Probability, Statistics and Applications of Mathematics to Other Disciplines
1. Probability Theory and Statistics in Italian Curriculum
Probability theory and statistics were introduced into the Italian curriculum comparatively recently, as illustrated in the table below, where Y stands for the year of introduction and A for the ages of the student population. Asterisks (*) indicate that curriculum has not been officially introduced but is in a phase of extensive experimentation[1].
| Y | A | Probability Theory and Statistics contents |
|---|---|---|
| 1985 | 6-11 | statistical representations; combinatorial games aimed at a classical definition of probability |
| 1979 | 11-14 | consolidation of above subjects; applications to other subjects (i.e. Genetics, Economics) |
| 1986* | 14-16 | simple spatial events' probabilities and operations, conditional probability and independence; combinatorial theory and ideographs as instruments; descriptive statistics |
| 1992* | 16-19 | consolidation of above topics; multivariate descriptive statistics; regression and correlation; discrete and continuous one- and two-dimensional distributions, Normal distribution and error in measuring; the Law of Large Numbers; inferential statistics: parametric estimation of simple models |
The curriculum also include notions (more or less developed according to age group) about the following topics:
6-11: Observation and statistical surveys and their representations. Linguistic aspects. Development of judgment and estimation of odds under uncertainty through games of chance, in the context of the classical definition of probability as the ratio of the number of favorable and possible cases in chance situations which are considered "symmetric".
11-15: Consolidation of above topics, introduction of a few simple applications to concrete problems. One important objective is to give students a critical sense and investigative method in non-deterministic situations.
15-16: Analysis and interpretation of various forms of data (tables, diagrams,...) and a first analysis of the various definitions of probability with respect to real life situations. This is part of a teaching theory that encourages students to study events in everyday life rationally and critically and to arrive at logical conclusions in conditions of incomplete knowledge or in probabilistic contexts.
16-19: Statistics and probability theory to familiarize students with modeling under uncertain conditions in different contexts: physical, biological and economic. Scientific, historical and philosophical background on above topics, also as axiomatic formalizing of theory. Linking statistical problems and probability by way of the Law of Large Numbers and the error curve to introduce examples of inference and elucidate aspects of and problems concerning parameter estimation.
1.1. The Unusual Characteristics of Probability and Statistics
In order to discuss developments in teaching research on these subjects it is useful to speak about some of their particular characteristics. Probability theory and statistics began as specific subjects only 300 years ago, have been taught at the university level for only 150 years, have been considered in relation to psychology for 50 years and were introduced into curricula about 35 years ago[2]. So they began relatively recently and at a time when mathematics was able to offer powerful research tools. This is why they did not follow the normal process most basic subjects undergo, and did not enjoy that gradual period of growth in which a specific language is formulated and teaching methods are shaped and perfected. This is only one consequence of a larger question. The fact that probability has no objective reality on which to base itself, the attempts to construct it artificially and, more generally, the deterministic prejudices, psychological and scientific, all hindered its development and interfered with society's acceptance of a non-deterministic approach to reality. It isn't easy to accept the role of statistics and probability as fundamental to science, and it has been only partially recognized that this basically inductive method is the origin of a great number of research subjects and the means by which to learn more about them.
More generally, also because of specialized research, it is hard to affirm a philosophy that seeks to avoid the creation of artificial barriers between subject matters, within them or between ways of reasoning.
The words of Charles Sanders Pierce, a nineteenth-century American philosopher, give us an insight into another aspect of this question: "This branch of Mathematics (probability) is the only one, I believe, in which good writers frequently get results entirely erroneous".
