1. Definition of the field of arithmetic in mathematics education
We may start with the classic definition of arithmetic as the "science of numbers", and may consider the whole of the activities regarding teaching-learning numbers as the "activities relevant to arithmetic in mathematics education". A less accurate but richer and more suitable definition may be suggested by the fact that the "science of numbers" includes several subfields, from investigation into particular classes of numbers (natural numbers, for instance) and their properties to historical and epistemological investigation into the "science of numbers" itself. Within each subfield we may then consider specific issues: for instance, the methods (which are often interdisciplinary nowadays) employed to deal with conjectures concerning prime numbers; or the distinction between "number" and "numeral" which suggests that algebra is seen as the science of general numerals or as the generalization of arithmetic; or the applications and generalizations which the knowledge acquired in the field of numbers (applications both inside and outside mathematics - see for instance the applications to the so-called "coding theory") has undergone. It is just in relation to the connections with other subfields of mathematics and other sciences that arithmetic, besides being the "science of numbers", might be considered as well the "science of algorithms".
These observations on the complexity of the "science of numbers" (both inherent complexity and in relation to the connections with other subfields of mathematics and other sciences) are useful to prevent a too narrow view of arithmetic education as it has recently happened when an exclusive relationship between natural numbers and set theory was considered.
These observations seem also suitable to include historically and culturally meaningful issues, which have more or less direct implications on the teaching process, within the field of research into teaching-learning arithmetic. We are, for instance, referring to observations involving a concept of the natural number set which considers it both "potentially" and "actually", not only as a "global" set but also as a set of "individual" numbers having specific properties. And again, we are referring to the cultural value of conjectures on natural numbers (see, for instance, the "Goldbach's conjecture" ) to build up a concept of mathematics capable of emphasizing its developing aspects through its open problems.
We are also referring to a revaluation of arithmetic as a field to add to geometry in order to develop argumentation and proof. And mostly, we are referring to the importance that, in our opinion, historical and epistemological studies of arithmetic have in suggesting possible explanations of students' difficulties (as in the well-known article by G. Glaeser of 1981[1], continued by Hefendehl-Hebeker in 1991[2]); and to the analysis of the connections between typical problems of the "theory of numbers" and other subfields within and outside mathematics (such connections may suggest interesting didactical situations - see, for instance, "Fibonacci's numbers" and their relationships with combinatorics and biology).
2. Main fields of investigation into teaching-learning arithmetic carried out world-wide in the last twenty years
Within such a complex frame, the investigations into teaching-learning arithmetic carried out world-wide in the last twenty years have especially devoted themselves to the following subjects:
We may assume that more than 70% of the articles concerning arithmetic published by the leading international mathematics education journals since 1975 has dealt with these issues. These areas of research have actually become of great importance for different reasons:
However, it is necessary to point out that other areas of study (of great importance because of their educational implications as they deal with issues which the current teaching has failed to convey properly, or because they involve the teachers' educational choices) have not been dealt with throughly. In particular, we are referring to:
3. Peculiarity of the Italian studies in teaching-learning arithmetic from the 1980's and on
In order to understand this peculiarity, it seems fit to refer to some specific aspects of the Italian history of mathematics at the end of last century and during the first three decades of this one. During that period some of the leading Italian mathematicians (such as Enriques and Peano) took active part in the international debate about the foundations of mathematics with original contributions concerning the "science of numbers" and its foundations. As known, during the same years, other European countries (in particular Germany) took part in the same debate as well. A characteristic of the Italian situation is the direct connection between the investigation into the foundations of the "science of numbers" and educational transposition (the proposed axioms and carefully considered epistemological issues directly influenced the more widely used textbooks - the most meaningful example is certainly the high school geometry textbook written by Enriques in collaboration with Amaldi).
This tradition and the richness and variety of the contributions brought by the Italian mathematicians have, first of all, created favorable conditions to have the university training (in courses such as "elementary mathematics from a higher viewpoint") for mathematics teachers include various issues concerning the foundations of arithmetic. But the same issues, re-thought in terms of connections with educational choices, and problems and theories of learning (heuristic, problem solving etc.) have recently fed and keep feeding the debate among Italian mathematics educators. All this may explain why a great number of works published by Italian mathematics educators in the last few years deals with questions concerning the foundations of the "science of numbers" and the "elementary" re-formulation of some of its complex issues. Some of these works, because of their reference to a strong historical tradition, are difficult to compare with the contemporaneous international studies especially in relation to the cognitive dimension within the study of teaching-learning problems. On the other hand, the wider and wider connections with international research have implied, in other works, a gradual, and greater attention to subjects of study which lack a generalized, institutionalized (within the mathematics environment) tradition of research and educational care in our country.
