnet/custom/files_uploaded/uploaded_resources/503/cloc k.swf Event 2/16/23 March 2011 Target Group(s): Trainee maths teachers Participants: 24
LESSON PLAN The first seminar: 1. Motivational tasks and their solutions 2. Introducing the idea of B-day mathematical competition and teaching materials 3. Homework assignment Procedure Motivational tasks Task 1 Prove that the connecting lines that connect numbers 1 and 4, and 2 and 9 on the clock face are perpendicular to each other. 45 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Task 2 It’s just after twelve o’clock and the two hands on the clock form an angle of 90 degrees. What’s the time? The students in both groups developed several different solutions to the given tasks. They were working in pairs and later discussing the solutions. In both cases the students found a universal approach to understanding and solving the tasks. In Task 1, it was the relationship between central and peripheral angles in a regularly divided circle, which a clock face represents. The students also discussed the importance of illustration, in other words, the essential role of a picture whose correct size and sketch are important for finding a task‘s solution. In Task 2, the universal approach was identified in using coefficients of movement of small (hour) and big (hour) hands in relation to time. Students discussed the accuracy question while expressing numeric results of the tasks, in case the result was a certain time. For example: 12 hours 16 minutes 46 seconds where 46 seconds is the result rounded off the fraction of the minute. It was stated that it is suitable to use coefficients and numerical expression of parts of an hour or a minute by a fraction rather than a decimal number. (e.g. it is better to express 46,1538 minutes as 46 minutes, that is 46 minutes and 9 seconds). Students were given homework consisting of two parts. The first part of the homework was to revise the steps leading to the solution of Task 1. The second part of the homework was to study the ”In the Hands of Time“ material and to solve the problems set in the text. The second seminar: During the second seminar the solutions of the task s from the material ”In the Hands of Time“ were discussed step-by-step. Students explained the solutions they had prepared for the homework; they modelled some critical hands positions by the applet on the interactive board. A solution to each task was summarised either by a student or a teacher, so that it was clear what the correct result and an appropriate answer to each question posed within each task were.
46 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
The third seminar: In the third seminar students reflected on their experiences from the previous seminars in which they had been dealing with tasks about time. The students wrote a brief evaluation where they summed up their interest in the given tasks. They also expressed opinions about the possible use of these tasks with their future students. The following table presents the opinions of 22 pre-service teachers as they wrote them during the third seminar. 1
2
3
4
5
I would try to solve these problems with my students because I think students would be interested in them. I wouldn’t use all the problems we had in our seminars because some of them were quite demanding. Because I consider the problems demanding, I would try to solve them just with students at secondary school or in higher grades of primary school. They are very useful because they present new teaching approaches. I would only deal with the easier problems. The problems were interesting, creative, but in my opinion, too demanding for pupils and students at primary and secondary schools. Maybe I would solve some of them during the lesson but only if time/my plan allowed. They would be used to liven up a lesson. The problems were good to wake up one’s thinking and also because they had to be solved in everyone’s own way (partially). I think it’s a good idea to solve these problems with students but it’s also important to make students solve some problems on their own. If the solution is revealed during the lesson, the problems become meaningless. Interesting problems that make students think. I would solve them just during seminars because of lengthy solutions and number of options to be discussed. Students would feel a change if they worked with an interactive board because it would make the lesson memorable and unusual. On the other hand, it could also be interesting for students with weaker results in mathematics.
Based on the students’ opinions it is obvious that they were attracted by the problems and they considered them to be interesting. At the same time they observe that the problems are time-consuming from the point of using them during traditional mathematics lesson. Almost every problem requires longer and more complex consideration in comparison to any problem traditionally solved at higher grades of primary schools and at secondary schools. Moreover, the problems were considered demanding because of the knowledge necessary for solving them. The students also noticed that it is necessary to model different situations that describe the positions of the hands on a clock face in spite of the fact that the context is familiar. After all, almost all students wear watches. Another aspect that shows the complexity of solutions is the arithmetic of the hexadecimal number system. It is not acceptable to express the values of time in decimal numbers, so it is necessary to work as accurately as possible with fractions and rounding numbers.
47 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Conclusion In conclusion, we can state that the problems about time based on the ”In the Hands of Time“ material were positively accepted by the pre-service teachers. While solving them, the students realised the relationships between arithmetic, geometry and algebra from a different point of view. They were able to perceive their own mathematical knowledge in a new way. Through the model of hours, they have revealed the necessity of complex - from the point of view of school mathematics non-traditional use of mathematics knowledge, when solving real-life problems. We just have to believe that the teaching material and the problems will be included in their own portfolio of mathematics problems and that they will try to solve them with their future students.