It is in this perspective that Popper denies the importance of inductive learning processes and science constructions, and it is for this reason that scientific psychology has only recently accepted probabilistic models as necessary for an interpretation of human behavior. Great doctrines such as behaviorism, Pavlov's theory of conditioning and Gestalt psychology do not refer explicitly to probabilistic matters. However, the relatively recent tendency to see concepts of probability from a subjectivist point of view is being met with slow but gradual recognition by mainstream culture. To give an example close to home, the vast scientific and philosophical production of Bruno de Finetti[3], mathematician and epistemologist, was accepted with difficulty. He was an illustrious scientist, who, among other things perfected a model of mathematics learning and devoted himself assiduously to the question of didactics. One unusual thing about him, but nonetheless interesting in this context, was that he understood the importance of some of the aspects of instinct, the subconscious and intuition, and even spoke about a wild boar having a sense of probability while looking for an escape route during a hunt.
1.2. International Research on Teaching Probability Theory and Statistics
These particular facts and other characteristics of these two disciplines (cf. Barra, 1994), which mainly result from their not having been accepted and spread, have greatly influenced didactical research on them. It seems that this research does not even take these factors into consideration and is concentrated mainly on the first inklings of probabilistic and statistical thought in a person. Thus historical researchers, following on what was done initially in scientific research, can only envisage relatively simple cases of objective probability and stochastic independence in probabilistic matters which are much more complex (cf. Barra, 1992). In this way Piaget and Inhelders (1951)[4] conducted their research on probability only as the ratio between probable and possible cases, looking for deductive and explicit consistency in children's' answers to the probabilistic questions they posed. To be consistent Piaget and Inhelders, had to link probabilistic reasoning to the concept of ratios, and so they were led to underestimate the value of intuition at the pre-logical and pre-operational level, arriving at the conclusion, for example, that a good knowledge of combinatorics is essential. Since it is generally considered to be too difficult for beginners, Piaget and Inhelders concluded that it is not possible until the stage of formal operations is reached, i.e. mid-adolescence. By contrast, Engel (a mathematician) et al. (1976)[5] describes a number of games which can be very effective in introducing the necessary concepts to children much younger than this and, before him, the psychologist Estes (1950 and 1964)[6] reported that people are able to estimate probability from the very first stages of cognitive development. Others who criticized Piaget's theory are: English (1991)[7], who reports the presence of combinatory capacities very early on, Nortier (1991)[8], who emphasizes the role of diagrams in teaching-learning of combinatorics in primary school and Caredda e Puxeddu (1993), who study the relationship in learning between concepts of probability and fractions. However, as Green (1982)[9] demonstrates, there is a (well-known) danger that a student reared on an 'equally-likely diet' will always attach a probability of 0.5 to each of two mutually exclusive and exhaustive events based on any probability experiment, irrespective of how different events' probabilities really are. Jolliffe and Sharples (1991)[10] also notice, in studies of undergraduates in the United Kingdom and in New Zealand, a tendency for students to choose the "events are equally likely" option. However, Konold (1989)[11] argues for individual reason according to the outcome of individual trials and uses a probability of 50 per cent to mean that no prediction can be made. Bandieri takes a critical view (1990). Piaget's 'beads in an urn' type of studies are carried out by Falk et al. (1980)[12] e Falk (1983)[13] researches young people's preferences in chance experiments. In line with this we also find Hirst (1977)[14] and, above all, Fischbein et al. (1991)[15], with very interesting research conducted on 618 elementary and junior high school students in Pisa. Aside from analyzing certain preferences of the students, they also study the various types of difficulty the students encountered, among them those of a linguistic nature. Caredda and Puxeddu (1991) do research on the same subject for the elementary school level, and Benedetti and Di Cataldo (1990) address this aspect and others with respect to 15-17 year olds.
Again Engel (1970)[16] maintains that "students should not consider probability as a branch of mathematics, but as a way of viewing reality. We teachers should always have this in mind and, from this point of view, probability is second only to geometry". However, while he is convinced of the existence of a natural geometrical intuition, and speaks of a combinatorial way of thinking and a statistical way of thinking, he reports a lack of any kind of probabilistic intuition, the foundation for which must be given by formal education. For studies on the logical or probabilistic abilities that can be considered spontaneous in primary school children, see Mazzoni et al. (1991). While Engel is essentially showing what should be done, Freudenthal (1973)[17] sets out to convince us that it can be done. Finally, studies on "probability learning" seem more linked to frequentistic conceptions of probability, and refer to a situation in which the responses of the subject are not constantly reinforced but randomly, with a particular relative frequency. (For a very clear account of probability learning, see Millward, 1971[18]).
