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Educational Studies in Mathematics, Special Issue on 'Teaching and learning mathematics in contexts', 1999

ABSTRACT. A framework for discussing (through the concepts of "model" and "field of experience") the nature of the relationships between contexts and the formation of mathematical knowledge is proposed, and some educational, cultural and cognitive problems of situated teaching-learning are tackled within it. This framework is supported and illustrated with references to the curricular innovation and educational research carried out by the Genoa group, of which the authors are members.

Over the past two decades, the failure of "new mathematics" has contributed, together with other factors, to the development of a "movement", grounded in theory and practice, which has focused renewed attention (in the planning of mathematics curricula and in the study of concept formation) on the uses of mathematics and its out-of-school applications. The Genoa Group for Research in Mathematics Education, has taken part in this movement.

In Boero et al. (1995) we and other researchers of the Genoa Group explored the question of teaching in/with contexts as part of the relationship between "mathematics" and "culture". In this paper we take the viewpoint of curricular research, and explore issues in epistemology and cognitive psychology which we regard as being intertwined with it:

•	forms of connection between mathematics and contexts,
•	normative and descriptive aspects of mathematization,
•	risk of predominance of mathematical point of view,
•	control over shifts between different layers of meaning, and other problems.

We will propose a framework based on the epistemological concept of model and the didactical one of field of experience in order to tackle these issues, which we will illustrate with some brief examples based on the projects Genoa Group has carried out at various school levels.

1.1. Introduction

When speaking about "teaching/learning in context" (also known as "situated teaching/learning") referred to a given set of disciplinary concepts, the word "context" is used to indicate a situation or activity in which these concepts are introduced and applied in a meaningful manner (that is, meaningful as regards to the situation or activity itself). This way of teaching contrasts with other ones where the separate concepts are introduced in an abstract exposition, while their combination and application to situations are postponed to example phases, to the teaching of other subjects, or to post-scholastic experience.

To clear the epistemological and didactical problems of the sense to attribute to the adjective "meaningful" we shall use two conceptual tools: model and field of experience.

1.2. Models …

Modelling is usually a form of metaphor (Boyd & Kuhn, 1979), i.e. a shift between semantic spaces with a non-empty intersection. It can involve abstraction or idealization. In the case of logic, model means particularisation of a theory (shift from the semantic space of "syntactical" objects to that of mathematical structures); for instance, the structure of whole numbers is a model of the theory of groups. Both meanings of "model" (abstraction-idealization and particularisation-exemplification) exist even in common language.

We shall concentrate on the former use, that is the model as a simplified representation of a portion of reality (of interest in a certain problem) used for: improving the "visualisation" of some aspects (using a scale reproduction instead of the original diagram), generalising properties (grammar rules), allowing comparisons (comparing different regions using population density instead of population and area), etc.. From this point of view the theory of groups is a model for the structure of whole numbers. The model concept can be represented as in the diagram in Fig. 1, which makes the representations generally used in the literature on mathematical modelling (e.g. see Gilchrist, 1984) more detailed.

For example, let R be a quadrangular field that we wish to evaluate economically and AR the extent of that field. If we note that the field is more or less rectangular, we can consider the lengths of two consecutive sides as the factors ER that determine the extent. The arrow ER  AR representing this dependence is dotted, as we overlook other factors: protuberances and depressions, the fact that the shape is only roughly rectangular, etc..
Let us associate the lengths ER to two numbers, a and b, which express the lengths in a particular unit U; ER  EM is also dotted as the measurements are not exact.
Let us associate to AR a·b; this is AM. The arrow EM  AM is not dotted; the value of a·b depends exactly on a and b; AR  AM is dotted because it is affected by the simplifications and approximations used in the choice of ERs and the association of EMs.
Hence, the model M is the multiplication (or rather the formula S=a·b, if S expresses the extent in squares whose side is U). 
Let us take another example. Let R be the relationship that exists in the English language between how a noun is used for expressing one object and how it is used for expressing more than one, and AR the way the noun changes in appearance passing from one use to the other. We shall consider as the meaningful elements (ER) the final parts of the nouns and their changes. The model is made up of a collection of statements such as: «nouns ending in -ch, -s, -sh, -x form the plural by rewriting: noun  noun+"es"»; EMs are the final segments of the strings that constitute the nouns and the rewriting rules; AM is the formal procedure for string transformation obtained by applying the rules. This is another example of simplified representation: the "exceptions" (epoch  epochs, basis  bases, fish  fish, codex  codices, etc.) are overlooked. By considering only the strings, we have lost aspects related to pronunciation (if -ch is pronounced -k, -s is used instead of -es), the origin of nouns (foreign words follow different rules), changes in use, etc..