48 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Hungary: Talent Day Context On the 10th of March, 2011, a “Talent Day” was organised in the school “Közgazdasági Politechnikum Gimnázium és Szakközépiskola” (Budapest, Vendel u. 3.). It was primarily a dissemination event for out-of-school target groups, but there were additionally two other goals to meet: (1) to provide an opportunity for teachers from other schools to be engaged in IBL activities, and – not independently of the previous goal – (2) to provide an opportunity for future trainers of our PD programs to get to know how IBL is understood and practiced in this secondary school. The event was open to everyone, however, some constraints on the number of attendants of different activities was limited due to the lack of space in laboratories. Event Date: 03 March 2011 Target Group(s): Prospective trainers of IBL PD-programmes Participants: 8
During the Talent Day, there were one physics and one integrated science lesson organised. The teacher attendees participated in the integrated science lesson. The main topic of the complex science lesson was: the pudding probe. It was about the gel state. Students had to mix water and concentrate in order to make the most valuable food from them. They used calorie tables and other data about foods, and they had to search for these data sources on the internet. They worked in a kitchen-like environment, making the experience more close to real-life situations. The following teachers participated as guests: Kissné Gera Ágnes, Illés Éva, Vantsó Erzsébet, Holzgethán Katalin, Nagy Mária, Kocsis Márta. Since the event preceded the 2nd of April (The day of the first training of IBL trainers event) this opportunity served as an experiential basis for studying IBL. A group discussion led by Gábor Veres and Ágota Somogyi followed the complex science lesson.
49 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Conclusions Recommendations
and
The event proved to be useful in providing experience for studying the IBL approach. It has become clear that when starting a regular PD session on the IBL approach, visiting school events will definitely help in deepening IBL concepts and in connecting and contrasting innovative practice with socalled ‘traditional’ teaching techniques.
50 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Romania: PRIMAS piloting Action Context This case study describes a series of piloting events in which teachers were involved in practising IBL methods with students at different locations (Gherla, Sanmartin, Odorheiu Secuiesc, Hodod, Beltiug, Batanii Mari, Estelnic, Cluj Napoca, Miercurea Ciuc, Târgu Mures, Satu Mare). A group of 14 teachers was selected at previous (“inform them” type) dissemination events. After an initial workshop each of them prepared a teaching unit (2-3 lessons), which was carried out with students, attended by 1-3 colleagues, videotaped, analysed in group and then used with another group of students. At the end of the activity the participating teachers formulated their concerns, remarks, positive and negative insights about IBL methods in general and about the analysed teaching activities in particular. The whole activity was carried out in April and May 2011 and it was intended to exemplify a basic cycle of a PD course. At the dissemination activities the participants were informed about the existing European context, about IBL and about the PRIMAS project. At the initial workshop they had an IBL task and a group session where they had to develop a common structure for their activities based on the IBL definition formulated by the PRIMAS project. At the dissemination activities there were more than 200 participants, but only 14 wanted to work in this experimental group. The aim of this group was to gain some authentic experience with IBL in the Romanian context, where students are not familiar with open-ended questions, or with group work at mathematics or science lessons; they do not usually use scientific calculators or computers at mathematics and science lessons, and they have a centralised and overloaded curriculum and a well-contoured traditional assessment system. In this way, the participants got fully involved in practising IBL methods and through them the project had a good dissemination in their schools (for students, teachers and parents) and the experience of this group can constitute a strong basis for the future CPD courses. Due to a strong collaboration with some regional and local associations (the Hungarian Teachers’ Association, the Farkas Gyula Association and the SimpleX Association) only the coordinators and their expenses were paid from the PRIMAS budget. Activity Date: April/May 2011 Target Group(s): Maths and physics teachers Participants: 14
As a result of participation in the event “School Days in Harghita County”, where the PRIMAS team made presentations and held four different workshops in different locations (Odorheiu Secuiesc, Toplita, Gheorgheni, Miercurea Ciuc) for more than 200 mathematics teachers, the necessity of a deeper insight arose. As a consequence, we organised an 51 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
experimental study group with 14 teachers. The action plan had three phases. The first phase consisted of a workshop on 10 April, where they had the opportunity to work on an IBL task as teachers. They had to develop a teaching unit for understanding the arithmeticgeometric mean inequality based on the comparison of different interest schemes (3x1.5 hours). After this they had to adopt a common strategy they would use in their future lessons and in developing future teaching/learning tasks. In the last part of this phase each participant chose a teaching unit he was going to detail and transform to fit the IBL strategy (for a list of these units see Appendix 1). In the second phase each participant developed his own teaching unit and after several discussions with the coordinators and colleagues he/she implemented it with his own students. Each activity was videotaped and attended by a few members of the group to obtain a more personal insight and to have a better opportunity to observe students’ actions/reactions. In the next phase these activities were discussed in the group and the improved versions were implemented with other students. In the last phase the group had to analyse the improved version of the activities and to evaluate the progress. In almost all schools, after the regular activities, we organised Puzzle Houses, where the students and colleagues from the school (and occasionally from the neighbourhood) were invited to play with Happy Cube puzzles, Impuzzable cubes, a collection of more than 100 wire puzzles and other IQ games. At the end of these activities a short workshop was performed using the Happy Cube study. Results: six members of the group decided to support the PRIMAS team and to act as multipliers in future PD courses; another three members chose IBL as the main topic of their degree thesis (this is required in the Romanian system to obtain the highest level in undergraduate education). Two members had the idea of making a virtual environment for some teaching modules (the elementary operations) in order to offer a supporting tool for students and teachers. Three members renounced further developments - they will use IBL only if they obtain already-developed teaching materials. The accumulated experience and know-how can constitute a strong basis for a future CPD program. The development of the teaching materials will be continued and most teaching units will be published not only on the website of the project, but also in some local journals, or possibly in a Romanian PRIMAS publication. An overview of this activity, some units, comments, and conclusions will be 52 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
published in Romanian journals with a wide readership (over 3000 readers). From the next academic year a new course for pre-service teachers will be introduced at the Babeş-Bolyai University. This will focus on practical aspects of teaching mathematics with IBL methods. The Romanian PRIMAS team was invited to give lectures on IBL (a one-week PD program in July) in the framework of the “Bolyai Summer Academy”, organised by the Hungarian Teachers’ Association in Romania. Conclusions and Recommendations 1.