Fischbein (1975)[19], quoted above, who seems to be well versed in mathematics and with a good understanding of psychology, is especially suspicious of a logic which envisages a slow overcoming of determinism and rationalistic deductivism or one influenced by a particular image of probability. Fishbein sees probabilistic reasoning as a fundamental skill, which is based on intuitions, and which is necessary for man's adaptation. "The educational problem is not only that of building a set of mental skills for logical thinking. New intellectual belief, i.e. intuitions, have to be built. Their role is not only to suggest or to confirm inference. Their function is also to measure, logically as well as subjectively, the completeness or conversely, the incompleteness of an argument: 'I feel that something is wrong with my reasoning'. Such feelings are possible only if sound active inferential intuitions have been built." "Intuition expresses a profound necessity of our mental behavior." Inferential (or logical) intuitions are those which express the feeling of validity which accompanies logical operations. They are intrinsically involved in reasoning and can be primary or secondary intuitions. Primary intuitions are formed before, and independently of, systematic instruction. Spatial intuitions belongs to this category, as do the fundamental operational intuitions which regulate logical thinking. Secondary intuitions are those which are formed after a systematic process of instruction and they enable the individual to transcend primary cognitive acquisitions. They convey the products of social experience in the form, mostly, of scientific truths. The mathematician's intuitions fall into this category[20]. Good (1990)[21] also reports that a number of professional statisticians believe that the emphasis in high school teaching should be on the training of intuition.
For Fischbein intuitions are the result of the maturation of many experiences and are more or less connected to behavior according to the stability of interpretative models and the number of confirmations one has had in active contacts with real life. "Pre-school children possess a natural intuition of chance and instruction does not bring about any significant improvement in this respect. If appropriate instruction is given at the level of concrete operations, children can learn to compare odds by means of a qualitative comparison of ratios." Intuitions must be linked to logical structures through specific teaching methods in order to construct a probabilistic reasoning system, in that this facilitates probability estimation. It can only take place in an essentially unified type of reasoning and is subjective. This subjective probability must be recognized, nurtured and indeed exploited from the earliest stages. Probabilistic reasoning can be socially mediated and modified, and is far more useful to the teacher since it contrasts with the message that comes from Piaget's work about what not to teach children because they are not able to do it.
In a sense, Piaget and Inhelder's studies are completely constrained to finding what children are not capable of, whereas Fischbein is looking for evidence that children do have useful precursors of formal probability, in the form of intuition. Cavani et al (1990) also conduct their research on the subject of intuition. Fischbein discusses where to start from and how important it is to recognise existing precursor skills, to exercise and to develop them in order to provide a firm basis for more formal probabilistic ideas. In common with Evans (1989)[22] he favours experience-based learning rather than verbal instruction as a means of modifying probabilistic reasoning. "The difficulties encountered by the intelligence in acquiring and using probabilistic concepts are explained in part by an increasing tendency of maturing intelligence to seek univocal causal explanations."
