What is Realistic Mathematics Education? National Mathematics Conference Swakopmund

Presentatie titel
What is
Realistic Mathematics Education?
Rotterdam, 00 januari 2007
Swakopmund, May 2011

Europe
The Netherlands
Holland
Rotterdam

Rotterdam primary schools

April 12th, 1961

Hans Freudenthal
psychologist, mathematician, founder
of the Dutch New Math Movement.
Main idea: each individual
discovers mathematical structures in
its own living environment and creates
a personal concept of mathematics.
This is the principle of ‘Guided
Reinvention’.
Realistic Mathematics Education
http://www.fisme.uu.nl/fisme/en/

‘Mathematics must be connected to reality, stay close to children and be relevant to society, in order to be of human value.
Instead of seeing mathematics as subject matter that has to be transmitted, I see the idea of mathematics as a ‘human activity’. Education should give students the “guided” opportunity to “re-invent” mathematics by doing it.
This means that in mathematics education, the focal point should not be on mathematics as a closed system but on the activity, on the process of mathematization, going from the world of life into the world of symbols’
(Freudenthal, 1968).

1960
Mechanisticmathematics
Education in 95% of the Dutch
primary schools
MME
Context problems as a field of
application,
Procedure-focusedwayof teaching
in which the learning content is
split up in meaningless small parts,
Students are offered fixed solving
procedures to be trained by
exercises.
2011
Realisticmathematics
Education in 95% of the Dutch
primary schools
RME
Context problems as a sourcefor
the learningprocessand to apply
mathematicalconcepts,
Complex and meaningful
conceptualization of teaching and
learning,
Students are considered to be
active participants in the teaching
learning process, in which they
develop mathematical tools
and insights.

What is realistic mathematics education?
You enter a classroom. The teacher of this classroom gives her lessons according the basic principles of RME. How do you recognize this?
What is the influence of teaching according RME on the learning of the learners?
What can you expect of the learning proceeds by teaching mathematics education according to the principles of RME?
Does RME make the maths lessons more interesting, more attractive for the learners?
What does a lessonplan look like for a maths lesson
according RME?

What is Basic elements of Realistic Mathematics Education
Starting points are contexts and the learner experiences
Use of smart strategies
Learning by use of models and diagrams
Learning in interaction
Use of characteristic teaching materials

Learner centered mathematics education
Realistic
mathematics education
Starting point is the learners’ existing knowledge
Learning from the learners’ experiences
Learning from useful materials
Learning from pair and group activities
Learning by doing
Trial-and-error learning
Many similarities, small differences!
Contexts
Interaction
Models and diagrams
Strategies
Use of characteristic, structured teaching materials
Intertwining of teaching-learning trajectories

Contexts
Strategies
Models and diagrams
Interaction
Use of characteristic materials

Contexts in Namibian maths books

Contexts in Dutch maths books

What is a context for a child? (1)
What is the childrens’ reality?
What experience do they have?
What kind of context do they understand?

What is a context for a child? (2)
If we know what children are interested in, it must be easier for us, the teachers, to use this environment in educating children.
By doing so, the child will certainly be motivated to learn.
Why do we use contexts?
The child sees that you can use mathematics in daily life, that is to say, in your life.

What is a context for you?
How numerated are you?

Right or wrong?
Namibia is about 20 times larger than Holland
Right
Wrong

Right or wrong?
Holland has 20 times as many inhabitants as
Namibia
Right
Wrong

The mass of a slice of
bread is about …
20 gr
30 gr

The capacity of this
coffee cup is about …
100 ml
200 ml

The heigth of a grown-up
male giraf is at most
about …
5,25 meter
7,25 meter

The mass of a
grown-up dikdik is
about …
4 kg
7 kg

The surface of the
cricketfield is about …
7500 m2
15000 m2

Contexts
Strategies
Models and diagrams
Interaction

Strategies (1)
We know how we teach children maths....but....
Do we know how children think?
Do we know how children really solve the maths problems?
Is that in the way we told them to do it?
Should each child solve the problems in the exact way we told them to do so?

Strategies (2)
Please solve the next sum:
I’ve got a bottle with 1½ liter of lemonade.
I pour this lemonade out in glasses of 1/6 liter.
How many glasses can I fill?
And now:
3/4 ÷ 1/8 =

Strategies (3)
How many of you solved
3/4 ÷ 1/8 =
like this:
3/4 ÷ 1/8 = 3/4 x 8/1 = 24/4 = 6?