Different types of modelling are often considered simultaneously: if R represents weather conditions in locality X in the month Y of the year Z, we may consider R  M₁ and R  M₂, where M₁ is a graph of the daily maximum temperature related to the date and M₂ a distribution histogram of suitably classified daily temperatures. More sophisticated modelling may subsequently take place: when modelling the extent of the rectangular field, one can represent the lengths as indeterminacy intervals rather than as exact numbers.

The "model" diagram in Fig. 1 is itself a model. It summarises and illustrates only some aspects of the relationships between situations and their representation, and is not easy to apply to all cases (e.g. the distinction between AM and EM may be interpreted differently, in some cases the arrow EM  AM should be dotted, etc.).

Some apply the term "model" to the triple (M, R, {arrows}) (particularly the arrows from R to M), or to the triple (M,R,S), where S is the subject that constructs and/or interprets the arrows (for references, see Blum & Niss, 1991 and Norman, 1993). We have adopted the more common usage of applying it to the outcome of "modelling", i.e. the pair (EM, AM), a structure which can be associated to different pairs (ER, AR): S=a·b can be applied not only to the field of the previous example.

1.3. … and Contexts

1.3.1. Reality

Models are built by using material artifacts (objects, signs, etc., like squared paper, the computer,"", "·", histogram rectangles, colours, etc.) or mental artifacts (the concepts "noun", "multiplication", "frequency", etc.) which in turn are often models themselves (histogram rectangles to indicate amounts, "noun" to indicate particular words, etc.). We use the term artifacts after Norman (1993): this seems to us more expressive than other terms like "prosthesis tools" used by Bruner.

The reality R (see 1.2) is not necessarily a material object or a natural or social phenomenon. It may be a situation that is presented in a modelled form (e.g. a graph of a certain phenomenon), or a reality made up of a set of models ("being a group", "being increasing", "being singular", "being vectorial", etc. are models which have, like R, structures, functions, names, physical entities, etc.).

1.3.2. R  M and M  R

The use of models as artifacts for the construction of other models and the further modelling of models (i.e. the existence of metalevels of representation) characterises higher, human forms of reasoning. In our mental activities we frequently pass, more or less consciously, from one level to another, not only in the sense of R  M (reasoning based on pure numbers to find the area of a surface, employing grammatical rules to orient oneself in linguistic behaviour, etc.) but also in the sense of M  R (considering multiplication as a model for calculating the area of a rectangle in order to remind oneself of its commutativity, thinking up an example in order to reconstruct a specific rule for forming the plural of nouns, etc.).

1.3.3. Disciplines

Currently, the processes of modelling, reasoning at different levels of abstractions and so on are for the most part organised into disciplines or refer to them; they are supplied with specialised languages and procedures that standardise the forms of models, their internal processing, the links between different types of models, etc.. Furthermore, there are socially shared forms of knowledge which are not "accredited" but developed and passed down at popular level, into crafts, etc. (with their own models: proverbs, procedures, etc.).

Essentially, disciplines differ in that they organise models that refer to different types of phenomena. Mathematics is an exception: currently, it is characterized not by its area of application (it is no longer the language of physics) but by the kind of artifacts it employs; basically, its organisation into sectors is linked to artifact type, not to application environments (even though there are some borderline sectors - mathematical physics, information theory, econometrics, etc. - characterised in part by relationships with certain disciplines or technologies, and there is logic, which has mathematical activity as its area of application).

In mathematical modelling (i.e. when artifacts are mathematical objects), the process ER  EM is called mathematization. But also the use of mathematical artifacts in building a physical, biological, … model (for instance a physical law or a bionic model) is a mathematization. In a modelling activity which includes mathematization, the selection of ERs and the choice of EMs can involve assumptions, intentions, perceptions, intuitions, etc. which interlace mathematics' and other disciplines' aspects. With regard to this interaction and the nature of R, our "working definitions" of modelling and mathematization differ from the ones of Blum & Niss (1991): their R and AR are a real world and a real model which are outside mathematics, and they consider only mathematical models, whereas we want take into account also the mathematization of mathematics (see 1.3.1; cf. Freudental, 1968) and the presence of interferences with other disciplines.

