S. Ongstad, Oslo University College

S. Ongstad, Oslo University College

Language and communication in curricula for mathematics education.

The problem of disciplinarity versus discursivity

Mathematics education begins in language, it advances and stumbles because of language, and its outcomes are often assessed in language (Durkin, 1991:3)

On the relationship between the concepts of language, communication and disciplinarity

I will oversimplify a complicated matter indeed by claiming that a traditional definition of an utterance actually will help us combine and contrast language, communication and mathematics. To utter requires structured elements of materiality, forms, letters, sounds and the like. How these are connected is described by syntax. The forms represent or relate to something, to phenomena, things, contents, concepts etc. and is described by semantics. These combined form/content aspects are addressed, used, acted and is described by pragmatics. Different kinds of grammar may combine these aspects differently and thus end up with different perceptions of what is language and what is communication and how they relate.

In any case, whenever uttering, both more in general and within disciplines and school subjects, such as mathematics, language, communication and disciplinarity will accordingly always occur simultaneously. But paradoxically they both can be, and tend to be focused separately. Academic disciplinarity then will most often be built around the reference or the content, that is, the epistemological part. Without going into details here, this reciprocal view may help us understand roughly the paradox of the complementary aspects in question as both joint and separate. With this simplistic but basic view in mind I turn to the kind of subtle and dangerous surgery we were up to in our four studies.

Tendencies from the studies

The four national curricula in mathematics we studied have some, but not many explicit references to language and communication. These are mostly found in introductions where the discipline is related to learners, culture and society. The closer one gets to specific goals and aims, which are often found as 'bullet points' in the texts, the more weight is put on mathematics as such.

While the English curriculum has rather clear references to language and learning across the curriculum at primary level, split and imbalance between mathematics and communication are significant at lower part of secondary education. The tendency to separate mathematics and language is even found in the Swedish and the Norwegian curricula. The Romanian curriculum seems a bit more willing to se communication as part of mathematics as a didactic enterprise.

The English curriculum stresses language more than communication, a tendency visible for instance in prioritising clarity and preciseness when working with key concepts. The Swedish curriculum is similar as the introduction gives some space for language and communication, but more or less drops this announced interest in the other parts. The Norwegian curriculum is similar, but nevertheless different. I will return to that.

In all four countries basic skills or key competencies have been given rather strong priority in the overall general, national curriculum. Only the Norwegian curriculum has followed up this intention in the school subject parts by giving the competencies a key role in all the disciplinary curricula and hence also vis-à-vis mathematics as a discipline. A crucial difference between the four curricula then is that the Norwegian curriculum explicitly has altered the traditional but basic relationship between disciplines and more overall curricular aims (that is, basic skills). In principle all school subjects now more explicitly are servants for something outside the discipline itself. In all curricula it is outlined for each school subject specifically what the five prioritised competencies would mean in terms of --- disciplinarity. For mathematics these are:

Being able to express oneself orally in the mathematics subject means making up one's mind, asking questions, reasoning, arguing and explaining a process of thinking using mathematics. This also means talking about, communicating ideas and discussing and elaborating on problems and solution strategies with others.

Being able to express oneself in writing in the mathematics subject means solving problems by means of mathematics, describing a process of thinking and explaining discoveries and ideas; one makes drawings, sketches, figures, tables and graphs. Furthermore, mathematical symbols and the formal subject language are used.

Being able to read in the mathematics subject means interpreting and using texts with mathematical content and content from everyday life and working life. Such texts may include mathematical expressions, graphs, tables, symbols, formulas and logical reasoning.

Being able to do mathematics in the mathematics subject is, needless to say, the foundation of the mathematics subject. This involves problem solving and exploration, starting with practical day-to-day situations and mathematical problems. To manage this, pupils must be familiar with and master the arithmetic operations, have the ability to use varied strategies, make estimates and assess how reasonable the answers are.

Being able to use digital tools in the mathematics subject involves using these tools for games, exploration, visualisation and publication, and also involves learning how to use and assess digital aids for problem solving, simulation and modelling. It is also important to find information, analyse, process and present data with appropriate aids, and to be critical of sources, analyses and results. (LK06)

What we can register then is that mathematics in this curriculum on the one hand is described as a means in the introduction and also, as seen above, competencies are actually spelled out. On the other hand there is nevertheless a significant schism in the different main parts of the curricular text. When we arrive the concrete aims, all references to language and communication have evaporated. Disciplinarity has not only just survived in the process, but it could fairly be said that the density and gravity of mathematicallity, that is the weight of mathematics as such, has kept the bullet points untouched by any external pollution.

