Mathematical Modelling in School and in Teacher Education –

Conceptions and Examples

Modelling

Santiago de Chile, 11th of January 2013

Prof. Dr. Rita Borromeo Ferri, University of Kassel (Germany)

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Rita Borromeo Ferri

Modelling

“From what we know about the human brain and its functioning, it appears that modellingengages human cognition in a process to which 

it is best suited.“ (Lamon, 1997, 35)

Outline

1 Long‐term development of modelling competency2 Modelling in Primary School3 Modelling in Secondary School4 Modelling in Teacher Education5 Summary of the two lectures and central message

Primary School

Secondary School

Teacher Education

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Rita Borromeo Ferri

Modelling

1 LONG-TERM DEVELOPMENT OF MODELLING COMPETENCY

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Rita Borromeo Ferri

Modelling

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Mathematical Modelling as a compulsory competency within the standards from primary to secondary school.

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Rita Borromeo Ferri

R M

mathematical contents (guiding ideas) (5)

levels of performace (3)

Mathematical competencies (6)

Demand of task: translationReality ↔Mathematics(“mathematical modelling”)

Modelling

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Modelling: a lot of competencies are needed!

Long‐term development of modellingcompetencies is needed from primary school up to university (teacher education).

Modelling

Rita Borromeo Ferri

Modelling

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For a long‐term development of modelling competency the following is important:

• suitable modelling tasks (which include all sub‐competencies)

• quality reality‐based teaching at all levels (lecture yesterday)

The teacher is important!

Rita Borromeo Ferri

Modelling

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Rita Borromeo Ferri

Primary School UniversitySecondary School

Learning of mathematical modelling starts in primary school…

Modelling

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Rita Borromeo Ferri

Primary School

University

Secondary School

pre-service

teacher education

in-service

teacher training

University educators and teachers should be trained for teaching mathematical modelling and so work as “multiplicators”!

Modelling

2 MODELLING IN PRIMARYSCHOOL

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Rita Borromeo Ferri

Modelling

„The Big Foot“ (Lesh/Doerr 2003) (up from grade 3)

Please help to catch the thief of the jewels and help the police!Look at the footprint and find out the height of the person.

Justify your answer!

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Rita Borromeo Ferri

Which competencies are needed?Primary School

Secondary School

Teacher Education

Modelling

Competencies „The Big Foot“:• knowledge, skills and conceptions about:  arithmetic 

operations, handling with decimal numbers,       measurement, proportional thinking

• finding assumptions independently• validating: Can a person have such a height?• communicating: reading and presenting the results

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Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

Modelling

A lot of substantial mathematical activities become visible; example:

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Rita Borromeo Ferri

We measured the Nic‘s foot and it is 22.5 cm…and on the picture it is 40 cm, and now we double of 22.5 cm and…

Primary School

Secondary School

Teacher Education

Modelling

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Rita Borromeo Ferri

results:

• 1.72 m

• 1.65 m

• 2.69 m

• 1.87 m

Sönke, 9 years (grade 3):

„Oh no! 2.69m can not be right, because the highest person of the world was not so high!“

The teacher should encourage validation activities at the beginning!

Later on the learners do it on their own!

Primary School

Secondary School

Teacher Education

„There was a lot of mathematics in this task! We had to think about many things before we began to calculate.”

(Esra, grade 4)

„We did something, what the police is doing. That was not like: “Do subtraction or division”. We had to search for the mathematics – that was great.” (Arturo, grade 4)

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Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

Modelling

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Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

Modelling

Pippi Langstrumpf is able to lift up her horse.

How many students of your class are needed to do it like Pippi?

Modelling Problem for grade 3 (7‐8 year old)

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Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

Modelling

A zebra‘s weight is about 230 to 400 kg. And a horse then about 600 kg. Every one of us can lift up about 10 kg.So, we need 10 of our class to lift up the horse.