1.3.4. Contexts

In a modelling activity, R is the context of AR, i.e. the setting or the conditions in which the aspects we want consider are found or take place.

The context of modelling includes the reasons for the choice of AR and the artifacts that one has at his disposal and means to use to build M. In Blum & Niss (1991) this context is described as a problem to solve, but the usual "didactical" meaning of problem solving can run down the complexity of the interests of the various "solvers "involved in a real modelling activity.

Contexts can be internal to a discipline (see 1.3.1). For an example, the context R of derivative concept can be the rate of the change of a function, while the context of formulating this concept can be the need of a tool for building the differential equation concept.

1.4. Situated teaching-learning

Among arguments in favour of including real situations of use and modelling activities in mathematics instruction (cf. Blum & Niss, 1991, de Lange, 1996) we can distinguish the ones that emphasize how referring to reality offers opportunities for forming critical and socially and culturally aware people, and the ones that focus on its potentialities in promoting an effective and balanced construction of mathematical concepts. In this paper we are considering especially topics that are linked to the latter ones.

In traditional teaching disciplinary concepts are not presented as models, but only as things to study. In example phases, if present, there is only AR as context of AM; it is usually introduced verbally, and in most cases is (1) a make-believe context used as an alternative to symbols for setting out a stereotyped disciplinary problem, or a quiz-type question (as in puzzle-magazines). Rarely, especially at secondary level, AR is (2) explicitly presented as "educational" simplification of real-life problem; type (2) contexts are sometimes used also at the beginning, as a springboard for the abstract introduction of concepts. Case (1) presents merely a caricature of the phase ER  EM; case (2) involves the phase of choosing ERs, but the delimitation of AR (i.e. its context, R) is usually neglected.

When discussing situated teaching, by context we usually mean the context of the modelling activity (see 1.3.4) in which the concept is introduced (it is introduced as M or a model used as an artifact to build M).

By referring the introduction of concepts to meaningful contexts, situated teaching aims to develop students' skills in managing the interaction between the various linguistic and semantic levels through which knowledge is organised and applied: in particular, the significant aspect of teaching how to employ mathematics for solving problems does not lie so much in pupils' internal mastery of the symbols used to build the model of a situation, but rather in their understanding of or confidence with the process by which the problem is transformed into a (more) formalised situation and subsequently further transformed (through internal processing and/or transfer to other - more formalised or less formalised - situations that are easier to solve) until the solution becomes clear.

Both the choice of the contexts and their didactic management (teacher's interaction with the students, organisation of learning material, etc.) are important to this effect. A context can be presented in various ways: (partially) experienced, evoked by drawing on student experience, etc. or described verbally and graphically as it happens for examples in traditional teaching. What's important is to exploit the potential cognitive resources of working with a non-prepacked situation for a prolonged period. By reorganizing (and referring to mathematics learning) arguments from Norman (1993), we can say it is well suited for:

1.5. Fields of experience

When the Genoa group was set up in the late seventies and tackled the context problem, it was pursuing a desire for school reform linked to the experience of its members, and drew on a varied background (study and reflection on the history of culture, skills in mathematical logic and applied mathematics which clearly revealed the cultural and technical shortcomings of "new mathematics", etc.). In teaching, it was natural for us to extend the definition of "mathematics" and "culture" beyond the knowledge that today's mathematicians and "learned men" possess to include the ways of thinking and reasoning by which this knowledge is developed and applied more or less explicitly in various human endeavours. Only at a later point was this outlook coupled with cognitive considerations (see 1.4).

"Educational theories" were approached in a non-rigid manner, considering them as "models" (for instance, we adopted some concepts developed by the French school of mathematics education - like "didactical contract" or "tool-object dialectics" - even though we did not share their reference to Piaget's constructivism; cf. Boero, 1993). When, to provide a framework for planning and describing the evolution of situated teaching in the classrooms, a specific "model" of our own was established, we chose a flexible one in order to avoid the risk of tunnel vision inherent in all modelling activities, particularly in cognitive research (cf. Norman, 1993, Ch.5): persisting in interpreting R on the grounds of a short-sighted or prejudicial selection of ERs, in trying to fit one's own model overlooking or disregarding other factors that would put the whole modelling process in doubt.