Regarding the dissemination: a. Concrete, relevant and detailed examples and cases are needed (especially for attracting people to CPD activities). All activities have to be prepared in detail and in advance, so this requires a well-organised team. b. There are some typical negative reactions (time-related, assessment-related and students’ behaviour-related). To overcome these, some longitudinal studies would be necessary, or the absence of these studies has to be compensated for somehow (by presenting student/teacher interviews). c. The efficiency of “inform them” type dissemination events is very low in comparison with an “involve them” type event, but both are necessary. A halfday workshop is sufficient to illustrate some IBL principles and to co-opt people into a short-term CPD activity, but a deeper exercise (one PD cycle) is useful in co-opting colleagues in longer PD programs. (In Romania the majority of PD courses are not practice-oriented and they are usually very short – some colleagues refused to participate in a longer CPD). d. Some participants appreciated the personal outcomes of this activity as enlightening not only for the aspects of IBL but also for some difficulties from their previous teaching practice; but they formulated a lot of concerns regarding a long-term use of the IBL methods. Most of these concerns were related to the huge amount of background work in preparing these activities.
2.
Regarding the teaching activities: a. A major difficulty (especially with high-school students) is that if students are not familiar with group work or with making inquiries, time-control needs to be very efficient (at the beginning), otherwise they may waste their time. b. The teacher’s role has to be understood via the classroom processes and in concordance with some long-term development processes. It is not sufficient to support classroom processes, they have to be integrated consciously into a long-term learning process.
53 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Annex: Teaching units (brief description): • Teaching centroids – the main aim of this unit is to understand the notion of the centroid for polygonal domains and for an arbitrary system of points and to understand the principle of decomposition. A basic problem is to determine (calculate and construct) the centroid of a homogeneous polygonal object (a piece of wood). • Colouring problems – the aim of this unit is to understand the usefulness of colouring in some covering problems and to construct non-standard colouring starting from a given situation. A basic problem is to determine the maximum number of 1× n pieces that can be placed on a rectangular board with given dimensions ( a × b ) without overlapping. • Logical puzzles (following Raymond Smullyan: What Is the Name of This Book?) – The aim of this unit is to develop a bridge between formal mathematical logic and some logical puzzles. • Operations and algorithms with natural numbers – This unit is developed to understand on an operational level the properties of the elementary operations and to understand the algorithms we use for calculations (addition, subtraction, multiplication, division) • Operations and algorithms with integer numbers – This unit is developed to understand the elementary operations with integer numbers. • Operations and algorithms with rational numbers – This unit is developed to understand at an operational level the elementary operations by rational numbers (how can you calculate the ratio of two rational numbers, etc.) • Extracting square-root – Discover an algorithm for square root extraction using visualization • Visual representations of algebraic formulas (proofs without words) – Discover algebraic formulas using visualisations • The concept of the area and The Bolyai-Gerwin theorem – Introducing the concept of the area by decomposition and understanding the Bolyai-Gerwin theorem. • Teaching divisibility properties using rectangular grids – The aim of this unit is to give a visual interpretation on rectangular grids for a series of number theoretic notions (lcm, gcd, etc.) and to understand some properties of these using the properties of the grid. • Combinatorial proofs for number theoretic theorems (Little Fermat, Wilson) – Studying combinatorial problems that imply number theoretic properties and obtain generalisations for the number theoretic properties by generalising the combinatorial setting. • Graph representations and probability – The main aim of this unit is to detail elementary problems of probability in a unitary manner using graph representations (how to discover the product rule for conditional probabilities using non conditional probabilities in a classical framework) • Puzzle making and classification • Puzzles and combinatorial problems
54 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Slovakia: Methods day for teachers Context The following case study describes the Methods Day for Teachers which was organised by a secondary grammar school named Saint Vincent de Paul in the town of Levice, southern Slovakia. The Methods Day for Teachers is organised once a year and its aim is to introduce new trends in education to participating teachers. The workshop on mathematics was led by members of Slovakia’s PRIMAS team. We were invited by organisers from the school’s management with the goal to discuss mathematics education, to become informed about the actual needs of teachers, to help them improve their lessons and to work with teachers on interesting materials. Project PRIMAS’s website, the project aims, activities and ideas were demonstrated. The principles of the IBL approach in mathematics were introduced and discussed. Teachers worked on two items of PRIMAS material. They also completed the PRIMAS WP8 questionnaires at the end of the workshop. Event Date: 26 April 2011 Target Group(s): Mathematics teachers Participants: 11
The Methods Day for Teachers started at 10.00am. E ach participant introduced herself/himself during the first part of the workshop. Every participating teacher teaches mathematics at upper primary or secondary grammar school in the southern Slovakia region. Eleven teachers from eight different schools participated. In the next part of the workshop the PRIMAS project website was presented. The ideas of PRIMAS, its aims and activities were also introduced to the teachers. Teachers were interested in all this information and asked several questions, which completed the presentation of general information about the PRIMAS project. The IBL circle was shown at the presentation and referred to as the main principle of mathematics education introduced by the PRIMAS project.
