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Figure 1: Modelling cycle (Blum et al., 2009) A realistic problem formulation is given at the start. To solve this, a modelling process needs to be applied which contains different steps that need to be undertaken. These sub-processes include to understand the task, to simplify the problem in building a mathematical model, to solve the problem in the chosen model and, finally, to interpret and validate the solution and the model used. A modelling process is often referred to as a modelling cycle as the process usually needs to be undertaken more than once. Advantages and disadvantages of the model have to be revised iteratively to find a better solution. The global modelling competence is defined as the ability to undertake a full modelling process and to have the knowledge of the procedure (Kreckler, 2015) and is necessary to be able to solve applied modelling tasks on the whole. While formulating and developing modelling tasks it is essential to keep with the four quality features of good modelling tasks named by Blomhøj & Kjeldsen (2006): 1. Comprehensibility and signification 2. Adequate challenge 3. Authentic problem 4. Open formulation Hence, it is important that modelling problems are comprehensible and significant for the students and that they describe an adequate challenge, i.e. the tasks should neither be too easy nor too difficult. It is also important for the problems to be authentic and to use authentic data. “An authentic problem is a problem posed by a client who wants to obtain a solution which is applicable in the issues of the client. The problem is not filtered or reduced and has full generality without any manipulations, i.e., it is posed as it is seen. A real-world or realistic problem is an authentic problem 112
which involves ingredients which can be accessed by the students in real life.” (Bock et al., 2012) Another essential point about modelling problems is, that they need to be formulated widely open such that various mathematical models can be built and such that the problem can be solved with respect to different approaches. Due to the lack of experience and many uncertainties how to implement mathematical modelling in regular school lessons or larger projects, teachers need to be trained and prepared for an efficient classroom practice. The question of how to do this efficiently has been addressed in various research projects as, for example, the project LEMA3. LEMA (2006-2009) was an international research project with the goal to develop a concept and materials for teacher training courses in the field of modelling and applications. Within this and other projects a spiral model has been proven to be efficient, meaning a continuous cycle of analysis, implementation and reflection. With each cycle the knowledge and experiences can be extended and improved. This concept of combining theory, actual implementation in the classroom and a reflection afterwards is also used in the teacher training courses carried out by KOMMS. 3 Concept of teacher training courses in the framework of KOMMS Teacher training courses in the framework of KOMMS are based on three main concepts: 1. Additional teacher training courses for the mathematical staff of single schools. 2. Training courses to be undertaken in the three modules: Implementation, mathematical topics and software. 3. Connection of theory (training courses) and practice (implementation to classes in school). To successfully educate teachers to become multipliers of modelling, training courses are not only offered for individual teachers from different schools, but also for the complete mathematical teaching staff of specific schools. From our experiences, this enhances the chances for actual implementation of modelling to the classroom of participating teachers as the communication with colleagues who have gone through the same learning is efficient and motivating. 113
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Editors: Katja Maaß Bärbel Barzel
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Conference Proceedings in Mathemati
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Contents 1 Conference: Educating th
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Inquiry based biology education in
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1 Conference: Educating the educato
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and at the same time, as profession
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In addition to exemplary, high qual
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2 Conference outcomes and conclusio
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One of the most significant develop
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2.5 Trends and needs in Europe to s
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necessary to delve even deeper into
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Strategies that effectively support
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• Common research question: In re
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3.2 DZLM - German Centre for Mathem
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4 Keynotes: Abstracts and Speaker I
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Teacher professional development in
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aspects of science education and ed
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EU countries (www.edumatics.eu). Al
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For several years, Manuela Welzel-B
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6 Papers 6.1 Track 1: Scaling-up wi
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3.1 The mathematical culture of cla
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literature and of a distributed fly
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proportional magnitudes - asking fo
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surface. Alternately two persons A
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subsets? Explain!” and “Write a
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Building capacity: developing a cou
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provision of professional developme
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teaching of specific concepts in sc
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• The use of repetition in some p
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Even, R. and Ball, D. eds., 2008, T
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the curriculum, or main part of it,
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its applicability, write a final th
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Page 67 and 68: “The meeting closed with a reques
Page 69 and 70: set-up. PRIMAS courses offered mult
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Page 73 and 74: 3.4 The facilitator Looking at the
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Page 77 and 78: a means of ensuring that students a
Page 79 and 80: environment and attitude that all w
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Page 83 and 84: 3.2.4 Feelings reported Teachers de
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Page 93 and 94: For this structure the Chair of Mat
Page 95 and 96: Acknowledgements This project has b
Page 97 and 98: Module 1: • Interpretation and de
Page 99 and 100: 3.3 Structure & Contents The whole
Page 101 and 102: One usual name for providers of suc
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Page 105 and 106: This could make the situation of mu
Page 107 and 108: used for continuous adaptation of d
Page 109 and 110: M2-idea was extended and combined w
Page 111 and 112: discussed, also representatives fro
Page 113 and 114: in B Herzog-Punzenberger (ed), Nati
Page 115: KOMMS: Teacher training courses fos
Page 119 and 120: smartphones in school is increasing
Page 121 and 122: References Blomhøj, M., Jensen, T.