The established model is the field of experience concept (Boero et al., 1995), seen as a sector of human culture which the teacher and pupils (or similar characters involved in a learning situation) can recognise and consider as sufficiently unitary and homogeneous (from the standpoint of their schemata).

We can identify three main components through which references to a field of experience evolve in teaching/learning situations (we use "component" in the sense of polarity - by extension of statistics and physics uses - rather than in the sense of "part"): an "external" component made up of the portions of "reality" (objects, phenomena, information, documents, procedures, etc.) that are the subject of reflection, discussion and analysis; a component "internal to the pupil" constituted by his/her schemata, conceptions (including the unconscious ones), emotions, expectations, etc.; and a corresponding component "internal to the teacher", which also includes learning goals.

Fields of experience can be related to contexts in different ways.

The field of experience related to the way production costs form may be the context for a learning activity, but may also be recalled in an activity involving other contexts, e.g. activities concerning the limitations of mathematical modelling (under what conditions and up to what production levels can a linear model be used), or the set of linear functions (for recalling an application example, for consolidating the roles of two parameters, etc.).

Even in cases where the field of experience is the context of a learning activity, it does not necessarily need to be handled comprehensively in the classroom. It can be evoked, if, in terms of the pupil's schemata, it really is already a field of experience (so that the modes of learning examined in 1.4 can be activated).

A field of experience can also be gradually built up at school within work on other contexts (e.g. the field of experience of linear functions dealt with in activities related to economics, technology, etc.). This may then become a direct context for a learning activity (for introducing the definition of the linear function concept, studying the properties of the structure of linear functions, etc.).

1.6. Examples of the use of fields of experience as tools to organize situated teaching

We shall briefly illustrate through a few examples how the choice (and the construction) of fields of experience and contexts (internal and external) have been developed in our projects (see Boero, 1997, 1996 and Gruppo Didattico MaCoSa, 1997 for documentation about primary, lower secondary and upper secondary school projects, respectively).

We have concentrated on wide-ranging contexts, presented to students in a manner that integrates verbal and graphical forms, references to life experience or non-scholastic knowledge (handling money, technology, observation of nature, etc.), and, in many cases, activities that deal directly with the situation itself. We choose wide-ranging contexts not only for the cognitive reasons we discussed in 1.4, but in order to avoid giving the idea that the mathematization of a phenomenal area is a mere summation of many (only) mathematical models and historically developed as a summation of problem solving activities (see 1.3.3, 1.3.4).

References to aspects of daily life are not reduced to the reproduction of realistic problem (such as "make a cake" or "plan a trip"): "bodily development" is a real matter which interests pupils but its mathematization is not a real life problem. The "external" components are related to the "internal" ones: pupils' needs/knowledge (adolescence problems, biological conceptions, mathematical background, etc.) and teacher's goals/knowledge (relating to pupils' problems, introducing to statistics, biology and statistics knowledge, etc.). In the mathematics education literature (in both examples and general considerations) this kind of interplay with students' life is usually left out of account (see for instance Burkhardt, 1994 and de Lange, 1996).

The various contexts, combined with "traditional" example-contexts (see 1.4) and practice activities (mixed in different proportions depending on the school level: from primary to secondary school we pass from an "interdisciplinary integrated approach" to a "mathematics curriculum integrated approach": see Blum & Niss, 1991), are arranged within pathways which take account of mathematical, cultural and motivational background and priorities. Contexts are referred not only to non-scholastic experience and knowledge (see 1.3.4).

2.1. An example

Having examined several general aspects of the rationale behind context-based teaching and the way it can be organised, we shall explore in greater detail a few issues concerning how to choose and deal with contexts. In order to introduce these issues we refer to an example: the money and calendar fields of experience (see 1.6), the first ones considered in our primary school project.

2.2. Mathematics incorporated into contexts or used for rationalising them

When choosing a non-mathematical context in which a specific mathematical concept is to be introduced, consideration must be given to which aspects of the mathematical concept (1) are incorporated in the context and/or (2) have a meaningful role in understanding it. Mutatis mutandis, similar considerations can be made about mathematics contexts (see 1.3.4).