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Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

Modelling

Teaching and learning principles for modelling lessons:

Systematical change of methodsGroup work/single work (Modelling is best suited for cooperative learning)

Ideal phases of the lessons:1) Presenting the task in the plenum.2) Co‐constructive work: single‐group‐single3) Presentation of the solution in the plenum4) Comparison of the solution und reflection

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Teaching and learning principles of this lessons (e.g.):

1. Structure of the lesson is rich in content through challenging modelling tasks.

2. Cognitive activation of learners:  stimulating of self‐independence: modelling can only be learned by doing.

3. Encouraging of individual solution of learners’.

4. Stimulating of learners’ metacognitive activities; in particular: accompanying and retrospective reflections.

Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

Modelling

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Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

Modelling

Criteria for a modelling task:

Modelling tasks should be:

• open

• complex

• authentic

• realistic

• a problem

• solvable through all phases of the modelling cycle

3 MODELLING IN SECONDARY SCHOOL

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Rita Borromeo Ferri

Modelling

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Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

Example: „Filling up”(original version 2002)

Mister Stone lives in Trier, 20 km away from the borderof Luxemburg. To fill up his VW Golf he drives toLuxemburg where immediately behind the border thereis a petrol station. There you have to pay 0.85 Euro forone litre of petrol whereas in Trier you have to pay 1.1Euro.Is it worthwhile for Mister Stone to drive toLuxemburg? Give reasons for your answer.

Modelling

Model of the mathematical modelling process: „Filling up“

mathematics

1 Constructing

2 Simplifying/ Structuring

3 Mathematising

4 Working mathematically

5 Interpreting

6 Validating

7 Exposing

mathematical model & problem

mathematicalresultsreal

results

real model & problem

situation model

real situation& problem

rest of the world

1 2

3

6

7

5

4

Mathematical Modelling

Model of the mathematical modelling process: „Filling up“

mathematics

1 Constructing

2 Simplifying/ Structuring

3 Mathematising

4 Working mathematically

5 Interpreting

6 Validating

7 Exposing

mathematical model & problem

mathematicalresultsreal

results

real model & problem

situation model

rest of the world

1 2

3

6

7

5

4real situation

& problem

Mathematical Modelling

0.85 € 1.1 €

Car:50 liters volume of the tank

8 liters per 100 km consumption

20 km

Where do you have to pay less ifyou only consider the additional

costs for the ride?

Model of the mathematical modelling process: „Filling up“

mathematics

1 Constructing

2 Simplifying/ Structuring

3 Mathematising

4 Working mathematically

5 Interpreting

6 Validating

7 Exposing

mathematical model & problem

mathematicalresultsreal

results

real model & problem

situation model

real situation & problem

rest of the world

1 2

3

6

7

5

4

CTrier = 1.1 €/l 50 l CLux. = 8 l

100 km0.85 €/l 50 l + 0.85 €/l 40km

S = CTrier - CLux.

Mathematical Modelling

Model of the mathematical modelling process: „Filling up“

mathematics

1 Constructing

2 Simplifying/ Structuring

3 Mathematising

4 Working mathematically

5 Interpreting

6 Validating

7 Exposing

mathematical model & problem

mathematicalresultsreal

results

real model & problem

situation model

real situation & problem

rest of the world

1 2

3

6

7

5

4

Mathematical Modelling

Model of the mathematical modelling process: „Filling up“

mathematics

1 Constructing

2 Simplifying/ Structuring

3 Mathematising

4 Working mathematically

5 Interpreting

6 Validating

7 Exposing

mathematical model & problem

mathematicalresultsreal

results

real model & problem

situation model

real situation & problem

rest of the world

1 2

3

6

7

5

4

Mathematical Modelling

Yes, it is worthwile for him to drive to Luxenburg

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Complex modelling problems should also be integrated!