Research on possible errors and misconceptions was carried out by Kahneman et al. (1982)[23] which was then elaborated upon by Hawkins and Kapadia (1984)[24], Falk (1988 and before)[25], and Bjorkqvist (1990)[26], critiqued da Shaughnessy (1992)[27], while Pollatsek et al. (1987)[28], preferred Konold's (1989)[29] clinical methodology for investigating student's cognitive and affective processes. As Bertrand Russell proposed in 1927, "popular induction depends on the emotional interest on the instances, not upon their number"[30]. According to Evans, (1989)[31] "The major cause of bias in human reasoning and judgment [of all kinds] lies in factors which induce people to process the problem information in a selective manner." This may arise because people use a selective mental representation of the information available, or because they selectively process it. Information selection may lead to the omission of relevant aspects or to the inclusion of irrelevant aspects. As Hawkins et al. (1992)[32] affirm, research has shown that people appear to have a fundamental tendency to seek information that is consistent with their current beliefs, theories or hypotheses and to avoid a collection of potentially falsifying evidence. This is known as 'confirmation bias'. Garfield and Ahlgren (1988)[33] point out related examples of 'interference' with 'sound' reasoning that can be caused by belief systems which run counter to orthodox science education: "It would be foolish to ignore such tendencies in interpreting responses to research questions" (Hawkins et al., 1992, p.110[34]). Perelli (1995) furnishes a very interesting view of psychological research in connection with probability learning, and, Hawkins et al. (1992), quoted above, presents a comprehensive, detailed survey, with greater reference to statistics. Both of these research projects are considered in this report.
On the Bayesian approach, Peterson and Beach (1967)[35] review the foundation research conducted in this area, and Lindley et al. (1979)[36] provide another source of ideas concerning research possibilities. Wonnacott (1986)[37] discusses people's facility with Bayesian and classical hypothesis testing, suggesting that the Bayesian approach is easier to teach and understand. Eddy (1982)[38] does research into probability misconceptions in Bayesian decision-making in medicine. The work of Scozzafava (1989-1995) on the subject is also very interesting: "a concrete teaching should start from the intuitive 'real life' meaning of probability as degree of belief". He furnishes a list of good examples, realististic (and simple), which demonstrate the inevitability of subjective judgments and conditioning to actual information in order to convincingly assess probability. In particular Scozzafava (1990c) offers a critical look at the so-called orthodox approach to statistics. Other good examples can be found in Gilio et al. (1994).
In terms of work with computers, one of the few research studies done on the subject, which points out the difficulties of evaluating software, is that of Green (1990)[39]. Microcomputer software catalogues have been produced by the Centre for Statistical Education (1986)[40], the Advisory Unit for Microtechnology in Education (1988)[41], and the British Computer Society (1990)[42], among others.
1.3. School and Society
We have been speaking about individuals so far, but the greatest problems seem to lie elsewhere. Fischbein holds that preconceived deterministic notions in teaching and in culture in general have a negative effect on intuitions which instead need to be reinforced, or modified if they contrast with recognized scientific truths, by way of conscious, systematic teaching. If they are not reinforced, these intuitions "survive in adults and in highly developed cultures only as a mere residuum". Hawkins and Kapadia (1984)[43] also report that children can have a sort of inferential probabilistic intuition, which is subjective, from the time they are very small. This is indispensable for language development and for understanding one's surroundings. These authors also state that schools and society are often responsible for discouraging these emerging probablistic conceptions by furnishing incorrect cognitive strategies, unsuited to a non-deterministic interpretation of our surroundings. Bandieri (1990) is also in agreement.
Piaget's pedagogical model is thus enhanced and inserted into the wider context of school and society. Since research in the area must be firmly aware of this, some researchers have turned their attention to philosophical and epistemological issues (Barra, 1990-1991) or have tried to give a general outline of the available literature (Malara, 1990), pointing out what is erroneous (Villani, 1994). Vighi et al. (1988) attempt to understand the reasons for epistemological obstacles, bias, misunderstandings and erroneous conceptions so often found in probabilistic reasoning. Rossi and Scozzafava (1993), give examples of the misleading (even if unconscious) use of erroneous conditional probability in the media. In NDR Modena (1990) learning obstacles are shown to be parallel to those found in the history of probability theory, which affected the link between probability theory and physics as well.