Page 123 and 124: Figure 1: The model of implementati
Page 125 and 126: Therefore following crucial challen
Page 127 and 128: as useful. Majority (91 %) of teach
Page 129 and 130: How to determine what teachers shou
Page 131 and 132: 6.2 Track 2: Blended learning conce
Page 133 and 134: process, compared to the usual scho
Page 135 and 136: Figure 1: Elements of Activity 1 Th
Page 137 and 138: The supervision of the teacher educ
Page 139 and 140: Promoting teacher trainees’ inqui
Page 141 and 142: 1.3 Tablet support Furthermore, Cas
Page 143 and 144: Figure 3. Blended-learning concept
Page 145 and 146: Measuring the distance of each step
Page 147 and 148: Figure 4: Interaction graph of the
Page 149 and 150: Bell, T., Urhahne, D., Schanze, S.,
Page 151 and 152: Online training courses: Design mod
Page 153 and 154: about 1000 course participants, whi
Page 155 and 156: mandatory). This is followed by the
Page 157 and 158: Research project WOLF is about deve
Page 159 and 160: Positive aspects timely, content-re
Page 161 and 162: References Bruder, R. & Sonnberger,
Page 163 and 164: Figure 1: An activity from the Corn
Page 165 and 166: The PD process is illustrated in Fi
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and for students with English as an
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5 Future research The Cornerstone M
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A Virtual Mathematics Laboratory in
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informatics/IT). The synergy betwee
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than 70% would commit essential err
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computer, Theme of the month (Figur
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References Branzov, T., 2015, Viva
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Supporting the teacher role during
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worksheets and encouraged teachers
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Using Blended Learning in a Math Pe
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2.2 Blended learning and the HU uni
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At the beginning of the weekly cycl
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3.3 Evaluation and reflection The n
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4.2 A perspective on flexibility No
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4.3 Experience We are going to inco
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6.3 Track 3: Disseminating and scal
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processes and on important science
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Figure 1: Model for studying differ
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An IBL application from the inside
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3.1.2 The classroom management In t
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and what part it plays in the cover
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Figure 7 3.2.3.4 Task 4 At the fina
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Moreover, they pointed out that thi
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Appendix: The activity (Worksheet)
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f = 60 / 80cm Hole aperture Ø ≈0
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a) The image size in a camera obscu
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D(m) h(m) h(m) Image height when f
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pose questions, explore situations,
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Figure 1: Worksheet for the Parking
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mathematical domains and related to
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5 minutes 10 minutes 5 minutes 25 m
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groups, the work of individual stud
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(Retrieved at January 9 th , 2015 f
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equired to spend 440 hours (within
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Therefore the Steering Committee as
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References Michels, B., Kruger, J.
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7. Development of competencies 8. E
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classroom environment or not be ver
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4 Conclusions The project of raisin
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(Settlage 2007, p. 204), that is to
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were engaged in the process of defi
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emerging. The students are expected
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5.1.3 PTs’ design and implementat
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eflected on their experience and in
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a designer to the students. This is
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throughout their studies. The use o
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6.4 Track 4: Professional learning
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2.1 Objectives and areas of experti
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Teacher PD in the Czech Republic an
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• and last but not least the call
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3 Teacher training in the Czech Rep
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Samples of course topics for primar
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Professional qualification of teach
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Three practical work phases Phase 1
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All courses are evaluated by the DZ
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Competencies in Mathematics and Sci
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up to upper secondary level. The pr
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materials and interim reports are e
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Professional Learning Communities:
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presentation of professional develo
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and included participants from Spec
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In each intake participants exhibit
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Figure 1: The course content has be
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Figure 3: There have been sufficien
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Professional Learning Communities -
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Lesson study Observation classroom
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studying their response to particul
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Learning on three levels In the kno
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study as one form of clinical resea
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A study of collaboration between ma
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4 Towards an effective community of
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students and providing professional
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Figure 1: The structure of the JCU
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JCU learning community and (2) a gr
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At the moment this paper is written
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In 2014, 144 students were selected
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Van der Valk T., 2014b, “Connecti
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Figure 1: Lesson study cycle (adapt
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Work experience in Degree + extra i
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I learned that… from.. IMTPG Alan
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References Batanero, C., et al. (19
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Content focused peer coaching and t
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group simultaneously receives an in
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Wanzare, Z.O. (2007) The Transition
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Mathematics

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