We may so trace the rather complex picture of the areas of study carried out in Italy from the end of the 1980's. The studies deal with one or more of the following issues:
Subjects B) and C) are rather widely dealt with by the works we have gone through. This may be explained by the importance that the matters concerning the renewal of the teaching of specific subjects have traditionally had among mathematics educators in Italy where the institution is basically stationary but the system allows room for individual teacher's and group's initiatives. A number of works deals with subject A) giving particular care to problems of framing and technical-cultural, historical and critical revision of contents.
Often subjects B) and C) are both implicitly and explicitly connected to the issue above.
Generally it is worth pointing out that, in accordance with the Italian tradition within the field of mathematics education, a great number of works deals with more than an issue. For instance, a work dealing with an activity concerning D) may include a proposal of type B) with a description of the test carried out (C) and the analysis of the students' behavior and results of learning (E).
It is somehow a result of this tradition that there are fewer works dealing with subjects D) and E), and that such issues have gained only recently ground within the community of the Italian mathematics educators (see Arzarello and Bartolini Bussi, to appear).
4. Italian studies in teaching-learning arithmetic from the 1980's on
4.1. Studies to revise the knowledge to be taught (numbers and arithmetic operations) in relation to problems of didactical transposition
In relation to the induction principle we will be reporting about two papers. In the first, Ferrari (1989), from the Pavia Group, analyzes the mathematical content of the induction principle and describes its intuitive aspects. A brief history of the principle itself is outlined and some didactical suggestions for junior high school are made. In the second paper (Lolli, 1994) some epistemological remarks are made to point out that the "induction principle" and the "order invariance principle" are equivalent for operations on finite sets (see for the cognitive aspect, "order irrelevance principle" in Gelman-Gallistel, 1978[3]).
'I sistemi di numerazione' (Capelo A.C. et al., 1990, from the Pavia Group) describes different possible number representations of integer and fractional numbers also in reference to past civilizations (the Sumerians, Egyptians, Greeks, Romans, Incans, Mayas). Numeration systems are analized from a mathematical (polynomial, almost polynomial, pseudo polynomial systems) and current language point of view.
The set of papers on 'I numeri decimali' (Ferrari, I, II, III, IV, 1992) stresses historical and foundational aspects: the (I) deals with algebraic and order structures of finite decimal numbers; the (II) deals with numbers with decimal point justifying their written expressions and rules of operations with them; the (III) introduces the decimal approximation of order r, with an error less than 10 -r, for a rational number a/b. The existence and the uniqueness of such an approximation is also proven; paper (IV) describes the periodicity of decimal numbers and the relationship between periodic numbers and generating fractions.
'I numeri primi' (Ferrari, 1991) while describing classic themes of elementary number theory (including various formulas which "produce" prime numbers, and Mersenne and Fermat prime numbers) also discusses some problems still open. It has educational aims and suggests activities for students of different levels of schools.
In the group of Rome, Cannizzaro (1993) synthetizes classic and recent results in a new perspective to interpret the evolutionary process of the formation of the number concept. The difficulties more often met by the students are interpreted on the basis of the works of different scholars such as mathematicians - refer to Peano and Enriques; psychologists - refer to Piaget, Davydov and Vygotskij; and educators - refer to van Hiele. Bernardi et al. (1991) produce materials for teacher training by presenting epistemological and didactical considerations.
Scimemi (1992a, 1992b) deals with some basic problems of indeterminate analysis which are recommended to junior high-school teachers as a good tool to introduce arithmetic. After briefly discussing what a good arithmetic problem should be like, some examples of problems creating linear diophantine equations are given whose solutions are obtained by algorithms of continued fractions. A very simple program in BASIC is added. Continued fractions are recommended as a subject to be enclosed in high-school and undergraduate curricula, as a good opportunity for teachers to introduce various basic concepts such as induction, limits (epsilon-delta arguments), approximation etc. Some classic applications are also given and some modern developments, such as control theory and chaos, are briefly mentioned. Scimemi (1994) also explains the connections between combinatorial analysis and arithmetic through a famous theorem of Euler theorem which states that the number of partitions of n in which all parts are different equals the number of partitions of n in which all parts are odd.
