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Blum , K. Houston and S. Carreira Ed. Assumptions and context: Pursuing their role in modelling activity. Establishing a community of practice in a secondary mathematics classroom. In Leone Burton Ed. Journal of Mathematics Teacher Education ,. Mathematics Education Research Journal , 30 1 : 7 - Galbraith, Peter Out of the frying pan: into the fire of post-global financial crisis GFC university management. Higher Education Policy , 26 4 : - Galbraith, Peter Scoring points: goals for real world problem solving.

Australian Senior Mathematics Journal , 26 2 : 51 - Galbraith, Peter Models of modelling: genres, purposes or perspectives. Journal of Mathematical Modelling and Application , 1 5 : 3 - Australian Senior Mathematics Journal , 24 2 : 29 - Stillman, Gloria , Brown, Jill and Galbraith, Peter Researching applications and mathematical modelling in mathematics learning and teaching. Mathematics Education Research Journal , 22 2 : 1 - 6. Galbraith, Peter System dynamics: A lens and scalpel for organisational decision making.

OR Insight , 23 2 : 96 - Galbraith, Peter and Stillman, Gloria A framework for identifying student blockages during transitions in the modelling process. Galbraith P. Journal of Educational Administration , 42 1 : 9 - Australian Senior Mathematics Journal , 17 1 : 62 - Cretchley, P. New Zealand Journal of Mathematics , 32 Supplementary : 37 - Goos, Merrilyn , Galbraith, Peter , Renshaw, Peter and Geiger, Vince Perspectives on technology mediated learning in secondary school mathematics classrooms.

Journal of Mathematical Behavior , 22 1 : 73 - Goos, Merrilyn , Galbraith, Peter and Renshaw, Peter D Socially mediated metacognition: Creating collaborative zones of proximal development in small group problem solving. Educational Studies in Mathematics , 49 2 : - Mathematics Education Research Journal , 12 3 : - Burghes, D. Pemberton, M. Australian Senior Mathematics Journal , 14 2 : 4 - Perspectives on Educational Leadership , 10 5 : 1 - 2.

Higher Education Policy , 12 2 : - Australian Senior Mathematics Journal , 12 2 : 20 - International Journal of Science Education , 19 4 : - Goos M. Educational Studies in Mathematics , 30 3 : - Clatworthy, Neville J. Teaching Mathematics and its Applications , 10 1 : 6 - Teaching Mathematics and its Applications , 6 2 : 55 - Educational Studies in Mathematics , 12 1 : 1 - Students and real world applications: still a challenging mix.

Mathematics Education: Yesterday, Today and Tomorrow. Modelling as real world problem solving: Translating rhetoric into action.

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Models of modelling: Is there a first among equals? Softly, softly: Curriculum change in applications and modelling in the senior secondary curriculum in Queensland. In: R. Hunter , B. Bicknell and T. Applications and modeling in mathematics education: Progress to celebrate - so much more to do. In: Niss, M. Some thoughts around disciplinary mathematics for teaching. A framework for success in implementing mathematical modelling in the secondary classroom. In: Watson, J. In: C. Galbrait , W. Mathematics for teaching: Some issues, some reflections.

In: M. Bulmer , H. MacGillivray and C. Varsavsky , Proceedings of Kingfisher Delta ' Applications and mathematical modeling: Meeting the challenge. In: S. Koen , A. Maclean and K. Geiger, V. Choosing and using technology for secondary mathematical modelling tasks: Choosing the right peg for the right hole. From description to analysis in technology aided teaching and learning: A contribution from Zone Theory. In: L. Bragg , C. Campbell , G.

Herbert and J. In: J. Gimenez , G. FitzSimons and C. Matching the hatch: Students' choices and preferences in relation to handheld technologies and learning mathematics. In: B. Barton , K. Irwin , M. Pfannkuch and M. Thomas , Mathematics Education in the South Pacific. Convergence or divergence? Students, maple, and mathematics learning.

Mathematics or computers? Confidence or motivation? How do these relate to achievement? Metaphor or model? Muddying the terrain of learning organisations. In: P.

Ledington and J. Digging beneath the surface: When manipulators, mathematics, and students mix. Computers, mathematics, and undergraduates: What is going on? Bobis , B. Perry and M. Mitchelmore , Numeracy and Beyond.

