1. Open-ended modeling tasks in Dutch math education Summerschool Math Education 2013 Dédé de Haan & Monica Wijers d.dehaan@uu.nlm.wijers@uu.nl

2. Realistic Mathematics Education (RME) Guided reinvention Meaningful contexts, tools and models Math as human activity (Freudenthal, 1991) Intertwining learning strands Own productions and constructions August 2013Summerschool - Open ended problems2

3. How is RME integrated in Dutch math education? meaningful contexts, tools and models: textbooks learning strands intertwined: textbooks guided reinvention: textbook / teacher More difficult to integrate into the curriculum: Own productions and constructions, modeling, math as human activity Higher order thinking skills (HOTS) August 2013Summerschool - Open ended problems3

4. Higher order thinking skills in Mathematics education August 2013Summerschool - Open ended problems4  Process skills  Modelling  Problem solving  Communicating  Information assessment skills Assessing the reliability of numerical information and its relevance for the solving of a problem  Research competencies Constructing and assessing of a model based on given information, adjusting the model after testing it.

5. Examples for practicing and assessing HOTS in RME in secondary school Lower secondary school: Mathematics day Upper secondary school: Mathematics A-lympiad Mathematics B-day Mathematics A-lympiad Mathematics B-day One whole day mathematics as problem solving August 2013Summerschool - Open ended problems5

6. Assignments 3 different assignments (Mathematics day, Mathematics A-lympiad, Mathematics B-day) work in teams during the day and present the process and your results on a poster at the end of the day August 2013Summerschool - Open ended problems6

7. Schedule 9.15 – 12.15work on assignment in small groups, including, 10.00 – 10.30whole group clarifications on the assignment around 10.45coffee, tea, etc. 12.15 – 12.30 plan posters 12.30 – 13.30lunch 13.30 – 14.00finish posters 14.00 – 14.30discuss posters in small groups (same problem) 14:30 – 15:00 discuss task characteristics in mixed groups 15.00 – 15.30tea break 15.30 – 16.30summarizing 16.30 - 17.00questions & closure August 2013Summerschool - Open ended problems7

8. Start working on the assignment! Check for the assignment on Elevators: How many seconds does it take the elevator to start at floor 1; stop at 3 and 4 and run to 7 and stop there? Answer: 53 seconds or 63 s (if you include the time it remains on 7) August 2013Summerschool - Open ended problems8

9. Discussing posters 4 groups of 3-5 teams each (with the same assignment) Hang posters on wall Present and discuss (30 minutes) Max. 2 minutes 1st team presents poster Max. 4 minutes others react and discuss Repeat Reflect in mixed team (30 minutes) What did you notice? See: questions. Characteristics: Differences and similarities August 2013Summerschool - Open ended problems9

10. Three assignments today Elevators (Math A-lympiad): design an “elevator-system” using 6 elevators in an office- building with 20 floors and 1200 employees that brings everybody to his/her workplace as soon as possible Paper folding (Math B-day):repeatedly fold two strips of paper in a prescribed manner. Sharp folds will result in interesting geometrical figures, that consists of straight parts that bend at a right angle on every folding line. The goal is to explain, and predict, the patterns. August 2013Summerschool - Open ended problems10

11. Security is an art in itself (Mathematics Day/Mathematics A-lympiad). A museum for modern art will organize a big exhibition with pieces of a number of great artists in the field of modern art. The organization is busy with preparations for the exposition, and they meet with difficulties concerning security. Your team has to figure out the best way (economically and artistically) to secure the exhibition (using camera’s and flexible walls August 2013Summerschool - Open ended problems11

12. Tea-break August 2013Summerschool - Open ended problems12

13. Summarizing What were the three assignments about? What are their characteristics (similarities, differences)? How to judge the products August 2013Summerschool - Open ended problems13

14. Characteristics Whole day contest For teams At school Open ended problem Different skills More than calculation Structure of task Start (closed) – body (analyse, use math) – end (create) August 2013Summerschool - Open ended problems14

15. Task characteristics, similar and different August 2013Summerschool - Open ended problems15 Accessibility Math-Alympiad/Mathematics Day: real/authentic situation simple/simplified/modelled starting question Math B-day: interesting situation (from mathematics) simple/simplified/modelled more starting questions

16. Task characteristics, similar and different August 2013Summerschool - Open ended problems16 Use of mathematics Math-Alympiad/Mathematics Day: math = not very explicit different strategies & representations can be used meaningful use of basic mathematics Math B-day: math explicit (= context) representations (often) already chosen develop/re-invent new math concepts/skills

17. Task characteristics, similar and different August 2013Summerschool - Open ended problems17 HOTS Math-Alympiad/Mathematics Day: modelling/representing reasoning – making assumptions; context provides conflicting constraints communicating – report (in context) Math B-day: mathematical modelling reasoning – prove; generalize; create/re- invent/discover/use new math own constructions

18. August 2013Summerschool - Open ended problems18 (inherent) Differentation Math-Alympiad/Mathematics Day: different solutions and approaches (open ended) different levels of using math justification creativity Math B-day open ended, but not as open ended as the A-lympiad different levels of mathematical reasoning research creativity Task characteristics, similar and different

19. DICUSSION AND PRACTICAL INFO August 2013Summerschool - Open ended problems19

20. Judging the work General criteria with task Teachers’ own criteria (rubrics, e.g.) August 2013Summerschool - Open ended problems20

21. Different from other mathematics-tasks Ordering/ranking (4x) Combined ranks August 2013Summerschool - Open ended problems21

22. Finals August 2013Summerschool - Open ended problems22

23. Thank you! August 2013Summerschool - Open ended problems23 http://www.fisme.science.uu.nl/alympiade/en/welcome.html http://www.fi.uu.nl/nl/onderbouwwiskundedag/ http://www.fi.uu.nl/wisbdag/

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25. Use math in non-mathematical (current) situations Sense making mathematics Mathematics Day for lower secondary August 2013Summerschool - Open ended problems25

26. Mathematics A-lympiad for upper secondary high school Problem-solving and modeling Real-world problems Apply mathematics in real-world problems August 2013Summerschool - Open ended problems26

27. Mathematics B-day for upper secondary high school Problem-solving and modeling More oriented towards abstract math, less in ‘real life’ August 2013Summerschool - Open ended problems27

28. Example of a context for design August 2013Summerschool - Open ended problems28

29. Example of a context for design August 2013Summerschool - Open ended problems29 http://www.london2012.co m/medals/medal-count/ Can you think of an assignment based on the medal-count of the Olympics 2012?

30. Focus of this workshop: Mathematics A-lympiad The real world mathematics team competition with open ended assignments for upper secondary students (aged 16-18) in mathematics A, since 1989mathematics A August 2013Summerschool - Open ended problems30

31. Task characteristics planning and scheduling elevators, trains, christmas buffet, tournaments decision making: define and weigh factors credit cards, designing roads analysis and design of a ‘system’ darts, pausing schemes (http://www.fi.uu.nl/alympiade/en/welcome.html) August 2013Summerschool - Open ended problems31 for a team competition Types of tasks open-ended real world different skills structure

32. RME in upper secondary school New curricula 1987: mathematics A (and B) New content (domains & topics) Contexts and models Math A prepares for Economics, Sociology, Psychology Math A prepares for Economics, Sociology, Psychology Math B is the ‘old’ Math August 2013Summerschool - Open ended problems32