55 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Mathematics in McDonald´s (a Slovak version) was presented as the first item of material (available at www.primasproject.eu). Every participating teacher got one copy as a handout. Teachers worked in pairs. They were asked to check the text and to eventually correct mistaken parts. The material was evaluated in the ensuing discussion. Teachers evaluated material as very interesting, with some problematic parts. A problematic example of arithmetic of one part of the texts was corrected. Teachers were surprised by the variety of mathematical tasks using the context of McDonald´s fast food. They consider that the context would be interesting for their pupils, because this type of fast food is popular among children. Teachers also expressed the possibility of using the material in their teaching. The second item of material presented during the workshop was the “Problem situation” based on the geometrical situation illustrated by the picture (see appendix). The first part of the task: Write down as many expressions for the objects in the picture as you are able to think of was completed individually by teachers on a small piece of paper. Then the whole group wrote the expressions on the blackboard. We discussed the results and sorted expressions together according to different criteria. The second part of the task: Complete the table working out the missing values was competed gradually. Teachers again worked in pairs and completed the table on the blackboard with the missing numbers. They were very active, captivated by the given numbers and solved the problems with a certain amount of enthusiasm. After completing the table they were quite happy and satisfied with the results. They commented on the difficulty of mathematical problems in particular columns of the table and discussed the forms of result numbers (square roots instead of decimal numbers). After the workshop teachers filled in the PRIMAS WP8 questionnaire. Conclusions and Recommendations The Methods Day for Teachers was a very important and useful event. Feedback given by teachers showed that the teachers appreciated the atmosphere of the day. The lack of interesting sources for teaching mathematics was confirmed by participating teachers. PRIMAS project materials were very welcome and teachers asked for more materials and contacted lecturers after the meeting by e-mail. Teachers are open to new approaches in mathematics education and willing to continue in mastering their skills, especially the IBL approach, during occasional meetings and also in regular courses of professional development in the near future.
56 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
The Netherlands: Ecent conference 2011 Context Ecent is the national platform for science teacher education. They organize a yearly conference for science teacher educators. We introduced PRIMAS at the 2011 conference in a presentation and a workshop. During the presentation awareness of the PRIMAS project was raised among science teacher educators for primary and secondary schools. We described the history, background and aims of the project and demonstrated the PRIMAS website. In a workshop we worked together on some of the PRIMAS materials for teachers. We specifically used the PD module ‘Asking Questions’. The purpose of the presentation and the workshop was to raise interest in the PRIMAS project among Dutch teacher educators and to discuss the aims of the PRIMAS project and possibilities to implement PRIMAS PD modules in current practice. Moreover, we hoped to find multipliers for the PD modules in the next years of the project to spread IBL in the Netherlands. Regional science centres in the East and in the South appeared to be interested. They asked for further information and plan to pilot some PRIMAS activities next year in their region in an in-service course for upper secondary science teachers. A teacher educator of the Utrecht teacher college is interested in using the PD modules for pre-service teacher education (lower secondary science education). Event Date: 12 May 2011 Target Group(s): Science researchers, educators and educational designers in the Netherlands Participants: 10
We started the 30-minute presentation with an overview of IBL research and best practice in the Netherlands. In the second part of the presentation we changed the perspective to Europe and introduced PRIMAS. In the one-hour workshop we used the module ‘Asking Questions’. In this module we exchanged the video activity with a Dutch video clip of a primary school setting. The final activity was replaced by a PRIMAS science activity3 in order to better align the content of the module to this target group. During the workshop we started a discussion on the video clip. Primary questions were: What kinds of questions are being asked? How do these questions promote students’ thinking? Which kinds 3
http://www.primas-project.eu/artikel/en/1116/Falling%2C+modeling/view.do See appendix.