Scimemi (1992c, 1993) explains the basic laws of sound consonance and their use in establishing classic scales and describes the works of some XVI-XVII century music theoreticians who gave empirical solutions to mathematical problems involving irrationals as approximations of rational numbers.
4.2. Analysis of the arithmetic content, planning of educational situations and related investigations
This area of study (where the issues listed at items A), B) and C) above, connected to item D), are found with a different emphasis depending on the kind of work) includes several works produced at Modena and Parma within the NRD respectively coordinated by N. Malara and C. Pellegrino, and by C. Marchini.
Marchini (1988) compares various ways - as required by state programs - of introducing natural numbers in primary school and shows what difficulties may be inherent to various approaches. The approach proposed is an algebraic one. Marchini (1991) studies and compares the properties of addition and multiplication tables of natural numbers written in different bases with the operation table of modular arithmetic. The formal properties for the ordinary sum and product algorithms are stressed.
Marchini and Morini (1990) report on an activity for a first grade class. Division and multiplication are introduced without any reference to addition by using body motions and drawings, while Marchini and Micol (1990), still by using body motions and drawings, introduce an activity involving relative integer numbers and operations among them for a II grade class.
Marchini and Pellegrino (1993), using a colloquial style, introduce a geometric algorithm to determine the gcd and lcm among natural numbers.
Marchini (1990) deals with an operation, deduced from gld and called "semidivision", which is an inverse of multiplication. Some properties of it are shown and the operation is also presented in an axiomatic form.
Pellegrino (1992) re-works out the text of a well-known geometry problem concerning the construction of regular polygonals inscribed in a circumference and put it forward as a dialogue. In addition some important properties of the gcd and lcm concepts are discussed without using the "fundamental theorem of Number Theory".
Iaderosa (1994), from the Modena Group, deals with the subject of divisibility in the natural number realm with reference to grades from VI to VIII, according to the national Italian syllabus. The Author discusses the following: i) cultural value, qualifying aspects and aims of the subject at issue; ii) educational difficulties, frequent mistakes and their possible explanation: iii) proposal of educational strategies. In addition, it is outlined the way leading textbooks deal with divisibility.
Gherpelli and Malara (1994) outline some aspects of a project carried out by their research group which was aimed at stimulating algebraic thinking in grades VI-VIII. For them the number is a mathematical object studied by means of a series of problem situations - inherent to mathematics - in which the student is asked not only to single out properties and regularities but also to discuss the matter under specific hypotheses, formulate counter-examples and give demonstrations.
Navarra (1994), from the Modena Group, tells of an activity developed in some V and VI grade classes. Such activity concerns the positional system as a tool for investigating the multiplication algorithm by comparing different techniques and representations, such as the so-called "a gelosia" method (XVI century), Nepero's small sticks (1671) and Genaille's small sticks (end of XIX century).
4.3. Studies on educational proposals of "situated" teaching-learning for the meaning of numbers, arithmetic operations and written algorithms
Among the studies focusing on items B), C) and D) above, it is worth mentioning those carried out by the Genoa Group coordinated by P. Boero. In general, they aim at analyzing and enhancing situated teaching-learning mathematics through educational projects covering the whole syllabus for primary (I-V grades) and comprehensive (VI-VIII grade) school; potentialities and limitations related to this educational orientation are investigated as well.
Three studies on arithmetic have been carried out within the primary school project, which analize the students' approach to written algorithms and the building up of the concept of decimal number in suitable "fields of experience" (i.e. meaninful contexts).
Boero et al. (1989) analize III grade students' spontaneous strategies to deal with division problems, and the teacher's role in helping the students reach a universal, written algorithm.
Gazzolo and Rubini (1994) describe the building up of the decimal number concept and the use of the same algorithm to deal with the division between decimal numbers (for V grade students).
Boero (1994) considers the conflict between the Genoa Group's approach to reach a written algorithm and the traditional "transmission" of the standard algorithm for division.
In VI grade, the properties of natural numbers in the "field of experience" of arithmetic are important in relation to the potential quality of the mathematical activities (such as approaching mathematical proof and introducing algebraic language). A teaching experiment regarding these activities is described and discussed by Sibilla (1994).
