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Syntax and Semantics in Algebra Jean-François Nicaud, Denis Bouhineau, Jean-Michel Gélis

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Syntax and Semantics in Algebra Jean-François Nicaud, Denis Bouhineau, Jean-Michel Gélis Syntax and semantics in algebra Jean-François Nicaud, Denis Bouhineau, Jean-Michel Gélis To cite this version: Jean-François Nicaud, Denis Bouhineau, Jean-Michel Gélis. Syntax and semantics in algebra. Pro- ceedings of the 12th ICMI Study Conference. The University of Melbourne, 2001, Australia. 12 p. hal-00962023 HAL Id: hal-00962023 https://hal.archives-ouvertes.fr/hal-00962023 Submitted on 25 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1 Syntax and Semantics in Algebra Jean-François Nicaud, Denis Bouhineau Jean-Michel )*lis I IN, University of Nantes, France IUFM of Versailles, France $bouhineau, nicaud%&irin.univ-nantes.fr gelis&inrp.fr This paper is the first chapter of a cognitive, didactic and computatio- nal theory of algebra that presents, in a formal .ay, .ell /no.n elements of mathematics 0numbers, functions and polynomials1 as semantic ob2ects, and expressions as syntactic constructions. The lin/ bet.een syntax and semantics is realised by morphisms. The paper highlights preferred semantics for algebra and defines formally algebraic problems. 3ey.ords4 syntax, semantics, formalisation, algebraic problem. Introduction This paper describes a theory of algebra .ith cognitive, didactic and computational features. 5e 0the three authors1 have a bac/ground in mathematics and computer science, and some /no.ledge in mathematics education. T.o of us have been secondary mathematics school teachers. 5e all have been for long designers of computer systems helping students to learn algebra, involved in the Aplusix pro2ect 6Nicaud 1778, Nguyen-9uan et al. 1775, Nicaud et al. 1777;. 5e have been engaged by this activity to elaborate a theory of algebra .ith cognitive, didactic and computational features. The reason .hy .e present this theory here is that .e thin/ that the interest of the theory goes beyond the design of systems and that the theory can be a contribution to mathematics education. The first phase of this theoretical .or/ is described in the J.M. )*lis PhD dissertation 6)*lis 1778;. It is a technical 0hard to read1 document specialized for polynomial factoring. 5e are no. enlarging the problem domain and re.riting the theory in a more easy to read style. This paper is the first chapter. Syntax and semantics are present every.here in algebra. This is emphasized by researchers in mathematics education, sometimes explicitly, sometimes implicitly, i.e., in describing situations. Until no., no formalisation of these concepts has been built. This chapter is a tentative to build such formalisation. It is based of a theory of computer science called algebraic specification 65irsing 1770; or formal specifications. Several aspects of signification have been identified 6Arzarello et al. 2000, Drouhard 1772, Lins 2000;, this chapter is only devoted to the one generally called denotation by the t.o formers. This theory is presented to the T.elfth IAMI Study in the section Approaches to algebra. It is mainly a formal approach for syntax, semantics and problem solving in a general sense. Some aspects also concern the sections Why algebra? and Language aspects of algebra. Informal significations. Before entering the sub2ect, .e .ill say a little about some .ords that .ill be used in this paper, .ords li/e ob2ect, concept, abstract, concrete, language, syntax, semantics. These .ords .ill occur in formal and informal discourses. In formal discourses, they .ill be defined explicitly 0e.g., .e call conceptB1 or supposed to be defined else.here 0e.g., let us consider a set of conceptsB1. In the informal discourses, these .ords .ill more or less have the significations that follo.. Objects are things .e manipulate explicitly, physically or mentally. There are concrete ob2ects that .e can see and touch, there are abstract ob2ects that are .ell-formed mental constructions .hich are explicitly manipulated. As the readers of this paper are researchers and teachers, the authors call ob2ects .hat they .ould li/e the readers to consider as ob2ects. This 2 does not mean that they are ob2ects for students. A ma2or cognitive and didactic problem consists of deciding .hen and ho. some of these ob2ects are supposed to become ob2ects for students. Cb2ects are basically semantic. They have a raison d‘être that gives them signification. Concepts are mental constructions of various sorts related to ob2ects or to other concepts 0e.g., piece of furniture, temperature, number1. Words and sentences are assemblages of signs 0letters, symbols, sounds, etc.1 on particular supports 0paper, blac/board, screen, voice, etc.1. A language is a set of syntactic rules that indicates .hat .ell-formed .ords and sentences are. The role of a language is to allo. to describe ob2ects, concepts and relationships bet.een them .ith signs. For mental constructions, this role is fundamental, because it is the only .ay to communicate about ob2ects and concepts. 