As far as (1) is concerned, we note mathematics is usually incorporated in everyday objects which are not merely "things" but include functions of use, knowledge, etc. (e.g. money, calendars, rulers, thermometers, scales, watches, speedometers, standard containers, diagrams, playing cards, calculators, radios, cameras, and so on); it is also incorporated in the social relations that are inherent in these objects (cf. Bishop, 1988). The teaching effectiveness of references to these objects depends heavily on two factors (as we saw for money and calendar contexts): the extent to which the aspects of function and social relations have been taken into account, and the links that the class-work maintains with out-of-school modes of learning and knowledge transmission (see the end of 1.4). For further investigation, problematic aspects and references see 2.4.2, Walkerdine (1988), Nunes (1992), Boero et al. (1995) and Fitz Simons et al. (1996).

Also in case (2), the effectiveness of references to contexts depends on the success in reproducing in the classroom forms of training and communication (collective discussion, mutual correction, etc.) common in out-of-school environments (work, play, sport, etc.; for further investigation and references see Norman, 1993), so that children (with teacher's help) can use, explicit and compare their own mental strategies (which are often richer and more directly linked to pure mathematical concepts than standard scholastic ones) and build prototypes that they can use as a support for abstract mathematical reasoning.

Connections between the components (external and internal ones) change according to the features of the field of experience. In "money", didactic activities especially concern the external component. In "shadows", they especially concern the relationship between the phenomenon (external component) and the pupils' conceptions (their internal components). In other fields the components have further ways of linking together.

2.3. Normative and descriptive aspects

In situations where mathematics is incorporated, there is often overlap of so-called normative aspects (mathematics used for building the object) and descriptive aspects (mathematics for describing the object). For example, if we graph Italian railway fares in relation to mileage (see Fig. 2), the discovery that this step function can be approximated with a continuous piecewise linear function not only leads to concise modelling but also reveals the "logic" behind fare definition.

In money and calendar fields this overlapping is evident. It is present in the drawing field too. But this twofold aspect is especially present in the automation field of experience.

The relation between normative and descriptive aspects in this field of experience provides a context for learning activities of various levels: • interpreting unexpected outcomes (seeking to understand the underlying mathematics and then mathematically modelling the behaviour of a calculator or computer program that gives an unexpected value for a numerical term, or the behaviour of Derive when it makes a mistake in solving or graphically representing an equation, etc.); • learning to use paint or draw applications (What is a figure within these applications? When ruler and compass are abandoned in favour of computer graphics, what are the new basic "geometric" entities?); • exploring/describing the working logic of black boxes; • exploring/studying artificial languages to communicate commands and procedures to computer; • pointing out the role of mathematical modelling in creating automated systems, simulations, etc.; up to • focusing on formal and historically determined nature of mathematical proofs (which cannot be identified with absolute ideas of reasoning, truth, etc. but depends on a definition of rule concept; cf. Davis, 1978).

2.4. Some difficulties inherent in situated teaching

Two of the most common objections to situated teaching of a discipline are that (1) reference to real contexts involves the use of models from other disciplines and the consequent presence of interferences of goals, conceptions and languages which are difficult to manage, and (2) reference to particularsituations of use may inhibit the shift to abstract reasoning, misdirect it or hinder the transfer of abstract reasoning to other contexts. These objections in some cases are a priori, in others correspond to teachers' real difficulties.

2.4.1. The predominance of mathematical aspects

When presented with concepts inside a modelling activity, pupils may face additional difficulties in relation to the knowledge of the context. At the same time, teachers may need extra knowledge and skills in order to fully exploit the learning potential that the context offer (see 2.2). It is not easy to draw a line between mathematical and non-mathematical work, nor to strike the right balance (between artistic and geometrical aspects in drawing field, biological and statistical ones in the bodily development field, physical and geometrical ones in the shadow field, etc.). There is a risk of concentrating too much on the introduction of mathematical models and neglecting the specificity of the viewpoints, models, languages, etc. of other disciplines (or knowledge domains), running into possible cultural errors. It is important to allow for mental interference with concept construction in other disciplines and with the pupils' related conceptions and misconceptions.

Depending on how closely the choice of ERs and the ER  EM association are linked to mastery of R, a mathematical model may be either conceptual or formal. Conceptual models are based on careful identification of AR factors and of the relations between them, as it occurred in the mathematical modelling of classical physics. Formal (analogical or empirical, for some authors) models are those in which we seek to define M so that there is a behaviour analogy (an "isomorphism" according to Bertalanffy, 1968) between AM and AR, omitting to study (by reasoning about R) in what sense the ERs are factors of AR, as it occurred in modelling heartbeat as relaxation-oscillations (cf. Israel, 1996).

