1. Introduction
Logic was included in the official Italian curriculum a few years ago and consequently Italian research in this field is not as developed as in other more traditional subjects.
The curriculum for grades 1-8 (Italian primary and middle school) includes specific contents and methodological guidelines that are closely related to logic. The high school curriculum (grades 9-13), on the contrary, has not been changed for over 50 years, but large-scale experiments (that involve a considerable number of schools) concerning major changes in science and mathematics curriculum have been carried out. At this school-level, logic has been introduced in connection with computer science, with the purpose of promoting a general improvement of contents and methods in mathematics education.
Consequently, today logic is included in mathematics curriculum of almost all grades, from primary to high school, even if in different ways. A common feature is the emphasis on the educational value of the teaching of logic and the interplay with linguistic education.
The trends of Italian research in the teaching of logic can be summarized as follows.
2. Theoretical studies
A considerable number of studies are concerned with both curriculumr improvement and in-service training. A reason for this is the lack of specific training of both in-service and prospective teachers. Moreover, specific books containing teaching ideas can hardly be found; the textbooks that exist are often inadequate: even those that present the specific contents correctly generally regard logic as a specific subject, to be studied separately, rather than as an attitude[1] and do not comply with the guidelines included in the official mathematics curriculum. For this reason some studies present teaching units and reports of teaching experiments (with the related hypotheses and reference frames) for training purposes. This focus on teachers' training also agrees with the Guidelines for logic education[2] published by the Association of Symbolic logic: <<Most important, make sure the instructor is interested in and well grounded in logic>>.
2.1. General
In this section are discussed papers concerning the role of logic in mathematics education from a general point of view or focusing on topics without a close relationship to a specific age level.
We start with some papers of Lolli (1989). His theoretical and historical research on the interplay between pure and Applied mathematics and, in particular, on the role of proof in mathematics has produced results that are original and profound. From an educational point of view, his papers have contributed to make clear the nature of links between mathematical logic and human reasoning. This should help people to avoid two different misinterpretations: on the one hand, some oversimplified versions of fallibilist philosophy that deny the value of proof in mathematics, on the other hand, the identification of logic (and mathematical reasoning) with the teaching of few chapters of mathematical logic, such as the truth-functional definition of logical connectives. A careful analysis of the trends in the research on logic education in Italy have been carried out by Arzarello (1991), who has pointed out the links between logic and computer science in education. The links between logic and mathematics education are pointed out by Marchini (1989c, 1989e), Ferro (1993c), Bernardi (1989). In particular, Marchini analyzes the links between natural language and logic models and suggests some teaching ideas, whereas Ferro carefully sketches purposes and features of a first approach to logic.
Lolli presents a detailed account of mathematical induction, which includes an historical analysis and the explanation of the links between induction and different topics in mathematics and computer science. Bonotto and Zanardo (1990) discuss the interplay between artificial and natura languages in mathematics education. Marchini (1990b, 1990c) emphasizes the widespread use of substitutions in mathematics. In particular he gives a number of examples pointing out the role of substitutions in the definition and representation of relations. Bernardi has written some papers devoted mainly to teachers' training and to the suggestion of teaching ideas. In this section we mention his paper (1988) devoted to the discussion of the role of some particular logical games in mathematics education.
2.2. Curriculum
In this section are discussed papers concerning mathematics curriculum at different age levels.
Official curriculum for primary school include a lot of topics under the headline `logic'. In particular they include the language of sets, introduced as a tool to express some mathematical properties in a simple way but not as a requirement for learning natural numbers. Sets are presented related to classifications, with strong links between logical and set-theoretical aspects. They include some aspects of combinatorics and computer science and activities related to pattern finding and ordering. In the recent years in Italy, there has not been a large number of studies on these issues. Italian research in logic education related to primary school is often interested in pointing out the differences between natural language and the language of logic and to exploit the opportunities to improve linguistic competence. Natural and formal languages are regarded related to their specifying settings and purposes. Most studies do not consider any of them more important than the other, as far as mathematic education is concerned.