5e .ill also use forms for syntactic constructions. 5ords, forms and sentences are concrete 0.e can see or hear them1. They are basically syntactic. The sentence x represents y means that the syntactical construction x represents the semantic ob2ect y. 5ords, forms, sentences become ob2ects .hen .e describe them .ith sentences. This is a necessary confusion. In the case of algebra, algebraic expressions are basically syntactic4 they are made for describing numbers, polynomials, functions, etc. But an important part of the algebraic activities consists of reasoning on expressions. That means that expressions also are ob2ects. In this chapter, algebraic expressions only appear as syntactic constructions, they .ill also appear as semantic ob2ects in further chapters. Abstraction has t.o complementary meanings. First, abstract is used to Dualify non- concrete ob2ects, second, it is used to Dualify ob2ects obtained from others by omitting a part of them 0see models belo.1. There are several degrees of abstraction for each meaning. Theories and models. 5e consider theories as frame.or/s for scientific .or/s. There are informal theories, particularly in human sciences. There are formal theories, particularly in mathematics, physics, computer sciences. Theories have4 011 cognitive features .hen they concern human behaviour and /no.ledge, 021 didactic features .hen they concern human learning and teaching, and 0E1 computational features .hen they include mechanisms for calculating certain ob2ects and concepts. The theory described in this paper has cognitive features, but there .ill be no discourse on that point, .e simply al.ays consider algebra for humans. The theory has no advanced didactic features described in this chapter. It is a frame.or/ .e thin/ adeDuate for didactic constructions, but .e do not ma/e propositions about the teaching of algebra, except a general one in the conclusion. The computational features of the theory are indicated. Models can be vie.ed in t.o different .ays. First, as often in science, a model is an abstraction of a concrete ob2ect. The electric schema of a car is a model in this vie.. It does not include mechanic, aerodynamic aspectsF it does not include the details of electric components. Second, as it is in computer science, a model is a construction verifying a formal system. These t.o vie.s are complementary4 a model may be, at the same time, a construction verifying a formal system, and an abstraction of a concrete ob2ect. In this paper, .e .ill mainly adopt the second vie. of models 0construction verifying a formal system1. Assimilation. Very often, .e are ma/ing t.o sorts of assimilation in algebra. The first one is the syntax semantics assimilation4 the .ord representing an ob2ect is assimilated to the ob2ect. It is the case for representations of natural numbers4 12 is usually seen as a natural number, although it is a succession of t.o digits. This sort of assimilation is important4 it avoids saying continuously GrepresentsH F it allo.s to reason on the semantic ob2ects through the syntax, as mathematicians use to do 6Arzarello et al. 2000;. Io.ever sometimes it causes trouble because it hides syntactic constructions on .hich the reasoning applies. In education, this assimilation seems to be al.ays done implicitly. 5e are not sure this is the best .ay to teach algebra. The second sort of assimilation is the inclusion assimilation. It is a conscious assimilation that occurs .hen elements of a set are assimilated .ith elements of another, for E example, .hen numbers are assimilated .ith polynomials .ith degree zero. This sort of assimilation is important too, it avoids the continuous presentation of relations bet.een the assimilated ob2ects. It is sometimes necessary to undo for a .hile an assimilation, .hen one .ants to be clear on a particular point. Plan. The first section of the paper describes shortly .ell /no.n elements of mathematics that are numbers, functions and polynomials. They are described as semantic ob2ects, .ith a language for representing them. In section 2, a formal system is presented4 numbers, function and polynomials appear as semantic models of this formal system .hile a unified syntax for the expressions appears as a formal model of the formal system. Section E extends the constructions of section 2. In section 8, .e present our preferred semantics for algebra. Last, in section 5, .e define algebraic problems. Numbers, functions and polynomials In this section, .e describe shortly numbers, functions and polynomials. They are semantic ob2ects, .ith a language for representing images of functions.
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