Open-ended modeling tasks in Dutch math education

Summerschool Math Education 2013Dédé de Haan & Monica Wijersd.dehaan@uu.nl m.wijers@uu.nl

Realistic Mathematics Education (RME)

• Guided reinvention• Meaningful contexts, tools and models• Math as human activity (Freudenthal, 1991)• Intertwining learning strands• Own productions and constructions

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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)

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Higher order thinking skills in Mathematics education

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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.

Examples for practicing and assessing HOTS in RME in secondary school

• Lower secondary school:Mathematics day

• Upper secondary school:Mathematics A-lympiadMathematics B-day

One whole day mathematics as problem solving

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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

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Schedule

9.15 – 12.15 work on assignment in small groups , including,

10.00 – 10.30 whole group clarifications on the assignmentaround 10.45 coffee, tea, etc.12.15 – 12.30 plan posters 12.30 – 13.30 lunch13.30 – 14.00 finish posters14.00 – 14.30 discuss posters in small groups (same

problem)14:30 – 15:00 discuss task characteristics in mixed groups15.00 – 15.30 tea break15.30 – 16.30 summarizing 16.30 - 17.00 questions & closure

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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)

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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

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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.

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• 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

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Tea-break

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Summarizing

• What were the three assignments about?• What are their characteristics (similarities,

differences)?• How to judge the products

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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)

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Task characteristics, similar and different

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Accessibility• Math-Alympiad/Mathematics Day:

real/authentic situationsimple/simplified/modelled starting question

• Math B-day: interesting situation (from mathematics)simple/simplified/modelledmore starting questions

Task characteristics, similar and different

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Use of mathematics• Math-Alympiad/Mathematics Day:

math = not very explicitdifferent strategies & representations can be used meaningful use of basic mathematics

• Math B-day: math explicit (= context)representations (often) already chosendevelop/re-invent new math concepts/skills

Task characteristics, similar and different

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HOTS• Math-Alympiad/Mathematics Day:

modelling/representingreasoning – making assumptions; context provides conflicting constraintscommunicating – report (in context)

• Math B-day: mathematical modelling reasoning – prove; generalize; create/re-invent/discover/use new mathown constructions

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(inherent) Differentation

• Math-Alympiad/Mathematics Day:different solutions and approaches (open ended)different levels of using mathjustificationcreativity

• Math B-dayopen ended, but not as open ended as the A-lympiad different levels of mathematical reasoningresearch creativity

Task characteristics, similar and different

Dicussion and practical info

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Judging the work• General criteria with task• Teachers’ own criteria (rubrics, e.g.)

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Different from other mathematics-tasks

• Ordering/ranking (4x)

• Combined ranks

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Finals

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Thank you!

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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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• Use math in non-mathematical (current) situations

• Sense making mathematics

Mathematics Day for lower secondary

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Mathematics A-lympiad for upper secondary high school

• Problem-solving and modeling• Real-world problems• Apply mathematics in real-world problems

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Mathematics B-day for upper secondary high school

• Problem-solving and modeling• More oriented towards abstract math, less in

‘real life’

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Example of a context for design

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Example of a context for design

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http://www.london2012.com/medals/medal-count/Can you think of an assignment based on the medal-count of the Olympics 2012?

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 1989

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Task characteristics

planning and schedulingelevators, trains, christmas buffet, tournaments

decision making: define and weigh factorscredit cards, designing roads

analysis and design of a ‘system’darts, pausing schemes

(http://www.fi.uu.nl/alympiade/en/welcome.html)

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• for a team• competition

Types of tasks

• open-ended• real world

• different skills• structure

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 B is the ‘old’ Math

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