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)
• Problem Solving
• Reasoning and Proof
• Communication
• Connections
• Representations
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)
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
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
A generalization of a concept
is an extension of the concept
to less-specific criteria
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.
3+4=7
3+–=7
7=3+_
If you add the dots
on the opposite
sides on this blue
dice the sum will
always be 7.
The sum of the
opposite sides are
always 7.
There are 3 dice.
I think: 3 multiplied
with 7.
Then I take away the
4 dots at the top.
(3 7) – 4
The numbers of
dice times 7
and then take away
the 4 dots at the top.
7n – y
• 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.
– • 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.
•
Following and assessing arguments
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manipulative
models
written
symbols
pictures
real
situations
spoken
language
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)
manipulative
models
We are three girls
having some
candies. We
share the candies
equally. I get one
third of them.
written
symbols
1/3
1
3
pictures
real
situations
spoken
language
One third
One of three
Every third
Understanding and utilising
different sorts of representations
relations between different representations, including knowing about their
strengths and limitations
Choosing and switching between representations
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 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
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.
How high is the building – and how do you know it?
Which competencies are possible for the pupils to develop?
How high is the building – and how do you know it?
Which competencies are possible
for the pupils to develop?
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)
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