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Linguistic competencies entailed by the concept of Mathematical Literacy
Seminar on Subject literacies and the right to quality education for democratic citizenship
and participationStrasbourg 28.09.2012
Prof. Dr. Helmut Linneweber-Lammerskitten
„Mathematical Literacy“ is more than „having basic skills in mathematics“
“Mathematical literacy is an individual's capacity to identify and understand the
role that mathematics plays in the world, to make well-founded judgements and to
use and engage with mathematics in ways that meet the needs of that individual's
life as a constructive, concerned and reflective citizen.” (OECD 2003, S. 24)
Linneweber-Lammerskitten 2
...and entails linguistic competencies
Expressions such as "identify and understand" and "well founded judgements"
refer to activities which have both cognitive and linguistic dimensions. One can
only identify, understand and judge insofar as one relates thought and language
to each other. A “life as a constructive, concerned and reflective citizen”
presupposes communication with others.
National educational standards in mathematics make abstract objectives more concrete and obligatory
3
e.g. „HarmoS educational Standards for Mathematics“ in Switzerland (cf. “Handout)”
The educational values targeted by mathematics teaching practices can only be realised if linguistic
competence is also targeted, since
• Linguistic competencies are constitutive parts of educational standards in
mathematics: Students can only reach the educational standards in mathematics at the end of compulsory
school, if they are able to name, to describe, to define, to explain, to argue etc.
• Linguistic competencies are preconditions for learning: Students can only successfully participate in learning mathematics if they have the linguistic
prerequisites to understand questions, problems, argumentations, etc. and are able to give
answers, to interact with others, etc.
• Linguistic competencies are preconditions for acting as an active, reflective and
intelligent citizen All these presuppose a certain degree of reflection, which can only be realised on a higher level of
language competence
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Three aspects concerning linguistic competencies
Cognitive and Linguisitic Aspects in Educational Standards (HarmoS)*) I
Knowing, Recognising & Describing• understanding, using and explaining technical terms• relating technical terms to mathematical objects, properties and relations and vice versa• identifying forms and pattern• naming and describing mathematical rules and laws in their own language• capturing mathematical states of affairs and describing them
…
Operating & Calculating• Carrying out calculations, transformations and constructions in written "standard form",
with notes or orally, with or without (technical) instruments
…
*) Linneweber-Lammerskitten, H. (2012). Items for a description of linguistic competence in the language of schooling necessary for teaching/learning mathematics (in secondary education). An approach
with reference points. Strasbourg: Council of Europe. Retrieved from: http://www.coe.int/t/dg4/linguistic/Source/Source2010_ForumGeneva/4_LIS-Mathematics2012_EN.pdf (August 2012)
Linneweber-Lammerskitten 6
Cognitive and Linguisitic Aspects in Educational Standards (HarmoS)*) II
Using Instruments & Tools• Using electronic instruments (calculator, Computer), works of reference, construction
instruments (compass, set square)• …
Presenting & communicating• Understanding calculations, transformations, constructions, argumentations of other
students• Presenting own calculations, transformations, constructions, argumentations in a way
that is comprehensible and traceable by others and appropriate with respect to the
mathematical object• …
*) Linneweber-Lammerskitten, H. (2012). Items for a description of linguistic competence in the language of schooling necessary for teaching/learning mathematics (in secondary education). An approach
with reference points. Strasbourg: Council of Europe. Retrieved from: http://www.coe.int/t/dg4/linguistic/Source/Source2010_ForumGeneva/4_LIS-Mathematics2012_EN.pdf (August 2012)
Linneweber-Lammerskitten 7
Cognitive and Linguisitic Aspects in Educational Standards (HarmoS)*) III
Mathematising & Modelling• Describing, interpreting and modulating (problem) situations (of daily life) in order to
solve them by mathematical means• ...
Arguing & Justifying• Forming assertions and giving reasons for them• Making thoughts and ways of calculating transparent and justifying them • Giving illustrative explanations for mathematical phenomena and laws• understanding and reproducing proofs and counterexamples • ...
*) Linneweber-Lammerskitten, H. (2012). Items for a description of linguistic competence in the language of schooling necessary for teaching/learning mathematics (in secondary education). An approach
with reference points. Strasbourg: Council of Europe. Retrieved from: http://www.coe.int/t/dg4/linguistic/Source/Source2010_ForumGeneva/4_LIS-Mathematics2012_EN.pdf (August 2012)
Linneweber-Lammerskitten 8
Cognitive and Linguisitic Aspects in Educational Standards (HarmoS)*) IV
Interpreting & Reflecting on Results
• Checking results for truth and for adequacy with the original problem
• Reflecting whether a result or an approach can be used for future problem solving
• …
Experimenting & Exploring
• Exploring mathematical situations and searching for mathematical laws
• Expressing conjectures and trying to support or falsify them by thought experiments
• ...
*) Linneweber-Lammerskitten, H. (2012). Items for a description of linguistic competence in the language of schooling necessary for teaching/learning mathematics (in secondary education). An approach
with reference points. Strasbourg: Council of Europe. Retrieved from: http://www.coe.int/t/dg4/linguistic/Source/Source2010_ForumGeneva/4_LIS-Mathematics2012_EN.pdf (August 2012)
Linneweber-Lammerskitten 9
Question for presenters to address:
1) Are there any language requirements specific to your subject area?
– Natural language mixed with formal language (& formal metalanguage) and graphs,
diagrams etc.
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DMK/DPK (Ed.) Fundamentum Mathematik und Physik. Bern: Orell Füssli p.15
Question for presenters to address:
2) How do you see the relationship between your specific subject area and the possibility
of formulating a general framework for the language of schooling?
- A general framework seems possible but only on an abstract level.
- On a less abstract level one must recognise that labels also used in ordinary
language have deviant meanings in mathematics and even within different areas of
mathematics:
- with respect to objects: Menge, Körper, Seite,
- with respect to activities: to define, to prove,
Linneweber-Lammerskitten 11
Question for presenters to address:
3) What do you think is the specific contribution of your subject to such a framework?
– Natural language mixed with formal language (& formal metalanguage) and graphs,
diagrams etc. (cf. 1)
– Situations, expressions, discourse functions, strategies etc. directed to “overarching
ideas” (PISA 2003)*): quantity, space and shape, change and relationships, uncertainty
– Situations, expressions, discourse functions, strategies etc. directed to “characteristic
mathematical competencies“ (PISA 2003): thinking and reasoning, argumentation,
communication, modelling, problem posing and solving, representation, using
symbolic, formal and technical language and operations, use of aids and tools
Linneweber-Lammerskitten 12
*) OECD (2003). PISA 2003 Assessment Framework: Mathematics, Reading, Science and Problem Solving Knowledge and Skills - Publications 2003. Retrieved from: http://www. oecd.org/dataoecd/46/14/33694881.pdf (June 2011).