Researchers seem to be aware of the danger of repeating the errors made when set theory was included in the curriculum of primary school. For this reason logic is not generally presented as a specific mathematical subject and, in particular, direct introduction of truth-tables is avoided, because this topic may be regarded in a way that is very narrow and also poor from a cultural viewpoint. Some papers present `logical environments', similar to Smullyan's logical puzzles, where the semantics of propositional logic is introduced in a natural way in order to allow students to learn a correct usage of truth-functional connectives through examples. In most papers logic is regarded as a tool to better understand natural language rather than to teach pupils reasoning.
These research trends, that have been independently developed, perfectly agree with the ASL guidelines on primary and secondary education: <<Everyone needs to be able to tell, at some intuitive level, the difference between a valid argument and an invalid one. One needs to be able to give some simple arguments, and to spot logical fallacies in others. Just how much one needs to know depends, of course, on many factors. A mathematician, for example, presumably needs these skills honed to a finer edge than someone in a manual vocation.
Recommendation: Promote and facilitate logical (i.e. analytical) reasoning at an early age.
The notion of a correct proof and the method of debunking fallacious proofs by means of counterexamples should be introduced as early as possible. It is not necessary, or even advisable, to introduce specific courses in logic. Rather, the recognition of valid and invalid arguments should be an integral part of education in the sciences (mathematical, physical, biological, and social), and the humanities quite generally. After all, part of what it means to be literate involves the ability to distinguish valid reasoning from invalid reasoning.
Strategy: Ages 5-9: Integrate logical matters on "good" and "bad" arguments into other material in a completely informal way with effective "inquisitive" teaching techniques.
Among the papers we mention Ferrero (1991) who present some ideas to promote the learning of linguistic connectives. The focus is on natural language and the situations suggested to promote learning are closely related to pupils' experiences. Navarra (1991) focuses on the relationship between logic and natural language too. In his proposals the reflection on language starts from stories invented by the teacher which are designed to promote accuracy in the use of connectives. At the same time, various representations are introduced such as Eulero-Venn and Carroll diagrams and different kinds of graphs and tables. Other papers are concerned with a general discussion on the official curriculum for primary school. Cannizzaro (1988) compares the official curriculum with the previous ones and with the curriculum for middle and high school. Marchini (1993a) discusses curriculum as well, criticizing some parts, such as combinatorial aspects, and points out the difference between logic and natural language. In (1993b) he deals with the introduction of sets in primary school, criticizes some current practices and propose some interesting teaching ideas.
The official curriculum for middle school (1979) first introduced topics related to logic in Italian school, such as the comparison between logical and probabilistic thinking and the recognition of analogies and differences between algebraic structures. The curriculum for primary school (1985) show traces of a wider perspective on logic education and of the related debate.
Most of the remarks on primary school apply to the reasearch related to middle school as well.
Here two different trends of research can be found that are clearly different from those of primary school. The first trend is concerned with the understanding of mathematical arguments and a first approach to mathematical proof through linguistic analysis, following the well-known opinion that a poor linguistic competence is a major cause of the lack of understanding of word problems. To avoid these obstacles, activities of classification, decomposition and recomposition of `difficult' words are proposed, together with the search for the Latin or Greek origins of some words.
The second trend is concerned with word problems and the so-called `logic of discovery' and the production of resolution strategies, euristic reasoning by means of conjectures to be validated or refuted by counterexamples. Discussions in class are often proposed in order to facilitate the analysis of the different conjectures and their validation or refutation.
The problem of assessing competence in logic is not very popular in Italy nor elsewhere, even if it deserves a careful analysis, in order to allow people to reproduce teaching ideas and to apply them in different school environments. No important study has been found on structures, regarded as environments where formal languages can be interpreted. Few papers deal with some properties of binary relations from both a syntactical and semantical viewpoint. The comparison with the recommendations of the ASL concerning teaching strategies for the ages 10-13 is interesting: <<Emphasize heuristic strategies in the spirit of Polya's How to Solve It. Take first steps towards recognizing the form of statements, formulating corresponding rules, and using interpretations. Give some word problems that have a distinctively logical component to their solution>>. In this regard it seems that there are only few studies focusing on formalization processes as steps to the learning of logic.