Finally, Prodi (1992), conducted research on basic cultural and educational criteria for planning curricula, activities and the teaching of more important concepts that might reduce the probability of committing errors, studying the problems that teachers run into in the classroom. Dapeuto et al. (1994) conducted analogous research at the high school level. Of utmost importance and delicacy is the question of teacher up-dating at the elementary school level (see Bazzini et al., 1990, and Perelli D'Argenzio 1991). Caredda and Puxeddu (1990 and 1992) researched the relation between games and learning probabilistic concepts at the elementary school level, and Cavani et al. (1990) conducted similar research on the use of fables. Pesci and Reggiani (1988) have proposed the teaching of elements of statistics and probability at the junior high school level, which has been developed and experimented with by a group of teachers since 1980. Pesci and Reggiani (1989 and 1990) also give a good survey of the research that has been carried out in Italy on teaching probability theory and statistics, and their work of 1990 deals with the question of the transition from elementary to junior high school and how to move from statistical description to prediction. Ferrari (1988-1991) reports on a project carried out by a group of researchers in Pavia on statistical activities involving the following subjects, of great interest today: the environment, time and the economy. Zanoli (1990) furnishes an overview of the literature currently available on the subject.
To conclude, it can be said that some of the research being conducted on methods for teaching probability theory and statistics-- at the international level, but especially in Italy -- could gain from increased contact with the field psychology, particularly necessary to the study of this kind of topic, and more focus on conceptual difficulties of a mathematical nature, looking for paradigmatic examples of improved teaching methods rather than examining learning-teaching processes.
It is worth noting that in 1982 the Cockcroft Committee, examining mathematics education in the UK reported that "some teachers, especially those who completed their degree courses some years ago, do not have sufficient knowledge of probability and statistics to teach in these areas effectively". The International Statistical Institute's Round-table Conference (Hawkins, 1990[44]), Training Teachers to Teach Statistics, concluded that although initiatives were being taken, with few exceptions their impact was relatively localised. Today, at the international level, the situation has not changed and may deteriorate further if we do not study how to introduce these concepts which are so fundamental to an interpretation of reality and correct thought. We must avoid a repetition of the errors made in some countries, i.e. Hungary, where these topics were given a great deal of space but were unsuccessful for didactic reasons.
2. Applications of Mathematics to Other Disciplines
Sometimes teaching/learning mathematics takes mathematics itself as its main point of reference: for some concepts (such as that of 'prime number'), a purely mathematically oriented construction activity works better than trying to find ways of connecting them to the physical world or to other disciplines. Other times we find different areas of mathematics that have similarities and can be made to interact. V. Villani (1985) [45] maintains that "the most innovative theories consist precisely in linking previously unconnected subjects together -- analytical geometry for example, which allows us to translate geometrical facts into algebraic formulas and vice versa..." In other cases mathematics is taught in connection with other disciplines, using various techniques which will be treated later on. "Moreover," Villani continues, "a good part of the reason for the existence of mathematics derives from problems inherent in the world of nature, and, in order to quantitatively interpret phenomena in the real world, the principal mathematical theories are indispensable."
The topic of this section, "Mathematics and other Disciplines", is unique in that it does not refer to one branch of mathematics alone: theoretically all branches of mathematics can be made to interact, and in particular circumstances, all those who have done research on the teaching of math can have contributed in some way to linking it with other disciplines. For many years the tradition of didactics in math in Italy have furnished examples of this at all levels. Among the first works in this field are certainly those of Emma Castelnuovo - for example,' Mathématiques dans la realitée' (French translation, 1980) and her latest book, Pentole, ombre, formiche (1993)[46]-, which inspired many of the works quoted in this report.
The need for schools to respond to pressing social issues (see Mathematics and Technologies Report) in the real world, on the one hand, and pressure from students to find new motivations for studying specific subject matters, led to an up-dating of curricula in the 1970's and 1980's in order to make the connection between school and the outside world explicit. The issue of ecology, which had a great impact on schools in Italy, has particularly led to this kind of change.
Recent epistemological thought on the complexity of nature and technology and peoples' interaction with them has led to new areas of study and serious reflection on the reductionism present in all types and at all levels of school and on the validity of teaching the various disciplines separately.
Research on links between the history of mathematics and general science and how to teach them has increased in the last decades (see Grugnetti and Speranza, in this book). If we look at the history we see, for example, that not only are some branches of mathematics rooted in others, some come from other areas of study, the humanistic and natural sciences.