“Modelling days and modelling weeks” (for secondary and upper secondary level (16‐18years)(University of Hamburg, University of Kassel, University of Kaiserslautern, Singapore…)

Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

Modelling

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Primary School

Secondary School

Teacher Education

Characteristics of the authentic examples and activities originating from applied mathematics or from 

industry; only little simplified; often no solution known,  often only problematic situation described, 

students have to determine or develop a question, which can be solved;

various problem definitions and solutions possible;

importance of own activities (no spectator sport); no fast intervention by the teachers resp. future 

teachers, experience of helplessness andinsecurity central aspects and a necessary phase

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Primary School

Secondary School

Teacher Education

Examples of recent modelling days

How can the mixture of chemicals in swimming pools be optimised?

How can 3 helicopters in a ski area be positioned in an optimal way to help injured people fastly?

Traffic lights versus roundabout traffics – what is the best arrangement for the traffic?

Modelling

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Primary School

Secondary School

Teacher Education

University students (becoming mathematics teachers) are coaching the learners at school.

Necessary bridge between School and University and teacher education!

4 MODELLING IN TEACHER EDUCATION

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Rita Borromeo Ferri

Modelling

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Important: supply teachers with knowledge and abilities to teach mathematical modelling successfully, beginning in university education. 

Specific competencies and attitudes are necessary.

Questions:o Which competencies are necessary for teaching

mathematical modelling?o How can future teachers be prepared in university courses 

for teaching modelling at school?

Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

Modelling

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Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

Guiding principle for university seminars:

If we want our students to teach modelling in an appropriate way (correspondence between content and method, cognitive activation, reflection, formative assessment) we as lecturers have to conceive our own teaching in exactly the same way (correspondence between content and method, cognitive activation, reflection, formative assessment)

Modelling

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Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

Conception of seminar

Part 1 (theory) – 3 lessons

Part 2 (practice) – 3 lessons

Part 3 (theory and practice) –4 lessons

Mid‐Term‐Evaluation

Part 4 (presentations) – 2 lessons

Part 5 (reflecting the seminar) –one lesson and Evaluation

Theoretical background about modelling

Solving and creating modelling problems

Analysing solution processes; Modelling competencies;Types of teacher interventions; Methods

Presenting experiences from school

Questionnaire

Modelling

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Rita Borromeo Ferri

Primary School

Secondary School

Teacher Education

“It is good that we are learning not only modelling as a subject in this seminar, but also the methods how we can teach this at school!”

(Katja, 6th semester)

“Testing the modelling task in grade five was important and helpful for my understanding of modelling and the practical transformation in school. […] It was good to have a chance testing modelling problems at school.” 

(Birgit, 7th semester)

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Rita Borromeo Ferri

Model for teaching applications & modelling(Borromeo Ferri &Blum, 2009):

Theoretical dimension

Task dimension

Instructional dimension

Diagnostic dimension

a) Modelling cyclesb) Aims &perspectives of modellingc) Types of modelling tasks

a) Multiple solution of modelling tasksb) Cognitive analyses of modelling tasksc) Development of modelling tasks

a) Planning lessons with mod. tasksb) Carrying out lessons with mod. tasksc) Interventions, support and feedback

a) Recognising phases in mod. processb) Recognising difficulties and mistakesc) Marking tests

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5 Summary of the two lectures and central suggestions

Rita Borromeo Ferri

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Encouraging results from different studies:

Mathematical modelling can be taught and learned successfully and sustainable from primary school up to university, if

it is implemented into every day teaching and into teacher education

teachers and university educators have necessary competencies

criteria for quality teaching have to be considered 

Rita Borromeo Ferri

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Rita Borromeo Ferri

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Central suggestions:

1) Mathematical modelling is important (e.g. for understanding the world better), so make it possible for your learners!

2) Mathematical modelling can be taught and learned!

3) Political guidelines are important: standards and curricula, which include modelling and also assessment of modelling.

4) Teachers and university educators are important! Modellingshould take in important role in teacher education and teacher training!

Teaching Modelling

Muchas gracias!

Rita Borromeo Ferri

Email: borromeo@mathematik.uni-kassel.de