[For those of you who have more special interests in these matters I would like to remind you about the three 'separate' contributions by Birgit Pepin, Mihaela Singer and myself where we in different ways and from different points of departure fabulate around and discuss different interfaces of language, communication and mathematics.]

What and where is 'disciplinarity'?

All curricula have rather brief descriptions of what we could call 'disciplinarity'. These are mostly given as short, concrete cues or bullet points. This well known 'discourse' is dominated by long rows of disciplinary concepts, nouns that most math teachers are familiar with. These nouns are accordingly not explained, and function implicitly as part of speech acts or 'doings' that seemingly give clear directives for the content of the syllabus and what to do with it. By the same token though the non-hierarchical structure of the bullet points avoids making clear which ones are more important.

So even if all four national curricula in the general parts clearly have given priority to the needs and the interests of the learner and society, the discipline of mathematics seems able to maintain its own agenda, its 'fachlichkeit' within this 'horizontally' structured, curricular framework. Clear general ambitions of introducing language and communication in curricula for mathematics are, therefore, generally not really backed up.

Our small scale studies can not indicate why this is so. But since all four ministries seemingly have been serious about the implementation of such aims, it would be worthwhile reflecting together here over the reasons for the drift away from the intentions. There are reasons though for believing that mathematics as a discipline, probably not just in Norway, has more problems than most other school subjects integrating language and communication with mathematics.

The profile of the elements of the school subject is often conceived as a 'skyscraper' rather than as a row of 'terraced houses'. Within such a 'vertical' paradigm there is probably less room for seeing mathematics education as a compound of elements of aspects from other fields of knowledge, such as for instance knowledge from language studies and sociology. If this is the case, it is quite thought-provoking that among an increasing number of researchers in mathematics education, language, communication and semiotics, in short, discursivity, is seen as crucial or perhaps even inevitable. A symptomatic case could be Mogens Niss' article from 2003 where he outlines significant competencies in mathematics. Niss is interestingly one of the key architects designing PISA tasks in mathematics:

(...)

(...) Handling mathematical symbols and formalisms such as

• decoding and interpreting symbolic and formal mathematical language, and understanding its relations to natural language;

• understanding the nature and rules of formal mathematical systems (both syntax and semantics);

• translating from natural language to formal/symbolic language

• handling and manipulating statements and expressions containing symbols and formulae.

(...) Communicating in, with, and about mathematics such as

• understanding others’ written, visual or oral ‘texts’, in a variety of linguistic registers, about matters having a mathematical content;

• expressing oneself, at different levels of theoretical and technical precision, in oral, visual or written form, about such matters (Niss, 2003:8-9).

Challenges?

A problem with educational understandings of mathematical concepts is that a more cognitive and hence a semantic view has dominated among educators, who are mostly recruited from the disciplinary side. Even if socio-constructivism seems relatively strong in mathematics in many European teacher educations, this direction seems to have had less impact on how the different curricula have phrased the core concepts that are important for integrating language and communication elements in mathematics as a school subject.

In my view the learning of mathematics is in some sense permeated by constant discursivity, mathematicallity must in some sense be communicated, any pragmatic mathematical act is in some sense an utterance, any semantic concept is in some sense both mathematical and linguistic. Mathematics is accordingly not just a language in its own right. It is even a 'languaging', discursive activity, and it contributes culturally to 'language' over time as do all other disciplines and school subjects.

Ongoing research in mathematics, and everyday use of mathematics in education and in society as a whole, thus contribute communicatively to culture in an integrative way, weaving together disciplinary and discursive aspects. Which aspects will be most important, relevant and adequate in different contexts, is a crucial educational challenge. However, that can hardly be convincingly addressed without a more subtle, integrated conceptual framework through which these questions can be understood and negotiated.

It will be a challenge for curriculum designers and test constructers to find a reasonable and valid conceptual balance between specific mathematicallity and general discursivity. A key question will be which concepts are adequate for describing overall communicational patterns within school subjects, and which ones are just valid and relevant in certain sub-fields. That should be answered by relating explicitly to disciplinarity while investigating communicational conditions for school subjects across borders in Europe.

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