57 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
questions do you recognise and which questions are new for you? Could this lesson be improved? During the discussion we collected aspects of questioning and presented the five principles of questioning for IBL from the module. The educators agreed with these principles as being of importance for creating an IBL classroom culture and extending a teachers’ repertoire. New suggestions were: •
Establishing ‘horizontal’ interaction (between students). In the video clip the teacher asked questions and valued most of the answers.
•
Having students pose the most important questions that have to be answered. The students should also become responsible for the main questions. A characteristic of this aspect is: Ownership.
•
Organising answering by students (balance between boys & girls, shouters & silent students, talented & less gifted…).
•
Distinguish product-oriented and process-oriented questions and feedback.
We are going to include these aspects in a second version of the module ‘Asking Questions’. Feedback During the workshop, project partners talked with participants about the possibilities for using this module – and the other PRIMAS PD modules – in their practice. At the end of the workshop we had five minutes to discuss these possibilities plenary. We also asked all participants to fill in a questionnaire (WP8 formative evaluation document). According to the discussion and the answers in the questionnaires, all participants agreed to have learnt a lot during the workshop and are willing to use PRIMAS materials in their practice. We were able to make appointments with a representative of the Utrecht science teacher education college and a head teacher of a Dutch school who also participated in the workshop and agreed to have further contact with the two representatives of regional science centres. Conclusions and Recommendations The structure and the content of the PD module ‘Asking Questions’ seems to be suitable for the Dutch context. Some adjustments are necessary to enforce the primary and the science character. We will proceed with translating the other PD modules and will start with piloting on a larger scale all five PD modules in year 2011/2012 as a CPD course for pre- and in-service teachers.
58 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Annexe From the module ‘Asking Questions’ we changed activity 6 in the hand-out for teachers (in Dutch):
Design a lesson plan Study the plan on the following pages for an afternoon session on Falling, modelling and being smart. Improve the plan with aspects of asking questions (the five principles). Next, design a lesson plan for a lesson in one of your classes. Take care that you explicitly implement aspects for asking questions in the plan. The following suggestions for a lesson on "Sharing Petrol Costs" describe a possible approach to using the problems with students. This may take one or two lessons, depending on the class. These suggestions help to improve the plan on the following pages and for designing your own lesson plan. 1. Introduce the problem, and give time for individuals to think 10 minutes Issue each student with just one of the three problems. Explain that in this lesson you expect everyone to think things through and to contribute. Use the think-share-exchange pattern to have them think about the problem and to collect initial ideas on the board. Focus on process-aspects for solving the problem; not yet on final answers. Everybody should be able to start working on the problem after this discussion. 3. Students work on the problem 20 minutes Allow students time to engage with the problems. When they ask questions, ask them a question that offers strategic guidance rather than technical help. 4. Whole class discusses the approaches being used 10 minutes When most students have made significant progress with the problem, ask the students about the way that they are working. Anyone stuck? Anyone have suggestions for a fruitful way to proceed? (It may be helpful to record this part for later discussion). 5. Students have a second go at the problem 10 minutes Encourage students to return to the problem and continue working on it using some of the ideas that have been shared. 6. Ask
Whole class reports on their reasoning students to take turns at presenting
their
reasoning
to
10 minutes the class.
59 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Practical
research
We will perform a mini-inquiry in which we not only learn how to design and test "models", but also how we can improve our results by communicating. We work in groups of four people (sometimes groups of five). We work with 3 to 4 groups in a classroom (shifting tables and chairs a bit). All stuff needed is already in the rooms. Each group examines one of the four variables. This means that with 52 participants, three groups on average work on each variable. It may occur that these three groups have a unique approach and the end results are not necessarily the same. Together we form a research community. In between, we communicate about our approach and preliminary results. We prepare a presentation in which we process what we learned from the other groups. Global planning 13:45 - 14:45 Performing the task, preliminary result (method + conclusion) is listed on an A3 sheet. 14:45 - 15:00 Break in which we go shopping (with the A3 sheets) for other groups. Two delegates from each group go to other groups that have examined the same variable. The other two of the group remain behind to receive others and if necessary give explanations. Consider not only the end result but also the method. Of course you can discuss with your fellow researchers. 15:00 - 16:00 Processing of what we learned from the other groups (when needed you may perform another single experiment). We make a poster on which can be found: - Our hypothesis - Our approach, research method (a drawing can be clear) - Our conclusion (this could be a formula) It does not have to be a "nice" poster, we do not have enough time for that. Just write and draw clearly. 16:00 – 16:30 We take the posters to the classroom. There, four groups get the opportunity to present their findings. After that, we hang up the posters which can be viewed during free time. We also take pictures and publish them on the blackboard.