Teaching often trivialises the modelling process, concentrating on a "formal" approach even in cases where a "conceptual" approach would be more meaningful, for example when the role of mathematics in physics is reduced to graphic representation of data or to the search for an approximation function. In other occasions, the application of certain models is extended groundlessly, for example when the sum of vectors is introduced for consecutive displacements and then also used, without explanations, for simultaneous displacements.

Intertwining and confusion with physics inevitably occur in (situated or not) mathematics teaching: mathematics only became a discipline in its own right in the late nineteenth century, an evolution that, emblematically, corresponds with the transition from (Euler's) concept of function as a graph of a physical phenomenon to the first mathematical definition of function, as "a set of pairs such that…".

This is a fruitful combination in educational terms but can also create serious conceptual confusion if differences are not clearly focused at some point. Consider, for example, the subtle presence of the time variable in many geometric activities (when dealing with movements, defining or describing curves, etc.), as well as the presence of "physical" reasoning in activities on euclidean geometry (using point as particle or centroid as barycentre, using movements without an axiomatic or analytical definition,…).

A "dual" aspect (with respect to 2.3 and this section) is that of interference with (or separation from) the ways in which mathematical concepts and techniques are used or introduced in other disciplines. This area has not been the subject of in-depth educational research yet (it is worth mentioning Pollak, 1976); we started confronting this issue at secondary school level.

2.4.2. Transfer

In this section we argue (with references to previous sections) that the transfer problem involves not only situated teaching but all forms of teaching which are not limited to experiential development of certain hard skills, and that teaching practice which explicitly deals with interference between different types and levels of knowledge seems to be better equipped to tackle it.

•   In 1.4 (and 1.3.2) we observed doing mathematics (not only applying it) consists mainly in transforming problems from a language or an (internal or external) context into another. These transformations are of various kinds, also because the ways in which mathematical concepts formed are various.

On the one extreme many concepts do not emerge from generalisations but surface in response to one or just a few situations, corresponding to primordial experiences. Consider for example the structure of natural numbers (even a child can quickly grasp the abstract structure,  , drawing on the subjective sensation of time passing) and the concepts of function, order and soon. Only at a later point when the abstract structure has been perceived can analogies be drawn with other situations.

On the other extreme there are concepts which have not been conceived as refinements of intuitive ones: many concepts originated as models of situations that had already been transformed into mathematical form (the group concept, the polynomial function concept, etc.).

In between there are cases of a complex and discontinuous transition from intuitive or everyday concepts related to "common sense" (the line as a pen stroke, a piece of cord, etc.) to concepts related to a mathematical practice but not rigorously defined (the lines of descriptive geometry, or of Euclid's Elements), and to formally defined concepts (the line as an equation or system of equations, or defined vectorially, or defined implicitly by a system of axioms, etc.).

•   One didactical aspect of these considerations is how to take into consideration the variety of forms of relation between mathematics and situations of use, at which we already hinted in 2.3 and 2.4.1.

Educating students to use concepts requires not so much tackling a lot of examples as dealing with some meaningful situations (prototypes: see 2.1 and 2.2) which point up the nature and the problems of the main kinds of transfer, enable students to grasp and experiment with shifts between different layers of meaning and can be internalized as emblematic stories (Schank, 1990): recollecting how the class dealt with a context may become a schema by which the pupil mentally arranges information, concepts, relationships between models and contexts (how to formalise, what the limits are of modelling, etc.), the strategies at play, the cognitive conflicts encountered, etc., as well as the feelings experienced. In order that this happen it is not enough to deal a lot of times with the same contexts: the apparent mastery of a kind of use can fade away shortly after, if teaching did not build communication channels between didactic activities and real situations of use (see 2.2) or other disciplines (see 2.4.1).

•   Another aspect is the control (during the shifts) of linguistic interference.

Interference between sectorial languages and natural language underlies many pupils' difficulties (cf. Pimm, 1994). This is true particularly of scientific languages, whose ambiguous terminology evolved before the "model" idea had emerged (scientific concepts as models instead of descriptions) and specialised languages had come into common use. The same is true (though less justifiably) for scholastic jargon, like the algebraic jargon that uses "to move right", "to cancel", etc..

