1. Open-ended modeling tasks in Dutch math & science education Summerschool 2014 Dédé de Haan & Monica Wijers d.dehaan@uu.nlm.wijers@uu.nl

2. Wednesday you used this model August 2015Summerschool - Open ended problems2

3. Today Experience working on an ‘inquiry based’ RME task for a length of time. August 2015Summerschool - Open ended problems3

4. Realistic Mathematics Education (RME) Guided reinvention Meaningful contexts, tools and models Math as human activity (Freudenthal, 1991) Intertwining learning strands Own productions and constructions Question: how does this relate to inquiry based learning? August 2015Summerschool - Open ended problems4

5. 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 2015Summerschool - Open ended problems5

6. Higher order thinking skills (21 st century) in Mathematics education August 2015Summerschool - Open ended problems6  Process skills  Modeling  Problem solving  Communicating  Creativity  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.

7. Dutch solution – since 1989 One whole day mathematics as ‘problem solving’, as a competition for teams Lower secondary school: Mathematics day Upper secondary school: Mathematics A-lympiad Mathematics B-day Mathematics A-lympiad Mathematics B-day August 2015Summerschool - Open ended problems7

8. Assignment for today You work in teams the whole morning and present the process and your results on a poster in the afternoon August 2015Summerschool - Open ended problems8

9. Schedule 9:15 – 12:00 work on assignment in small groups 12:00 – 12:30 finish and discuss posters 12.30 – 13.30 lunch 13:30 – 14:15 heuristics in solving 14:15 – 15:00 characteristics of tasks 15.00 – 15.15 short tea break 15:15 – 16:15 judging tasks, summarizing and closure August 2015Summerschool - Open ended problems9

10. Assignment: Elevators Elevators (Math A-lympiad): design an “elevator-schedule” for the 6 elevators in an office-building with 20 floors and 1200 employees that brings everybody to his/her workplace as quickly as possible. August 2015Summerschool - Open ended problems10

11. Start working on the assignment! Work in groups for about 30 minutes. Quick round for questions. How are you doing? Problems, questions? August 2015Summerschool - Open ended problems11

12. 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 2015Summerschool - Open ended problems12

13. On the poster Solution (partial solution is okay) Models; Representations; etc. Assumptions Solution process (maybe) Walk around look at the posters and note the representations! August 2015Summerschool - Open ended problems13

14. LUNCH BREAK August 2015Summerschool - Open ended problems14

15. Other problems: Fish and Peas Security in the museum Breaks HOW DO YOU START? August 2015Summerschool - Open ended problems15

16. Instructions For each task appoint one observer Read the task for yourself for max 5 minutes (do not start solving!!) Start thinking-aloud – in pairs The observer takes literal notes Do this for 10 minutes Change roles for every task August 2015Summerschool - Open ended problems16

17. Looking for heuristics New groups: all observers per task Compare notes Look for common strategies/heuristics Present two results! August 2015Summerschool - Open ended problems17

18. Results from group Museum Cameras 360 degrees – not 180 Where first camera Shading picture Fish Working through formulas – what are differences How many to catch & How often Do experiment Breaks Confused – language Graph not used by all groups Later: focused on optimizing August 2015Summerschool - Open ended problems18

19. Heuristics Polya (1945) “How to solve it” 1) Understanding the problem 2) Devise a plan: heuristics!* 3) Carry out the plan 4) Look back * Possible heuristics: next slide! August 2015Summerschool - Open ended problems19

20. Possible heuristics: - Could you imagine a more accessible related problem? - A more general problem? - A more specific problem? - An analogous problem? - Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? - Could you derive something useful from the data? August 2015Summerschool - Open ended problems20

21. REFLECTION task – student - teacher August 2015Summerschool - Open ended problems21

22. Relation HOTS – task design Dan Meyer, three-act tasks: “ Act one: Introduce the central conflict of your story/task clearly, visually, viscerally, using as few words as possible. Leave no one out of your first act. Your first act should impose as few demands on the students as possible — either of language or of math. It should ask for little and offer a lot.” Check if this is the case for our problems! August 2015Summerschool - Open ended problems22

23. Task characteristics (specific) August 2015Summerschool - Open ended problems23 Accessibility Use of mathematics (Natural) differentiation Open ended problem with a structure: Start (closed) – body (analyse, use) – end (create) Different skills More than basic math For teams Whole day

24. The student In one word write on a sticky note how you felt during the activity = ‘working on the task’. Were feelings mixed in your group? What do you expect how your students would feel? August 2015Summerschool - Open ended problems24

25. The teacher How did we (Monica and Dede) act as teachers? What would you as a teacher need to know/do/… in order to implement this type of activities in your classroom? August 2015Summerschool - Open ended problems25

26. HOW TO ‘JUDGE’? August 2015Summerschool - Open ended problems26

27. How to judge the products? Your suggestions: Correct calculations Quality of reasoning Clarity of describing the process Clarity of restating the problem Use of a model, use of representations Lay-out Our experiences ……….. August 2015Summerschool - Open ended problems27

28. Judging the work General criteria with task Teachers’ own criteria (rubrics, e.g.) August 2015Summerschool - Open ended problems28

29. Different way of judging Ordering/ranking (4x) Combined ranks August 2015Summerschool - Open ended problems29

30. Finals August 2015Summerschool - Open ended problems30

31. Thank you! August 2015Summerschool - Open ended problems31 http://www.fisme.science.uu.nl/alympiade/en/welcome.html

32. Security in museum Security is an art in itself (Mathematics Day/Mathematics A-lympiad). A museum for modern art organizes a large exhibition. The organization has to deal with problems concerning security. Your team has to figure out the best way (economically and artistically) to protect/survey the exhibition, using camera’s and removable walls. August 2015Summerschool - Open ended problems32