Scali (1996) analizes the potentialities of economics (within the primary school project) with regard to the building up of the number concept and the meaning of operations. "Situated" mathematics teaching-learning is useful but not sufficient to have students construct concepts and skills related to natural numbers, it is needed the teacher's mediating role. Some aspects of this role (involving a cultural, cognitive, social and semiotic mediation) are thoroughly looked into by Scali (1994).
More generally, the teacher's role is investigated by Arzarello et al. (1994). The Authors, who belong to the Turin Group coordinated by F. Arzarello, outline (by means of examples drawn from arithmetic) their lines of research into the subject; the Brosseau's theory on didactic situations is criticized, a new model is pursued and, at the end, learning mathematics is described as a result of social symbolic mathematical activities.
4.4. Experimental studies on the potentialities given by computers to develop arithmetic skills
This section especially comprises subjects listed under item B) and C) above and is well represented by some works carried out by the Modena Group. These studies belong to the junior high school project aimed at introducing the computer into mathematics education as a device which helps to think.
Pellegrino and Garuti (1989) outline the problems dealt with and the management of class activities. They report about a teaching experiment where the students, by using a computer as a learning tool, determined the number of divisors of a positive integer n, and this led them to consciuosly learn the following concepts: divisor of a number, prime number and factorization of a number.
Pellegrino and Garuti (1993) also report about a class activity which carefully combined a spontaneous approach to the recursion problem with the development of a reasoning about the positional representation of natural numbers and base changes.
Malara and Guidi (1988) describe a teaching experiment on algorithms by which one can compare or compute natural numbers represented as sequences of symbols. The aim is to show the possibility of representing natural numbers in different ways and make clear the distinction between numbers and particular representations of numbers, as well as between arithmetic operations and particular algorithms for them.
Malara and Garuti (1991) put forward three problem situations (the LOGO simulation of the odometer, the Russian multiplication and the sum of the first a hundred natural numbers) to study the process the student goes through to acquire the table as a bridge between the building up and/or analysis of an algorithm and its transposition to the computer.
4.5. Quantitative and qualitative research into the students' knowledge and the teachers' concepts about particular arithmetic skills
These studies chiefly deal with item E) (closely connected to items C) and D) above). During the time period considered (1988-1995) systematic investigation has been carried out by the Padua Group coordinated by C. Bonotto; a study has been started by the Naples Group coordinated by A. Morelli.
The Padua Group has studied the teaching of rational numbers and the learning difficulties related to them. The Group has worked out a specific didactic project for III-V grade students which has been tested in various classes (see Bonotto, 1991). The information gathered by testing students of age between 10 and 13 and not previously involved in the project has shown difficulties in mastering the meaning of decimals, the relationship between meanings and written conventions and between fractional representation and decimal representation, and finally in ordering sequences of decimals (see Bonotto, 1992, 1993 and 1995).
The rules, incorrect but somehow logical, and the way they work and the motivations supporting them, which the children applied in comparing decimals have been analyzed.
A hypothesis was then formulated according to which these results may depend not only on the inherent difficulties of the subject but also on the way teachers carry on their teaching and on their conception of what and how to teach. The work was then continued and an investigation into the methodological and didactical choices of teachers from the districts of Padua, Vicenza and Treviso was carried out. The results of this investigation, partly reported by Bonotto et al. (1994) have shown that the starting hypothesis was right. Little time and little energy are invested in building up meanings (all the care focuses on formal written rules and conventional aspects). In addition, the little usually done to build up meanings is too partial towards fractions seen as "operators", while very little is done to make connections between decimals and decimal measures (see Bonotto and Maddalosso, 1996). The teachers seem to put all their efforts in having the students acquire external representation systems (not coordinated either) (cf Vergnaud[4]). The only reference situations usually taken into account in primary school classes are the classic cakes (for fractions), while the operational invariants that teachers stress more are related to calculation rules and the formal properties of operations with rational numbers.
What the investigation with the teachers brings out for natural numbers is consistent with what stated for rational numbers: a great effort put into teaching rules of writing (but not much in dealing with the main references of writing), and an unbalanced concern about the meaning "cardinal" against very little attention paid to the meaning "measure", etc. More generally, the work done at school seems to have no connection with "the everyday knowledge in the arithmetic field", that is no connection with the rich experience the students gain about numbers before and outside primary school, which should instead be properly used, as Basso and Bonotto state (1996).
