## Open-ended modeling tasks in Dutch math

## education:

## the Math Alympiad

Summerschool Math Education 2012

Dédé de Haan & Monica Wijers d.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### How is RME integrated in Dutch math education?

### •

meaningful contexts, tools and models: textbooks### •

learning strands intertwined: textbooks### •

guided reinvention: textbook / teacherMore difficult to integrate into the curriculum:

### •

Own productions and constructions, modeling, math as human activity### •

Higher order thinking skills (HOTS)August 2012 Summerschool - Open ended problems Math

Alympiad 3

### 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.

### 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

Alympiad 5

### Examples for practicing and assessing HOTS in RME

### •

Primary school:Great Arithmetic Day

### •

Lower secondary school:Mathematics day

### •

Upper secondary school:Mathematics A-lympiad Mathematics B-day

One whole day mathematics as problem solving

### •

New theme every year### •

Often: Carrousel### Great Arithmetic Day

August 2012 Summerschool - Open ended problems Math

Alympiad 7

### •

Use math in non-mathematical (current) situations### •

Sense making mathematics### Mathematics Day for lower secondary

### Mathematics A-lympiad

### for upper secondary high school

### •

Problem-solving and modeling### •

Real-world problems### •

Apply mathematics in real-world problemsAugust 2012 Summerschool - Open ended problems Math

Alympiad 9

### Mathematics B-day

### for upper secondary high school

### •

Problem-solving and modeling### •

More oriented towards abstract math, less in‘real life’

### Characteristics

### •

Whole day### •

^{For teams }

### •

Open ended problem### •

Different skills### •

More than calculationAugust 2012 Summerschool - Open ended problems Math

Alympiad 11

### 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

### 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 2012 Summerschool - Open ended problems Math

Alympiad 13

### •

for a team### •

*competition*

### Types of tasks

### •

open-ended### •

*real world*

### •

different skills### •

^{structure }### 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

### 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?

Answer:

53 seconds

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

August 2012 Summerschool - Open ended problems Math

Alympiad 15

### Discussing posters

### •

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

Hang posters on wall### •

Present and discuss (30 minutes)• 2 minutes 1st team presents poster

• 5 minutes others react and discuss

• Repeat 3 times (4 in all)

### •

Reflect in own team (15 minutes)• What did you notice? differences and similarities

### Judging the work

### •

General criteria with task### •

Teachers’ own criteria (rubrics, e.g.)August 2012 Summerschool - Open ended problems Math

Alympiad 17

### Different from other mathematics-tasks

### • Ordering/ranking (4x)

### • Combined ranks

### Finals

August 2012 Summerschool - Open ended problems Math

Alympiad 19

### Short tea-break

### Formulate and write down 1

### characteristic of this type of task.

### Think about its implications for design.

August 2012 Summerschool - Open ended problems Math

Alympiad 21

### Task characteristics & implications

### •

Accessible• Real/authentic situation; simple/simplified/modelled;

starting question; ….

### •

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

### •

(inherent) Differentation• Different solutions and approaches; different levels of using math; creativity; constructions; …….

### •

Structure of the task• Start (closed) – body (analyse, use math) – end (create)

August 2012 Summerschool - Open ended problems Math

Alympiad 23

**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”

### Example of a context for design

### Example of a context for design

August 2012 Summerschool - Open ended problems Math

Alympiad 25

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

Can you think of an

assignment based on the medal-count of the

Olympics 2012?

### Thank you!

http://www.fisme.science.uu.nl/alympiade/en/welcome.html

### 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’ MathAugust 2012 Summerschool - Open ended problems Math

Alympiad 27