# Open-ended modeling

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## the Math Alympiad

Summerschool Math Education 2012

Dédé de Haan & Monica Wijers d.dehaan@uu.nl m.wijers@uu.nl

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Guided reinvention

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Meaningful contexts, tools and models

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Math as human activity (Freudenthal, 1991)

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Intertwining learning strands

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Own productions and constructions

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meaningful contexts, tools and models: textbooks

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learning strands intertwined: textbooks

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guided reinvention: textbook / teacher

More difficult to integrate into the curriculum:

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Own productions and constructions, modeling, math as human activity

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Higher order thinking skills (HOTS)

August 2012 Summerschool - Open ended problems Math

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### Higher order thinking skills in Mathematics education

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.

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### How to assess these HOTS?

Higher order skills cannot be tested with relatively small tasks (like time constrained tests).

August 2012 Summerschool - Open ended problems Math

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Primary school:

Great Arithmetic Day

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Lower secondary school:

Mathematics day

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Upper secondary school:

Mathematics A-lympiad Mathematics B-day

One whole day mathematics as problem solving

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New theme every year

Often: Carrousel

### Great Arithmetic Day

August 2012 Summerschool - Open ended problems Math

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Use math in non-mathematical (current) situations

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Sense making mathematics

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Problem-solving and modeling

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Real-world problems

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Apply mathematics in real-world problems

August 2012 Summerschool - Open ended problems Math

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Problem-solving and modeling

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More oriented towards abstract math, less in

‘real life’

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Whole day

For teams

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Open ended problem

Different skills

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More than calculation

August 2012 Summerschool - Open ended problems Math

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### Focus of this workshop:

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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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

August 2012 Summerschool - Open ended problems Math

for a team

competition

open-ended

real world

different skills

structure

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### Schedule

9.30 – 12.00 work on assignment in small groups , including, 10.00 – 10.30 whole group clarifications on the assignment around 10.45 coffee, tea, etc.

12.00 – 12.30 plan and prepare posters 12.30 – 13.30 lunch

13.30 - 14.00 finish posters

14.00 – 15.00 discuss posters in small groups 15.00 – 15.15 summarizing; judging the work 15.15 – 15.30 short tea break

15.30 – 16.30 about designing

16.30 Summerschool drink

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### Start working on the assignment!

Check:

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?

53 seconds

or 63 s (if you include the time it remains on 7)

August 2012 Summerschool - Open ended problems Math

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4 groups of 4 teams each: 1-4; 5-8; 9-12; 13-16

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Hang posters on wall

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Present and discuss (30 minutes)

2 minutes 1st team presents poster

5 minutes others react and discuss

Repeat 3 times (4 in all)

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Reflect in own team (15 minutes)

What did you notice? differences and similarities

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General criteria with task

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Teachers’ own criteria (rubrics, e.g.)

August 2012 Summerschool - Open ended problems Math

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### Finals

August 2012 Summerschool - Open ended problems Math

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### Think about its implications for design.

August 2012 Summerschool - Open ended problems Math

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Accessible

Real/authentic situation; simple/simplified/modelled;

starting question; ….

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Use of mathematics

Modelling = possible; HOTS = needed; Math = not too explicit; different math strategies & representations can be used; ……..

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(inherent) Differentation

Different solutions and approaches; different levels of using math; creativity; constructions; …….

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Structure of the task

Start (closed) – body (analyse, use math) – end (create)

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August 2012 Summerschool - Open ended problems Math

Let’s play Darts

The assignment consists of four parts:

Part A: the game and the rules

In which you look at how a game may proceed using the existing rules

Part B: the throw

In which you look at whether the pattern of throws can be a measure for the quality of the player, and in which you determine your own level of playing

Part C: the numbers on the board

in which you find out who invented the distribution of the numbers on the board and for what reasons

Part D: final assignment

in which you design your own “children’s board”

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### Example of a context for design

August 2012 Summerschool - Open ended problems Math

http://www.london2012.co m/medals/medal-count/

Can you think of an

assignment based on the medal-count of the

Olympics 2012?

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New curricula 1987: mathematics A (and B)

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New content (domains & topics)

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Contexts and models

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Math A prepares for Economics, Sociology, Psychology

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Math B is the ‘old’ Math

August 2012 Summerschool - Open ended problems Math