60 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Falling, modelling, and being smart When leaves fall from trees or when a piece of paper falls, this happens in a fairly unpredictable way. The leaf of paper zigzags to the ground and it is difficult to predict when and where it will land. But when we fold the edges of the paper, the movement suddenly becomes very predictable. The paper tray moves slowly and steadily to the ground. This is not a free fall; air resistance plays a major role. How can we describe this motion mathematically in a model? What is the influence of factors such as size and mass of the paper tray? How can we test our model experimentally, even without a stopwatch or balance? Although this activity is performed by simple means, the activity illustrates exactly what physical model thinking is. Let's try and see whether we can make a formula for the drop time t of a paper tray that is falling from height h. As altitude increases the time it will take the tray to touch the ground will increase as well. We assume that t ∝ h (where ∝ means proportional, so t = constant * h). Furthermore we expect the surface A of the paper tray to play a role. A paper tray with a larger surface will encounter more air resistance and thus will fall slower down. This implies that t ∝ A . Finally, we suspect a larger mass to fall down faster. So t ∝ 1 / m . When combining these results we get the following:
t=
k . A.h m
Where t = fall down time of the paper tray k = constant A = surface of the paper tray h = height to fall from m= mass
of
the
paper
tray 2
Now we have a simple formula, but maybe this one is too simple. Maybe we should have used A ,
A , or another function of A. The formula we now have predicts that a bowl twice as large with an equal mass and height has a twice as large fall down time. We can test this in an experiment. In this way we can also test other relationships between variables. Maybe another variable is lacking! Exercise Test by means of experiments whether the model (the formula) is correct by comparing different surfaces, masses, fall heights, etc. Can you do this without stopwatch, balance, and ruler? We substitute creativity and ingenuity for equipment! If necessary, correct the formula! 1 Surface 2 Fall height Variables 1 3 Mass 4 ………
61 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
United Kingdom: Introducing Inquiry-based Learning to trainee mathematics teachers in England Context Initial Teacher Education There are a range of different routes in Initial Teacher Education in England (ITE) in preparation for teaching. However, because teaching is a graduate profession, potential teachers in general either study initially for a first degree in a specific discipline such as mathematics or science, or follow a degree course that is situated in education as a discipline itself. The preparation of secondary teachers is predominantly of the former type, with over 90 percent following this route, which in most cases is an additional year of university-based study leading to a Postgraduate Certificate in Education (PGCE), whereas in the primary sector over 40 percent of entrants follow the latter route, which provides a broader range of subject knowledge appropriate to teaching as well as study related to teaching, and learning more broadly. These courses are predominantly based in Higher Education Institutions (HEIs) although a small number of entrants to teaching each year follow courses that are school-based (about five percent of post-graduates in both secondary and primary sectors). Although the design of the training/education in HEIs of those following courses towards teaching is in the hands of university lecturers and tutors, all courses lead towards Qualified Teacher Status (QTS) and to achieve this students must have met a range of statements of competence defined nationally. These “standards” are organised around three themes: (1) Professional attributes, (2) Professional knowledge and understanding and (3) Professional skills. Many of these are school-focused and assessment is carried out by teachers in schools working in partnership with university-based tutors. As part of the one-year HEI ITE courses students spend at least 18 weeks in schools that work closely in partnership with a local HEI, practising teaching under the guidance of a school-based mentor. Consequently for “trainee teachers” the focus is very much on day-to-day classroom practice and the school as a work place. In an ethnographic study Hodkinson and Hodkinson (1999, 275) found that the predominant focus of school experience for trainee teachers concerned the pragmatics of ‘teaching and implementation of national policies rather than those aspects of pedagogy, reflection and critical analysis’ traditionally encouraged by the HEIs. The different priorities are perhaps encapsulated in the tension often experienced in the use of the words “education” and “training” which are often used interchangeably in this context but with very different underlying philosophies. In general one would expect to find that trainee teachers are exposed to a narrow range of pedagogies in their placement schools with teachers very much focused on ‘teaching to the test’ that often focuses on routine application of rules and procedures rather than deeper understanding of a relational nature.
