1. Mathematical Modelling in the Dutch National Curriculum Bas Friesen Lorentz Casimir Lyceum Eindhoven, Netherlands 25 May 2016

2. Contents * The Dutch (mathematics) curriculum. * Integration of mathematical modelling into the curriculum and exams, backgrounds. * Overview of Dutch one-day modelling challenges. * Recent assignments from these contests. * Short modelling exercises in the national exams.

3. Dutch Curriculum Grade 1,2,3 (common curriculum): - Mathematics - Science: biology, chemistry, physics - Modern languages: Dutch, English, French, German - Classical languages: Greek, Latin (optional) - Society subjects: geography, history, economics - Arts: design, drawing, music - Sports Grade 4,5,6 (four study profiles): - Culture & Society emphasis on languages, history, artsmath course C - Economics & Society emphasis on economics, history, geographymath course A - Nature & Health emphasis on biology, chemistry, mathmath course B - Nature & Technology emphasis on physics, science, mathmath course BD

4. Dutch Mathematics Curriculum - General competences ABCD - AlgebraABCD - Differential calculus ABCD - Integral calculusBD - TrigonometryBD - Formal geometry in the planeBD - Analytic geometry in 3-spaceBD - ProbabilityACD - StatisticsACD - Complex analysisD - Discrete analysisD - Differential equationsD

5. Dutch Mathematics Curriculum General competences: * Information competences …… * Modelling & research competences Students are able to examine a problem, translate the problem into a mathematical model, use mathematical techniques within that model and examine the solution within the context. * ICT competences …… * Communication competences ……

6. Students are able to: examine and connect data, statements and results in a text, evaluate, separate relevant and less relevant information. connect the data with the formulation of the problem, choose a suitable approach, subdivide into separate tasks. transform the data provided in a text into a suitable mathematical model. determine whether a chosen model is adequate and if necessary adjust. determine whether additional data are needed and if so, which data. determine to what extent a model needs to be adjusted if data change. use suitable mathematical techniques within the model correctly. examine the solutions within the context as well as the accuracy of the solutions. reflect on the choices made for representation, used methods, process to the solution and the outcome and describe these reflections.

7. Mathematical modelling as part of the exam - In most schools a one-day mathematical modelling challenge is part of the school exams. - The national final exams for mathematics contain “quick” modelling exercises. One-day modelling challenges Dutch modelling contests: - Wiskunde Alympiade (Mathematics Alympiad) - Wiskunde B-dag (Mathematics B-day) - Onderbouwwiskundedag (Junior Mathematics Day) organiser: Freudenthal Institute, Utrecht University, Netherlands

8. Assignment Junior Mathematics Day Tennis balls (2016): Design a paper packaging with as little paper as possible packaging ten tennis balls as efficiently as possible. The less space used the better! * Groups of four students (2nd and 3rd grade). * Students work for 7 hours on the problem. * A few entry exercises (max. 1 hour). * Written report includes description of the problem, assumptions, calculations, drafts, pictures, clear description of work method, conclusion/solution. * Some guidance in the process (time management, cooperation, presentation) by the teachers, no mathematical guidance during competition. * For this specific exercise: no internet access.

9. National Junior Mathematics Day 2016 (all pictures from students reports) 4 balls: 7 balls: +3 balls: 10 balls:

10. Assignments Mathematics B-day Around the corner (2015): The assignment is centred around the question whether certain objects can be moved through a 1 meter wide corridor with a right-angled corner in it. Which mathematical shapes (line segment, rectangle, circle, …) can be moved through the corridor? Lights Out (2014): The assignment is about the game “Lights Out”. The game is played on a grid of 5 rows by 5 columns with a total of 25 lights. Some of the lights are switched on at the start of the game. The point of the game is to switch of all the lights. The final move (2011): The assignment is all about a game for two players. For example: there are 21 matches on the table. Each turn, a player must remove 1, 2 or 3 matches on the table. The player who removes the last match loses. What is the best strategy? Crossing a desert in a jeep (2001): You are planning a trip through the desert. Your jeep can carry fuel for only 1000 km, and the next petrol station is 3000 km ahead. What is the best way to get across? Try to develop a strategy that works for the final problem.

11. Example of a modelling excercise in the national exam

12. Bas Friesen (friesen.b@lcl.nl)friesen.b@lcl.nl Lorentz Casimir Lyceum:http://lcl.nl (Dutch)http://lcl.nl Mathematics A-lympiad:http://uu.nl/onderwijs/wiskunde-a-lympiade (Dutch)http://uu.nl/onderwijs/wiskunde-a-lympiade Mathematics B-day: http://uu.nl/onderwijs/wiskunde-b-dag (English)http://uu.nl/onderwijs/wiskunde-b-dag Junior Mathematics Day:http://uu.nl/onderwijs/onderbouwwiskundedag (Dutch)http://uu.nl/onderwijs/onderbouwwiskundedag Dutch final exams for Mathematics B: http://alleexamens.nl/alleexamens/vwo/?Vak=Wiskunde+Bhttp://alleexamens.nl/alleexamens/vwo/?Vak=Wiskunde+B (Dutch)