n = the number of dice

(1)

Thinking mathematically

Posing and solving mathematical problems

Modelling mathematically

Reasoning mathematically

Representing mathematical entities

Handling mathematical symbols and formalisms

Communicating in, with, and about mathematics

Making use of aids and tools

(Niss, 2003)

(2)

• Problem Solving

• Reasoning and Proof

• Communication

• Connections

• Representations

(3)

(Niss, 2003)

(4)

(5)

A tower of dice

(The number of dice 7) – the spots on the uppermost dice

(n 7) – y

n = the number of dice

y = the spots on the uppermost dice

7n – y

(6)

Thinking

mathematically

• Understanding and handling the scope and limitations of a given concept

• Extending the scope of a concept by abstracting some of its properties;

generalising results to larger classes of objects

(7)

(8)

A generalization of a concept

is an extension of the concept

to less-specific criteria

(9)

On this side of this blue dice I can see 3 dots.

The sum of the opposite sides is 7.

It must be 4 dots on the other side.

If you add the dots on the opposite sides on this blue dice the sum will always be 7.

3 + – = 7 7 = 3 + _

The sum of the opposite sides are always 7.

3 + 4 = 7

(10)

There are 3 dice.

I think: 3 multiplied with 7.

Then I take away the 4 dots at the top.

The numbers of dice times 7

and then take away the 4 dots at the top.

(3 7) – 4 7n – y

(11)

• Posing different kinds of mathematical problems

• Solving different kinds of mathematical problems whether posed by others or by oneself, and, if appropriate, in different ways.

–

(12)

• Performing active modelling in a given context:

structuring the field/ mathematising/working with(in) the model/

analysing and criticising the model/communicating about its results/

controlling the entire modelling process.

• Analysing and decoding existing models.

 

(13)

• Following and assessing arguments

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& $)*"

($!

(14)

manipulative models

written symbols

real situations

spoken language pictures

(15)

A representation is a sign or

a configuration (form, gestalt) of signs or objects.

The important thing is that it can stand for something other than itself.

(Goldin, Shteingold, 2001)

(16)

1/3 1 3

One third One of three Every third

We are three girls having some candies. We

share the candies equally. I get one third of them.

manipulative models

written symbols

real situations

spoken language pictures

(17)

Understanding and utilising

different sorts of representations

relations between different representations, including knowing about their strengths and limitations

Choosing and switching between representations

 

(18)

Handling mathematical symbols and formalisms

Decoding and interpreting symbolic and formal mathematical language, and understanding its relations to natural language

Translating from natural language to formal/symbolic language

Handling expressions containing symbols and formulae

7n – y

(19)

Communicating in, with, and about mathematics

Understanding others’ written, visual or oral ‘texts’ about matters having a mathematical content

Expressing oneself, in different forms at different levels of theoretical and technical precision

(20)

Making use of aids and tools

Knowing tools and aids for mathematical activity,

and their range and limitations

being able to reflectively use such aids and tools.

(21)

How high is the building – and how do you know it?

Which competencies are possible for the pupils to develop?

(22)

(Niss, 2003)

Which competencies are possible for the pupils to develop?

How high is the building – and how do you know it?

(23)

 

   

 

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