62 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
This case study describes a one-day event for PGCE secondary mathematics trainees in the final ‘enrichment’ period of their course during which the PRIMAS partner explored some of the ways of working, both mathematically and pedagogically, that mathematical modelling and inquiry-based learning promote. Prior to the day those attending were sent preparatory work (see below) to do; they brought their responses to the day and used them to stimulate discussion. An inquiry-based episode: Before the day please reflect on an episode (maximum size – a lesson) that encapsulates for you what might be thought of as “inquiry-based learning”. On a single side of A4 report very briefly what you - as teacher – and your pupils did and your thoughts as to why this encapsulates inquiry-based learning for you. Event Date: 08 June 2011 Target Group(s): Trainee secondary mathematics teachers Participants: 40
Forty trainee teachers in the final weeks of their initial training for teaching mathematics in secondary schools attended the day which was organised around a theme of considering the mathematical activity of mathematics learners. This was to raise general concerns of the experience of the learner in the mathematics classroom, which government inspectors characterise as being passive and leading to surface learning rather than deep understanding. “Pupils rarely investigate open-ended problems which might offer them opportunities to choose which approach to adopt or to reason and generalise. Most lessons do not emphasise mathematical talk enough; as a result, pupils struggle to express and develop their thinking.” (Understanding the Score, Ofsted, 2008a) The day was organised so that the trainee teachers would have opportunities to consider both mathematical activity for themselves and to consider issues in relation to pedagogies that might support such activity for pupils in their classrooms. An introduction to the trainees informed them of the curriculum development projects LEMA (Learning and Education in and through Modelling and Applications4) and PRIMAS (Promoting Inquiry in Mathematics and Science Education across Europe5). They were then asked, working in groups of four, to complete one of nine different modelling tasks that come from the LEMA professional development materials with the intention that these would immerse them in a new type of mathematical activity rarely found in schools. In addition to being asked to work on the modelling task trainees were asked to produce a poster of their work and were expected to make a presentation of this to the rest of the group. 4 5
http://www.lema-project.org/web.lemaproject/web/eu/tout.php www.primas-project.eu/
63 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Following this immersion in mathematical activity each group was asked to consider this activity and how they might summarise it in schematic form for their school students. Revolving door At what speed should a constantly revolving door rotate? …. at the entrance to a hotel?.... at the entrance to a supermarket?....
Race In a school playground there are two trees: one is small and one is large. There is also a straight fence. A group of pupils organise a race: each pupil starts at the small tree, then has to touch the fence before running to the large tree to complete the race. Where is the best place for a pupil to touch the fence?
The groups worked on their tasks with many feeling unsure of how to start, but after some advice felt secure enough to proceed. In many cases the groups worked with similar levels of mathematics as has been observed with students in lower secondary schools. The group working on the task “Race”, for example, felt unsure about how to proceed at the outset and eventually simplified the situation to consider the case of a straight-line fence with the two tress the same distance from the fence. They proceeded by making scale drawings of the situation, and by pooling resources they concluded that the best place for a pupil to touch the fence would lie at the point which is mid-way between the two tress. Time prevented them from making much progress with a situation where the trees were at different distances from the fence. In contrast, pupils have been observed taking a more geometrical view of the situation and realising that the place to touch the fence is such that a straight line drawn from one tree as object and the other as image gives a solution for all cases. Group feedback highlighted the need for pupils to make assumptions and to ask pertinent questions prior to getting started on any of the tasks. In terms of their developing pedagogy this raised lots of questions, and for most groups this was a
64 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
prominent part of their schema of the modelling process. Having considered their own understanding of modelling as a meta-competency the modelling cycle as introduced in the PISA assessment framework for mathematics (OECD, 2003) was introduced and discussed (see below).
Pedagogic issues were discussed prompted by use of video sequences of French primary school pupils working on a modelling task. In particular, this demonstrated how pupils of all ages can engage with modelling tasks – the same task, perhaps worked on at a more detailed and complex level, could be used by older pupils. Following this, the trainees were asked to share their initial understanding of inquirybased learning using their preparatory work and to write down some of their initial conclusions as to the nature of inquiry-based learning. This resulted in a discussion about whether inquiry-based learning in mathematics is closely aligned with mathematical investigations (briefly these might be considered as open-ended problems in mathematics in which students seek to explore mathematical structure in situations that are often situated in the world of mathematics itself). Morgan (2006, 230) points to the different discourses and values surrounding the use of coursework in mathematics at school, and the tensions that exist among them: “The discourse surrounding the notion of ‘investigation’ in school mathematics in England introduces values related to, among others, exploration, creativity, originality and the nature of mathematical activity that are at times in tension with the values of the dominant assessment discourse, including reliability and comparability.” Again, as earlier in the day, the trainee teachers were asked to explore some openended tasks that might be used to instigate inquiry-based learning in mathematics.
65 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Finally, inquiry-based learning pedagogies were again discussed prompted by use of the PRIMAS study modules on student-led inquiry with the group watching a video sequence from a lesson in which pupils used the ‘spirolaterals’ software.
Spirolaterals software from PRIMAS Video of a lesson with pupils working resource bank on the spirolaterals task. The ensuing discussion focused around needs in relation to teacher needs in the areas of questioning, working with groups, and resources. Analysis The session raised many developmental needs of this group of new entrants to the teaching profession. Notable among these were that these new entrants to the profession: • • • •
Had little or no experience of mathematical activity of this type during their own education. Were not always comfortable with how to proceed in open-ended tasks that support IBL – particularly in mathematical modelling tasks. Would like to implement IBL approaches in their future mathematics lessons. Believe that IBL can support the development of pupils’ problem-solving competencies, can be used with students of all abilities, and it is not necessary for the pupils to have extensive subject knowledge before they tackle problems.
66 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
• • •
Consider that the curriculum does not encourage IBL approaches and that text books that they use do not support such approaches. Do not know how to assess pupils’ work in IBL. Perhaps most important of these was the lack of confidence of the beginning teachers in tackling some of the tasks.