The study of Navarra (1993a) well represents the first trend. The paper deals with the teaching of De Morgan's laws. The author presents an interesting teaching sequence that starts from a story and introduces different kinds of representations, including Eulero-Venn diagrams. Students' difficulties are discussed as well. Discussions and some criticism on middle school curriculum can be found in Marchini (1994, 1995) and Bernardi (1990a), together with some teaching ideas.
In Italy there are great differences between the various kinds of high schools. Some of them have been designed in order to prepare students to get a job, other (called Licei) to attend university courses. Italian research does not take into account these differences very much, even though they strongly affect curriculum and methods. Often the studies regard a generic Liceo student, maybe because the curriculum appear richer and more stimulating.
The tradition of rational geometry has affected the Italian teaching style for a long time. Still today there are studies related to logic that start from the introduction of geometrical concepts and theories. Other subjects are related to logic as well: logic as an example of formal language and related to grammars, arithmetic in connection with the induction principle and its applications, analytic geometry as an example of interpretations of the language of geometry.
Mathematical proof in the frame of the hypothetical-deductive organization of classical theories has been the object of specific studies concerned with students' difficulties, the epistemic status of the propositions proved and the processes of proof construction. There has also been some study on the structure of proofs, which generally point out the superiority (from an educational point of view) of natural deduction.
The tentative curriculum for high school include logic not only as an attitude but also as a subject to be analyzed. Some papers try to present well structured teaching sequences, generally focusing on the differences between syntax and semantics, language and metalanguage. Other papers propose a problem-based teaching of logic and suggest some examples of problems. Some papers of this kind are not framed within any theory of learning, but focus on the obstacles related to foundational problems and are devoted to teachers.
The ASL Guidelines suggest, as regards teaching strategies at the ages 14-17: <<Teach the explicit use of logical notions and techniques to give proofs, counterexample, etc. mathematics courses are a natural place for the inclusion of such material>>.
Italian research agrees with the ASL recommendations. Also the tentative curriculum include many topics closely connected to the specific contents of mathematical logic. Two books deserve a particular mention: the proceedings of the meeting on the teaching of logic, held in Rome in 1988 and promoted by the school of logic, which presents a wide selection of Italian research on logic education and includes many contributions by important foreign researchers, and the proceedings of the training school of logic for high school teachers, organized by the Department of education and the AILA (the Italian Association of logic and Applications) and published in 1995. In this last publication the topics included in the tentative curriculum are dealt with from an educational viewpoint.
Nevertheless, some different opinions may be found among the studies on curriculum in high school. Dapueto (1989, 1992), argues that logic should not be taught as a subject. In his opinion students should learn to use different models and languages in order to understand and master a wide range of real-world phenomena. Ciceri et al. (to appear), Furinghetti and Paola (1991) and Furinghetti (1992) deal with mathematical proofs in high school, with a particular regard to the linguistic structure of texts and to the interplay between intuitive and formal approach. Speranza (1989, 1993) discusses the interactions between natural language, symbolism and logic as well, focusing on epistemological aspects. Bernardi (1993a, 1995) proposes some teaching ideas and suggests a problem-oriented teaching of logic. In this line he proposes a selection of problems. Bernardi and Tazza (1990) discusses student's difficulties in the learning of quantifiers. In this section we present two papers concerned with proof at university level as well. Boero analyzes the difficulties of first-year university students when dealing with mathematical proofs. In his opinion they fail to understand the reason why proofs are needed, or taught, in mathematics. He argues that in high school students learn proofs in the same way another could learn a poem and are not accustomed to construct mathematical statements themselves and recognize the relationships between statements and proofs. Bruno Longo (1992) surveys some errors of university students in the costruction of mathematical arguments. The errors are classified and some hypotheses are stated to interpret them. The papers dealing with mathematical reasoning at the beginning of university study seem to suggest that the ways students do and learn mathematic in high school are largely unsatisfactory. This criticism involves logic education as well.