From Lanciano (1988): "Some of the symbolic values of numbers originated from man's rapport with and knowledge of the firmament, of astronomy, and it was from these that we arrived at geometrical shapes", and again, "The connection between time rhythms and the geometrical organization of the space in the sky, which man was able to 'read', is mirrored in the later geometrical organization of space on earth....However, in subsequent eras, following the Pitagorians or the Egyptians and Babylonians, mathematics and astronomy were separated, and the original unit of knowledge was ramified to the extent that the initial relationship could no longer be seen...and so we not only lost the principal roots of vast parts of our culture but of mathematics itself."
On a more pedagogical note we wish to point out, following (Boero, 1990), that "the importance assumed by 'constructivism' in research concerning the learning of mathematics (Kilpatrik, 1987)[47] and by the problem of the so called 'context' in the 'constructivistic' approach. On the other hand, the epistemological and the cognitive analysis of the concepts in physics, astronomy, biology, etc. has emphasized the importance of the languages used to communicate and process the knowledge, with the aim of constructing mental models drawn from natural phenomena under consideration (Giordan, De Vecchi, 1989, Giudoni, 1986).[48]"
We should note in this context a greater attention to the linguistic problems, which is present in every step of the process of communication and teaching/learning.
All this has led to interdisciplinary practice, the interweaving and juxtaposing of related subjects in order to explore a given topic, problem or situation and, on the practical level, collaboration between teachers of different subjects. In this kind of practice some subjects will carry more weight than others, and we find an epistemological and temporal hierarchy, more or less evident, among disciplines that meet to study an object or phenomenon.
2.1. The situation in Italy
School programs in Italy, which have been up-dated at the kindergarten, elementary and junior high school levels, as previously mentioned, and which are in a process of experimentation at the high school level, have incorporated some of what we have been talking about and specifically state the question of interdisciplinary collaboration, especially regarding mathematics. We will cite a few:
Teachers at the nursery and elementary school levels, however, are often unprepared for teaching mathematics. Teachers of math, chemistry, physics and natural science, at the junior high school level, have degrees in one scientific subject. Teachers at the high school level are better prepared in terms of disciplines studied, but there are less research in link math and others disicplines.
As far as research at the university level is concerned, the need has been felt, in some cases, to combine research on math teaching with other subjects, particularly the sciences, in researching experimental teaching.
Besides what is going on at the public school level, several international conferences have been organized by mathematicians and others on mathematics and other areas of learning. The documentation from the JIES Chamonix 1992 [49] is particularly interesting.
A minority of math researches met with researchers and teachers from all the other fields of science and technology. Several national periodicals have also taken up the theme of the introduction of mathematics into other fields of study, among them: Epsilon (Paravia 1988-1995), L'insegnamento della Matematica e delle scienze integrate (Centro Ugo Morin) and L'educazione matematica (Cagliari), to which we will refer later on.
2. 2. Research and experimentation
It is clear, then, that different approaches can be taken to research on combining the teaching of math with other subjects, particularly the experimental and natural sciences. In some cases we need to go outside the field of mathematics -- to science, technology or the arts -- to find stimuli and models, and, vice versa, other disciplines certainly look to mathematics when searching for tools for description, calculus and symbols. There are two main approaches: one begins with mathematics and moves toward other subjects and, vice versa, one starts from another subject and moves toward mathematics.
mathematics other subjects
<---------------
The works presented here start from a particular areas of study either because the researcher is qualified in that subject, because the researcher wishes to teach it, because of circumstances and situations regarding private research on it or because the researcher is collaborating with other teachers on it. Examples of projects that start from astronomy and move toward mathematics are those of Lanciano (1990 and 1994 a), in which a great deal of importance is given to the study of how astronomy in its step by step development. Boero et al (1995 is an example of a project beginning with geometry, Scali (1994) with geometry and pictures, and finding areas of interest and examples from astronomy.
