Linguistic competencies entailed by the concept of Mathematical Literacy Seminar on Subject literacies and the right to quality education for democratic citizenship and participation Strasbourg 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-Lammerskitten2...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 Linneweber-Lammerskitten4 Three aspects concerning linguistic competencies

Linneweber-Lammerskitten5 use handout paper for cando descriptions in French and German

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: (August 2012) Linneweber-Lammerskitten6

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: (August 2012) Linneweber-Lammerskitten7

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: (August 2012) Linneweber-Lammerskitten8

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: (August 2012) Linneweber-Lammerskitten9

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. Linneweber-Lammerskitten10 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-Lammerskitten11

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-Lammerskitten12 *) OECD (2003). PISA 2003 Assessment Framework: Mathematics, Reading, Science and Problem Solving Knowledge and Skills - Publications Retrieved from: oecd.org/dataoecd/46/14/ pdf (June 2011).