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Integrating technology in mathematics learning: What some students say. Promoting collaborative inquiry in technology enriched mathematics classrooms. Teaching applications in technology enriched mathematics classrooms. Research-informed changes: Can research really reflect and hence inform the reality of the classroom? In: E. Efhymiades , Mathematics: What Changes? Mathematical modelling as pedagogy: Impact of an immersion program. The thinking-talking-technology connection.

In: 9th International Congress on Mathematical Education. ICME 9 , Tokyo , Swings and roundabouts: Riding the punches of university management dilemmas. Modelling cycle We describe the modelling process by a simple version of the modelling cycle. We start with a problem, which is to be solved using tools from mathematics. In the first stage the problem is described in terms of relevant non-mathematical concepts. During this stage one typically has to make some choices about simplifying assumptions.


The result of this stage is a conceptual model. This conceptual model is then translated into a mathematical model, which can be analyzed mathematically. The actual translation of the conceptual model and the original question into mathematics may also be subject to certain choices. Next, the mathematical solution is translated back into the context and language of the original problem. We call this interpretation. Finally, one validates the solution. If necessary, one starts the modelling cycle all over again, adapting one or more of the steps.

Role of mathematics in modelling The role of mathematics in modelling can vary considerably. It can be elementary or advanced. Sometimes computers are needed to aid mathematical analysis. The mathematics may involve calculus, algebra, geometry, combinatorics or some other field. The modelling problem can be well-defined with clear-cut data, a specific question, a standard mathematical model and ditto solution.

In such problems mathematics and context science merge into a very potent mixture. The interplay between mathematics and context is then especially fruitful with techniques like dimensional analysis, where mathematical algebra is applied to physical units. Conversely, a physical concept like velocity can be helpful to learn a mathematical concept like the derivative. For such models validation is a main point of concern. Most models are not built up from scratch anyway, but emerge as refinements and combinations of existing models.

Applications of mathematics in maths education Mathematics started as an applied science, dealing with practical problems in trading, measurement, navigation, etcetera. The separation of theoretical mathematics from the empirical sciences is a relatively recent phenomenon, brought about by the development of non-euclidean geometry around In the middle of the nineteenth century mathematical education followed this trend and its focus shifted from applications to logical reasoning.

They have a blind spot for applied mathematics and the role of mathematics in the sciences or daily life. If students never learn how to apply mathematics, then their mathematical knowledge is indeed useless. Furthermore, it is counterproductive if common sense, intuition and reality are not used to aid mathematical understanding. Modelling and the Dutch mathematics teaching programs Non-mathematical contexts have played an important role in parts of Dutch mathematics education since Since all mathematics programs for secondary education involve modelling.

The experiment which preceded the introduction of the new program indicated that assessment of open modelling tasks was a major problem and was avoided by many teachers. The modelling tasks in the national exams, too, paid little attention to conceptualization, interpretation and validation De Lange, To counteract this deficit, the Freudenthal Institute in Utrecht started organizing modelling competitions for schools where these aspects do play an essential role.

All these efforts have partially paid off: PISA shows that Dutch students perform well on modelling related tasks.

Teacher’s views on students’ cognitive modelling competencies

Students tend to neglect relevant concepts and work by trial and error. Sins also laments the lack of conceptual thinking and understanding of the purpose of modelling. Future maths education should address these weaknesses more effectively. Since many maths teachers in upper secondary education have only scant knowledge of applications of mathematics, post graduate courses for teachers should fill this gap. The essence of this framework is as follows. Anyone who takes up a complex task like mathematical modelling starts with certain knowledge not only mathematical knowledge like facts, algorithms, skills, heuristics, but also domain knowledge , aims and attitudes opinions, prejudices, preferences.

During the execution of the plan one monitors the progress on several levels, going back and forth between the stages of the modelling cycle. Metacognition thus plays an important role in modelling. We address the issues of aims and attitudes in the sections Goals, Authenticity, Dispositions and Epistemological understanding.

Knowledge aspects are dealt with in the sections Domain knowledge, Authenticity, and Computer modeling. We conclude with a discussion of decisions and monitoring in Monitoring and Assessment. It takes a lot of time and is difficult to assess Galbraith, a and Vos, So why should we take up modelling in mathematics education? First, students have to learn how to apply mathematics, to prepare them for their further education and their jobs, as well as for everyday life.