However, when different disciplines meet, this means that different researchers must meet, and it seems that they often have difficulty in understanding each other. Mathematicians are often 'accused' of this. In order to discuss things with researchers from other disciplines, mathematicians are often required to assume a point of view more flexible than usual, to be less precise and take a more intuitive approach, using everyday language -- which is not easy but open a new research communication in the scientific world at all levels.
This debate is particularly lively among physicists. Write C. Bernardini (1993)[50], "Serious prejudice is fostered when math is taught in an isolated context, using non-dimensional variables...(while in physics symbolic notations are always associated with something real)...even when speaking about geometrical similars teachers use a kind of language that has no immediate application to physics."
Other examples of combining math and physics are that of Garuti (1992) at the junior high school level and D'Antona and Pesci (1989) at the high school level. Garuti's project involves the teaching of 'functions' as 'formulas operating on the values of one variable, applied to models of real phenomena. During these activities, Cartesian diagrams were introduced and extensively utilized as mediating tools for heuristics, conceptualization and analysis of the relationship between experimental data and mathematical models.
Let us now consider the approach which searches for stimuli, food for thought and practical examples from disciplines that can also offer numerical data and be read in various languages (symbolic, graphic, verbal, etc.) in order to teach some area of math. Many math teachers employ problems typically related to other disciplines in order to introduce or deepen important mathematical concepts. For example the field of genetics offers interesting and socially relevant material such as the laws governing the heredity of basic characters for introducing combinations. An example of this is given in Boero et al (1995). Students study the shadows made by the Sun on horizontal planes and their change over time. This is particularly efficacious for work on measuring angles and lengths and similars. In this case students are concerned only with the details and precision necessary to carry out the prescribed mathematical problem, without considering the complexity of the whole context -- students are not asked, in fact, to think about the Sun as such, or the tacit astronomical aspects involved (i.e. relative differences in latitude or seasons of the year), but on the geometric model of its shadows. In this case the teacher is giving too much attention to geometry, and this fact is borne out by the author in another text (Boero, 1990): "In fact there is the risk that the objectives of gaining an insight into the mathematical concepts are prevailing over the scientific-experimental objectives".
Ferrari (1988-1991) suggests activities for learning statistics at all levels of elementary school -- for example recording everybody's weight and height or interviewing parents and grandparents to learn their social history and then doing qualitative and quantitative analyses of the information. In this kind of activity statistics are being applied to various disciplines, but they are considered only in terms of organizing the data gathered in the interviews (i.e. temperature is not seen in terms of meteorology or geography, but only in terms of data gathering).
Pesci and Reggiani (1989) suggest an activity for junior high school students which involves applying statistics to the economy, particularly to inflation and its measuring by means of price indices. The aim is to present an application of simple mathematical concepts and to review and amplify elements of statistics previously studied.
We shall now consider those situations in which methods or instruments formulas, statistics and scale drawings, from mathematics are helpful in studying natural or social phenomena. In this case mathematics is viewed primarily as a 'backup subject' and is 'utilized' depending not only on the scholastic level (students' age, level, adult training) but also on the amount of approximation required by the study of the phenomenon in question. For example in Lanciano (1993), mathematical instruments are explicitly indicated for astronomy, at the high school level, but in the same text is also indicated how mathematics can benefit from the study of astronomy. In (Lanciano, 1994b) it is in details treated some example in overcoming certain cognitive obstacles in constructing spatial concepts.
An example of work which makes use of statistics is that on cryptography furnished by Zuccheri (1992). This initiative is similar to research on learning in some ways, i.e. it analyzes conjecture and logical deduction and the various problem-solving strategies children adopt. Statistics in this case have to do with language (outdated words, frequency of letters in a language, etc.). The elements used which derive directly from statistics are: relative and absolute frequency, permutations and histograms, and from math: fraction comparison, biunivocal applications, modulus m congruities.