When using terms that are also part of everyday language, there is a risk of retaining meanings that have no place in the discipline concerned (in front of 2x=0 a pupil can transform it into x=-2 by a movement to right, if he/she believes a movement cannot make "2" vanish). Likewise, when we reason about a concept and picture it in a less abstract context, we add "data" that may prove misleading. We saw some examples in 2.1 and 2.2; here are some more. By materializing the triangle concept we are unable to represent it in both a non-isosceles and a non-scalene manner. When thinking of a continuous function F as a graphic representation, we actually include conditions on F (finite length, countable set of slopeless points,…), etc.. Even a mathematically-versed person may experience difficulty in: dealing with the idea that A has the same number of elements as B even when AB (if he/she interprets the sets in a finite universe); grasping complex numbers without an ordinal relationship (if he/she links the number concept to real numbers); realising that, even if 3x²+3x divided by x+1 "equals" 3x, (3x²+3x)/(x+1) is not equivalent to it (if he/she interprets the division as one between numbers); and so on.

•   If it is true that relation of mathematics with reality is often mediated by other disciplines' models, transfer problem of mathematics involves transfer problems of these ones too; for instance, in a didactical activity about the mathematization of a physical phenomenon, it may be necessary to tackle the difficulties which follow from the fact that "force" is not an abstraction of the intuitive concept of "force" (related to "push", "effort", etc.) but can only be grasped in relation to other physics concepts. On this subject links with educational research in other disciplines appear weak. Problems related to modelling, misconceptions, student motivation for schooling, etc. are often treated with a separate, unilateral approach within each disciplinary sector of educational research (for instance, many issues in common to physics education emerge from reading McDermott, 1993).

•   In conclusion, pupils have to be prepared to use and control the shift from contexts to abstractions and vice versa, must be confronted with the leaps and breakaways that occur (with respect to common sense, to other knowledge domains, and to their previous experience) into the development of new concepts. They must also be aware of the limits of mathematization and of the applicability of mathematical language (outside mathematics, interpreting the "angle" as an unlimited figure maybe "wrong"). They must become conscious that every modelling activity presents the risk of tunnel vision (see 1.5): confining oneself to those things that can be dealt with using the disciplinary tools one has chosen.

The source of difficulties does not lie in the reference to contexts but rather in disregarding them or in neglecting the consideration of their characteristics while planning and managing didactic activities.

2.5. Other issues and open problems

In 2.2-2.4 we analysed how different modelling activities can present different roles of mathematics, different forms of interference with other knowledge domains, and different ways of interacting of the three components of experience fields. In 2.1 we alluded to other issues too, which we do not go into in this paper; in particular:

•   the vital role of curricular research based on contexts: where contexts are chosen and developed not just in response to exemplification established prior to the use of concepts, this research may permit a debate on teaching objectives in mathematics and their updating;

•   the "potentialities" of referring to historical formation of concepts as a source of clues for identifying meaningful (today too) contexts and as a source of contexts (e.g. analysis of historical documents to explore the evolution of algebraic thought); but also its "limitations", if one overlooks the fact that the network of concepts of students and the social diffusion of mathematics differ widely from those of scholars of the past (e.g. experience with cars - knowing the speedometer, talking about acceleration, etc. - offers references for the introduction of mathematical analysis that were not available to Galileo or Newton).

Context-based teaching can modify the respective roles that the teacher and the pupil play in traditional teaching. We mention some important related problems which actually deserve more detailed examination (for some references, cf. Blum & Niss, 1991; Boero, Dapueto & Parenti, 1996; Sierpinska, 1997):

•   pupil and teacher can no longer rely on their familiar reference points (syllabus schedule, identification of basic skills to be tested, etc.): it is necessary to outline a new set of basic skills and a new balance between experiential and reflective learning, to develop new criteria for the assessment which exploit the opportunity for dynamic evaluation (day-to-day, in various activities,…) that differs from pre-set and ad hoc testing, but overcome the risk of prejudicial evaluations;

•   the motivations and viewpoints of teacher-researchers (involved in designing projects or in critical experimenting and evaluating them) differ from those of "normal" teachers; the non-neutral act of choosing contexts means assuming greater educational responsibility; managing the transition from contexts to abstraction, from individual ideas to shared knowledge, etc. and the interaction with out-of-school ways of training and communication arouse conflicts with the conventional wisdom of school education; all that makes the question of spreading situated teaching complex.