Strategies (4)
About the lemonade..
1½ liter of lemonade... let’s see:
With 1 liter you can pour out 6 glasses of 1/6 liter.
So half a liter is good for 3 glasses.
That makes 9 glasses together!
Important question:
What did they teach you at school?
They told you to solve like
11/2 ÷ 1/6 =3/2 x6/1= 18/2 = 9 ?

Strategies (5)
A book with 92 pages.
You are reading at the bottom of page 27.
How many pages to read when page 27 is finished?

Solutions
Algorithm calculation:
Addingbystringing, the ‘cashiersstrategy’:
27 + 3 = 30,
30 + 60 = 90,
90 + 2 = 92,
So: 3 + 60 + 2 = 65

Solutions
Stringing:
Splitting:
deficit
Combination of splitting and stringing:

Scale strategy:
Jumping toofar and back:
Smart strategies
92 – 27 = 92 – 30 + 3 = 65

Strategiescalculation up to 20 (1)
8 + 7
9, 10, 11, 12, 13, 14, 15
8 + 2 = 10, 10 + 5 = 15
5 + 3 + 5 + 2 = 10 + 5 = 15
7 + 7 + 1 = 14 + 1 = 15
8 + 8 – 1 = 16 – 1 = 15
etc

Strategies calculation up to 20 (2)
Maureen:
Thijs:
Luuk: “First, put three euro’s out of the six
to the seven euro’s;
thatmakes ten euro’s,
and threemakesthirteen euro’s.”
Hannah: “Six and six is twelve; and one
makes thirteen euro’s.”

Strategieslearning the times tables (1)

Strategies learning the times tables (2)

Strategies learning the times tables (3)
‘Stepping stones’

Reflection
Do you present various strategies in maths class?
Do children have preferences for strategies?
Is the children’s numberconcept sufficient to apply various strategies?
Do children choose their own strategy?
Do all children have to learn various strategies?

Contexts
Strategies
Models and diagrams
Interaction

Models and diagrams

Difference between € 54.- and € 37.-?

Contexts
Strategies
Models and diagrams
Interaction

Verticalinteraction

Interaction 1: Vertical interaction
The teacher explains - the learners ask questions – the teacher tries to activate the learners by asking stimulating and open questions.
teacher
child child
This type of interaction you see during instruction
lessons.
We call this vertical interaction.

HorizontalInteraction

Interaction 2: Horizontal interaction
The teacher has given a problem to the learners.
The learners try to solve this problem by cooperation, by thinking it through, together.
teacher
The teacher observes. He collects impressions.
Later, the teacher knows what to talk about during the evaluation.
We call this horizontal interaction.
child
child
group

Contexts
Strategies
Models and diagrams
Interaction

Use of characteristic materials (1)

Flash cards
Numeric cards
Number snake
Numbers in love
Sun game
Etcetera....

All these materials are meant to build structures in the minds of children.
The children have to obtain images of quantities.
If you don’t want them to keep counting, it is necessary that the children create their own world of numbers and quantities.
By using these materials, children will learn to control this tremendous world of numbers and, ultimately, understand the world they live in.

The Construction of Realistic Mathematics Education:
Realistic Mathematics Education always starts with
reality: How to solve the problem in this context?
This is the concrete level.
Realistic Mathematics Education supports the
thinking of children in models.
This is the model level.
Realistic Mathematics Education gives children the opportunity to abstract from their experience.
At the same time they know how it works and they can explain the action.
This is the formal level.

The levels of concept based mathematics education
formal level
model level
contextlevel

Advantages of Realistic Mathematics Education
Direct link with reality. Children recognize the items in their education as subjects they meet in daily life.
Because of the direct link to daily life, children get the possibility to use the matter they learned at school in their environment.
Children will be able to solve real problems based on what they learned at school. Not only with maths!
Children will not forget their lessons because of the knowledge that they can use their lessons in daily life.

Summary
Teaching mathematics in ‘the realistic way’ is teaching
mathematics by use of
Contexts
Strategies
Models and diagrams
Interaction
Structured materials

Let’s discuss this way of teaching maths
We will have to ask ourselves:
‘Do we want to teach our learners in this way?’
‘How can we teach our learners in the way of Realistic Mathematics Education?’
‘How can we work on a merge of Learner Centered Education and Realistic Mathematics Education, by taking the best of two visions?’

Thank you for your attention
‘Working together for excellence in mathematics
education in Namibia and in The Netherlands’
Jaap Griffioen, j.griffioen@hro.nl
Jaap de Waard, j.de.waard@hro.nl

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What is Realistic Mathematics Education? National Mathematics Conference Swakopmund

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What is Realistic Mathematics Education? National Mathematics Conference Swakopmund Presentation Transcript