Other examples of this kind of approach are those in which mathematics furnishes models for other disciplines, we can see the book of Israel (1986)[51]. Boero (1990) has this to say on the use of models: "We can say that mathematics provides the teaching of experimental sciences with mathematical models which can be classified in a number of ways"; we use the following classification:
Cannizzaro (1989) points out possible misconceptions that can arise in constructing and using mathematical models, in the relation between mathematics and real phenomena: mathematical models are seen by some not as a possible vision of reality, but as if a "perfect mathematics" existed that has to make compromises with reality. Others see it as an integral part of the physical world, which is regulated by it and think that is the inevitable way of describing reality or at least some of its phenomena. In other cases it's a question of going beyond a vision in which there is great dependency on cause/effect in order to see the parallel variations of phenomena. From an historical perspective the article cites modeling in the field of biology in the 19th century, recent developments of which have given birth to the theory of systems.
An example of interdisciplinary integration at the nursery school level is offered by Caredda, Pixeddu (1995). The "Nuovi Orientamenti" were introduced in 1991 which include six areas of experience, one of them being "space, order, measurement". The activity under analysis is a game using various types of colored dice which must follow different paths, involving movement in the real space of the classroom.
Ardizzone et al (1994) have researched the relationship between children and real space at the primary school level. Position, vision, perception and feeling in real space are the beginnings of geometry: the teaching hypotheses are presented in a proposal which goes beyond interdisciplinarity between 'multiple disciplines' to take a more global approach -- the rational and perceptive knowledge of an artistic and architectonic space such as the Pantheon in Rome. A further example, already mentioned, is that of perspectives (Menghini, 1988), mainly for teacher updating courses, on notions about linear perspective and homology in order to understand the use of perspective in paintings, for high school students and teachers.
Research on teaching in this area can be particularly fruitful and seems to be wide open. It involves looking at mathematics from the point of view of others and their request for clarification of language and pertinent illustrations, on the one hand, and on the other, all the new and unexpected results that emerge each time different disciplines come into contact and what can be learned from research on the history of mathematics itself.
References of part 1
Bandieri, P.: 1990, 'Il bambino la scuola e la probabilità', in Zanoli, C. (ed.), Calcolo delle Probabilità, Le Scienze, la Matematica e il loro insegnamento, vol. 27, n.4-5, 42-49.
Barra, M.: 1989, 'Il gusto dell'incerto - probabilità e statistica: cenni storici epistemologici e didattici', La vita scolastica 15, 12-15.
Barra, M.: 1990, 'Calcolo delle probabilità, statistica e conoscenza induttiva', Induzioni. Demografia, probabilità, statistica a scuola 0, 23-31.
Barra, M.: 1991, 'Aspetti epistemologici e storici relativi alla "legge dei grandi numeri" e alla "legge empirica del caso" a partire dai Greci', Induzioni. Demografia, probabilità, statistica a scuola 2, 17-32.
Barra, M.: 1992, 'Valutazioni della probabilità nella storia', Atti del Convegno di Ancona: Il pensiero matematico nella ricerca storica italiana, 143-174.
Barra, M.: 1994, 'Probabilità e Statistica nella scuola secondaria', Atti del XVI convegno nazionale sull'insegnamento della matematica: Probabilità e Statistica nella scuola secondaria, Notiziario della Unione Matematica Italiana, supplemento al n.7, 59-69.
Barra, M. and Bernardi, C.: 1995, 'Su un problema di probabilità, ovvero ... alla conquista del centro', L'insegnamento della matematica e delle scienze integrate, v.18A-18B, n.5, 533-549.
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| Mario Barra Dipartimento di Matematica, Univ.ersità"La Sapienza" P.le A. Moro 2, 00185 Roma, Italia Tel. (39)6 49913254 Fax (39)6 44701007 E-mail: barra@mat.uniroma1.it |
Nicoletta Lanciano Dipartimento di Matematica, Univ.ersità"La Sapienza" P. A. Moro 2, 00185 Roma, Italia Tel. (39)6 49913254 Fax (39)6 44701007 E-mail: Lanciano@gpxrme.sci.uniroma1.it |