3. Empirical research
In recent years Italian researchers have published fewer empirical papers in logic education than theoretical ones. This may be related to the need of making clear some theoretical issues before performing experiments (also to avoid the already mentioned misinterpretations of logic education). Nevertheless, empirical studies testing the different hypotheses will be required in order to go deeper into the topics.
Vighi et. al. (1991) deal with the existence of spontaneous skills in the fields of logic and probability. Their results suggest that children's thinking differs significantly from the corresponding models of mathematical logic and probability. Moreover, the role of teachers it is pointed out, as far as pupils' performances differs more from class to class than within the same class. (Take note that in Italy in each class children at different skill levels are included.) Rinaldi and Vighi (1992) explore the use of arrows to express mathematical relations, in particular divisibility. Costa et. al.(to appear) deal with geometrical reasoning and the role of pictorial representations. They argue that perspective drawing can work as a semiotic tool towards the statement of mathematical properties that shares some features of the statement of theorems.
In Italy, empirical research on logic education in middle school seem to follow different guidelines. Some papers focus on the first constructions of proofs in mathematical contexts. Iaderosa (to appear) takes into account the discussions on proof in mathematics education (from Balacheff to Duval). She is interested in the transition from the evaluation of statements to the production and understanding of argumentations and then of proofs. Her evidence suggests that pupils at this age level can understand and produce some mathematical arguments. She suggests that the difficulties with symbolic language may be even harder than for the transition from arithmetic to logical. Her study is a good example of research that presents both a theoretical frame that takes into account the most relevant opinions in literature and empirical results to test the hypotheses. Gallo and Grange (to appear) deal with the interplay among deduction, induction and cognitive development at the age of 11-14. Their research analyzes the rise and evolution of logic functions, following the positions of Wason and Johnson-Laird and criticizing Piaget's positions on propositional calculus as a model of deductive competence.
Margiotta describes in (1991b) her experiences with substitutions in middle school. Her research is connected to Marchini's papers (1990b, 1990c). Navarra (1990, 1992, 1993b, 1995) presents many examples of activities that closely connect logical thinking and the development of students' powers of expression. His evidence shows the productivity of his method, based on the construction of stories that force students to an accurate use of connectives.
Malara (1989) studies the introduction of implication (as a connective) and modus ponens (as a deductive rule). Boero and Garuti (1994) present a teaching experiment with students at the age of 13 concerning the production of geometry statements and the comparison with the statements given in the textbooks. The evidence shows that a constructive approach to Geometry statements is possible. Boero, et al. (1995) present an analogous study connected to the approach to statements and proofs of elementary arithmetic problems. Pellegrino and Iaderosa (1990) present a teaching experiment involving the use of Tangram. This study is more related to intuitive geometry, but it is also interesting from the viewpoint of logic as far as the activities have promoted students' reflections on ideas such as proofs, counterexamples, paradoxes and on the distinction between example-based argumentation and mathematical proof.
There are only few empirical studies on logic education in high school. Furinghetti and Paola (1991) deal with the problem of students' understanding of mathematical texts. Barnaba et al. (1992) present an a-priori analysis on different interpretations of the equality symbol in mathematics. There are also some examples of ill-usage of the equality symbol in textbooks. Some examples of student's behavior are given as well. Bartolini-Bussi and Pergola (to appear) are mainly concerned with the role of history and machines in the learning of geometry. Their paper may be interesting to the viewpoint of logic as far as the construction of geometry statements is concerned. In the same line, Bartolini-Bussi (1993) describes and interprets a short segment of a teaching experiment concerning the shift from the different levels defined by Balacheff in his study on the process of proof.
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