It might improve their understanding of mathematics as well. Second, modelling shows that mathematics is useful to scientists as well as practical problems solvers. He describes an experiment where two different tasks are distributed randomly among Swedish school children. Mathematically, the tasks are identical: to determine how many busses are needed if students have to be transported and each bus can hold 48 students.

Authenticity is also beneficial for motivating students. Dispositions about modelling Abstraction and generalization belong to the core business of mathematicians.

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Model building, on the other hand, depends critically on the characteristics of the context and the specific research question. In the minds of many students and teachers there is no connection between the subjects taught during maths class and the topics taught next door by the physics or economics teacher. We are not just talking about superficial problems like different notations, conventions or terminology, but also about deeply rooted opinions about mathematics and reality.

Students even think that using non-mathematical knowledge is forbidden Bonotto, As Schwarzkopf , put it: The students do not follow the logic of problem solving, but they follow the logic of classroom culture. This obviously impedes successful modelling in teaching of mathematics. Understanding what modelling is about is strongly related to dispositions about modelling Sins, He distinguishes between three levels. At the lowest level a model is considered a copy of reality. Students at the intermediate level understand that models are simplified representations of reality constructed with a specific goal.

Different goals may lead to different models. At the highest level attention shifts towards theory building: Models are constructed to develop and test ideas. Sins experiments show that a higher level of epistemological understanding leads to better models. Students at the highest level use their domain knowledge to analyze the relevant variables and the relations between them.

Most students, however, are at the middle level. They ignore domain knowledge, reason superficially and consequently produce poor models. Epistemological understanding Sins investigated the influence of epistemological understanding of modelling on the quality of models made by students. He advizes to make the goals of a modelling task explicit: what do we want to understand or which problem do we want to solve?

He proposes that the teacher presents reasonable models to his students who have to analyze and improve them. This way students learn about the tentative nature of models: They are not perfect copies of reality, since they often depend on choices, approximations and incomplete information. Furthermore, this adjusting of existing models and iteration of the modelling cycle gives a fairer picture of the modelling process as performed by experts, who of course have lots of standard models at their disposal and rarely start from scratch. It is not sufficient to just talk about modelling with students.

However, even if students have a sound epistemological understanding of modelling, in very open modelling tasks they still do not always understand what is given, what is asked and how to attack the problem. Domain knowledge Modelling typically concerns extra-mathematical contexts.

As a consequence, the maths teacher may find himself in an awkward position, since he cannot be an expert in all possible modelling domains, such as the natural sciences, computer science, economics, arts, sports or other specific not necessarily scientific contexts.

Professor Peter Galbraith - School of Education - University of Queensland

The same holds for students. We know, however, that lack of domain knowledge leads to poor models Sins, Furthermore, the teacher has to encourage the students to actually use their domain knowledge. Finally, the teacher has to be familiar with the modelling problem himself. In particular, he has to be aware that a problem can lead to several different models. Computer modelling Computers can be useful to in modelling, especially when the mathematics gets complicated. Using a graphic modelling tool it is easy to modify a model, run simulations and display the results graphically.

The representation of a model in such a tool reminds one of a concept map in the sense that it indicates the relevant variables and the relations between them. It facilitates exploring the limits of validity of a model. Simulation results may lead students to new research questions. Computer modelling is challenging and motivating for students, as long as the models are not too complicated and the software is easy to use. It also helps to turn abstract, theoretical models into something more concrete, which makes it easier to discuss these models.

Finally, experimenting using computer modelling helps students to understand and remember the phenomena and associated theory.

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  • Monitoring Monitoring the modelling process of a group of students can be very difficult. Different students make different and often implicit assumptions and simplifications, have different goals and use different data and notations. This makes monitoring the modelling process of a group of students very difficult if not virtually impossible Doerr, It is thus very important to force students to make all of the above explicit.

    The teacher can make life easier by inserting go-or-no-go-moments at certain points of the modelling cycle. However, even if everything is written down neatly, it can still be difficult for teachers and students to compare different modelling results. Are the differences due to different conceptualization or to mathematical errors? This problem can be moderated by discussing and comparing the various conceptual models with the whole group.

    Monitoring becomes much simpler if consensus is reached about the data, the goal and notations. This also facilitates understanding and comparing the different results, which in turn improves motivation and understanding Van Rens, ; Bonotto, If modelling is new to students it is advisable to have them record their modelling process in a pre-structured log. In this log they have to describe all data, assumptions, etcetera.