Bove et al. (1994) deal with an empirical study carried out by the Naples Group. They compare the results of a test submitted to 10-year-old students with the evaluation of their answers made by two groups of primary and lower secondary school teachers. They report that the comparative importance and the attainment of classic arithmetic topics (even if they are supposed to be mastered by the end of V grade) vary considerably from one teacher to another and from primary to middle school teachers.
4.6. Empirical studies suggested by historical, epistemological and cognitive considerations
This section comprises studies carried out by mathematics educators belonging to different Groups; what they share is the gathering of information about the students' behaviour and the interpretation of it according to historical, epistemological and/or cognitive hypotheses (therefore now it is especially the turn of items D) and E) above).
As to the representation of quantities in pre-school children, Aglì and Martini (1995), from the Bologna Group, deal with an empirical study about pre-school children's strategies employed to represent quantities and their spontaneous approach to number symbols. Eight different situations relating to games and practical subjects were systematically monitored and the children's written outcomes were classified.
With reference to zero, Bianchini and Velardi (1992), and Campedelli (1994) from the Florence Group have focused on problems related to the "special number" zero and compared the knowledge of students entering university with episodes and ideas taken from the history of mathematics. The Authors state that there are some difficulties about zero and infinity common to all students.
As to the difficulties met in getting over additive strategies in multiplication problems, Grugnetti and Mureddu Torres (1991) from the Cagliari Group deal with an experimental study on the students' behaviour while they try to solve a "ratio" problem out of close context with related rule. The analysis and interpretation of the students' answers are carried out at an epistemological, cultural, anthropological level and at a didactic, cognitive level.
In order to stress the influence of the analogical thinking in working out number properties and regularities, Bazzini (1994) from the Pavia Group describes and discusses a teaching experiment concerning two teaching units for primary school ("The calendar of the robots' realm", I grade, and "Multiples and not multiples of a given number", IV grade).
Mariotti et al. (1994) from the Pisa Group carried out an experimental research project aimed at investigating the evolution of the number concept. A questionnaire was drawn up and administered to students of different age classes. The paper discusses the results and some teaching implications. Previously, Mariotti and Bianchi (1991) had looked for the intuitive aspects of the divisibility relation in the realm of natural numbers. Fifteen classes of middle school students has been questioned. Both works may be related to the theory of intuitive models and the basic distinction between primary and secondary intuitions.
4.7. Transition from arithmetic to algebra: epistemological and cognitive studies
In Italy the transition from arithmetic to algebra has been particularly studied by the Genoa, Turin and Pavia Groups. These studies focus on the observation of students (E), which is led by considerations about the problems listed under item A) and D) above.
From VI to VIII grade, the transition from the learning of mathematics based on "meanings" built up in everyday situations to the study of inner regularities and constraints is a real "jump". Chiappini and Molinari (1994), from the Genoa Group, try to look into some aspects of this "jump" when it involves the approach to the multiplication and division of relative numbers.
The transition from sematically based behaviour to syntactic rules is also investigated by Arzarello et al. (1994), from the Turin Group. This empirical study deals with the algebraic nature of some problems about fractions, and looks into the analogies and differences between the students' strategies for manipulating fractions and some typical algebraic flaws.
As to the arithmetic-algebra transition, Reggiani (1994), from the Pavia Group, put forward some observations about the students' conventions used for relative numbers based on their mistakes and verbal reports. In the Author's opinion, the student's awareness of the conventions used in the arithmetic and algebraic fields is a crucial point of the arithmetic-algebra transition.
Cavallari et al. (1994), from the Pavia Group, carry out an experimental study into the equality sign in the arithmetic and algebra fields. A test was worked out and administered to V and VI grade students. The analogies and differences relevant to the results are described. Another test with VI grade students to look into their being aware of the existence of two different codes was carried out; the paper describes it as well.
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| Paolo Boero | Cinzia Bonotto | Lucilla Cannizzaro |
| Dipartimento di | Dipartimento di | Dipartimento di |
| Matematica | Matematica | Matematica |
| Università degli | Pura ed Applicata | Università La |
| Studi | Sapienza | |
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| Boero@dima.unige.it | Bonotto@pdmat1.math.unipd.it | Cannizzaro@mat.uniroma1.it |