Finally, it should be noted that general satisfaction with the day was summed up in two emailed comments: “All of us found the workshop the other day very useful and will definitely try using some of those scenarios for investigation/inquiry/enquiry in lessons! Open-ended tasks and modelling are useful for making pupils think more creatively and realistically.” And, “There were some really rather fun and thought-provoking activities in the work the other day! Many thanks!” Conclusions This one-off event introducing some of the issues surrounding IBL was facilitated by a PRIMAS partner who had worked with the group previously: this is clearly advantageous in getting down to some serious discussions relatively quickly. Also, because the session was run by a PRIMAS partner who had also worked on LEMA, they were able draw from the wide range of available resources with relative ease. Overall, it appears that an important part of the day was the dual focus on mathematical activity and pedagogies related to this. In particular, it was important that the participants were given enough time to work on some of the tasks to some depth and to consider the competencies and meta-competencies that these required. This reflection on mathematical activity was encouraged by the requirement of the groups to report their thinking back to the whole group.
References OECD (2003). Framework for Mathematics Assessment. Paris: Organization for Economic Cooperation and Development. Ofsted (2008). Mathematics: understanding the score. London: Office for Standards in Education.
67 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Malta: Introducing PRIMAS to heads of schools Context As part of the planning for the implementation of CPD for teachers, it was necessary to organise a meeting for the administrators of schools that had been targeted for the implementation of the project. In Malta state schools are organised into colleges, each college has a ‘College Principal’ and each school within the college has a ’Head of School’. Principals and heads of school have been identified as key persons in the implementation of the project in all countries, but this is especially so in Malta since the project will take place within five secondary schools (age 11-16). The permission and support of the heads of these schools was required. The meeting was intended to introduce these administrators to inquiry-based learning and the project. It was also intended to allow discussion with the heads of school concerning some practical arrangements that were required. The meeting was as short as possible in order to allow the participants to fit it into their busy schedules. Activity Date: 10 June Target Group(s): Heads of schools Participants: 8
The programme was designed to give heads of school an introduction to inquirybased learning which also served to address common concerns about inquiry. It attempted to answer questions such as: “What is IBL?” “Why include IBL?” and “How can IBL be implemented?” It was intended to show the benefits of IBL for students in order to motivate heads of school to support and encourage teachers at their school to participate in the project. The meeting also introduced the participants to t he project and outlined the main features of the project with special emphasis on the implementation of CPD for teachers. The plan for the implementation of the project in Malta was introduced giving details such as who the 10 multipliers were going to be and how each multiplier would work in a particular school with five teachers (either mathematics or science) who teach in that school. The meetings between multiplier and teachers would be held during school hours, within the school.
68 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
This was followed by a discussion about how the heads of school can promote and support the implementation of the CPD within the school as well as the integration of inquiry-based learning at school. The heads of school eagerly participated in the discussion and the following points were agreed: 1. Each head of school will organise a meeting for mathematics and science teachers. A member of the University Project Team will introduce the project to teachers, invite them to participate and answer any questions that they may have about the project. 2. The teachers who would like to take part in the project will be invited by the head of school to come forward before the end of June, so that arrangements will be made in their school timetable for the coming scholastic year to allow them to participate. 3. The heads of school also agreed to try to plan participating teachers’ timetables in such a way that they will all have a 1.5 hour slot at the same time where they are not teaching, so that meetings with multipliers can be held at that time. 4. The heads were also ready to give a reduced teaching load to teachers participating in the project to give them more time to participate in meetings with multipliers and to work on the implementation of IBL in their classrooms. The heads of school also expressed their concerns related to how IBL will be integrated in the current syllabi, time constraints and the high-stakes examinations that students are required to sit for. They appreciated the fact that the new curriculum framework for Malta, which has been published for consultation, stresses the importance of IBL at all levels of education. Conclusion The meeting was very successful because it helped to start the process of working within schools. The participants were all willing to participate and were already thinking of practical details required, such as possible dates and times for meetings with teachers and so on. They were all willing to encourage the teachers in their schools to participate and to provide their support throughout the project.
69 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Annex I – Examples of dissemination materials
Poster for teachers and other target groups (such as parents) 70 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Poster for a scientific target group
71 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Short flyer for teachers
72 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Brochure informing different target groups
73 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
PRIMAS Bookmark
PRIMAS calendar; cover page
PRIMAS ruler; cover page
74 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
Poster gallery, PRIMAS consortium meeting in Nitra, January 2011; photo 1
Poster gallery, PRIMAS consortium meeting in Nitra, January 2011; photo2
75 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
PRIMAS flyer in use, Slovakia
PRIMAS presentation, Romania
PRIMAS document wrapper in use, the Netherland
PRIMAS web page layout
76 The project PRIMAS has received funding from the European Union Seventh Framework Programme (FP7/2007‐2013) under grant agreement n° 244380.
