APPROACHING MATHEMATICAL THEORIES IN JUNIOR HIGH SCHOOL
Paolo Boero, Dipartimento di Matematica,Università di Genova
In most countries of the world, school education from grade VI to grade X is characterised by transitions: from compulsory education to jobs and/or vocational school education; from school-for-all to different schools (or curricula). Furthermore, we must take into account the transition from child to adult behaviours of learners within the 11-16 age range. Mathematics education plays special functions in these transitions: it must ensure the acquisition of crucial tools for life and many professions, it must prepare students for further studies in scientific and technological areas, it should pass over relevant components of modern scientific culture. Consequently, crucial elements characterising mathematics education in junior high school are:
- the evolution from empirical and episodical aspects, towards theories (from the WGA-2 document: "the theoretical status of mathematical knowledge, which on the one hand no longer has the empirical and episodical features of the elementary school, but on the other hand cannot yet have the character of a structured theory, like possibly at the senior level"
- the progressive differentiation between mathematics and the other scientific domains: "the transition from argumentation to proofs and from empirical statements to theorems within a mathematical theory as contrasted with respect to other scientific theories such as physics or biology"
In relationship with these "transitional" characters of mathematics education in junior high school, we need to consider not only the epistemological questions:
(what does it mean "theoretical status of mathematical knowledge"? How can we distinguish mathematical theories from other scientific theories?)
but also the related cognitive and didactical questions:
(how can students approach theoretical knowledge? How can they arrive to distinguish some features of mathematical theories and other scientific theories?).
In general, we can remark that mathematical theories and theoretical and cultural aspects of mathematics (like the systemic character of mathematical knowledge and the role of mathematical models in shaping scientific knowledge of the physical world - cf Boero, 1989) represent a challenge for mathematics educators all over the world. Neither abandoning them in curricula designed for most students, nor insisting in traditional teaching of them seem to be good solutions. Indeed, the former represents a dismission of school in its task of passing over scientific culture and models of rationality to new generations, the latter is scarcely productive and impossible to keep in today's school systems. By this way, an interesting and important area of investigation is opened to mathematics education research: on what theoretical and cultural aspects of mathematics should the effort be concentrated? Why must they be implemented in curricula? How to ensure a reasonable success in classroom activities about them?
My contribution will deal with some topics related to these questions:
- natural language and the approach to mathematical theories (Section 2);
- facilitating conditions for the approach to theorems (and proving) (Section 3);
- the need for teacher's mediation in the approach to theoretical thinking and mathematical theories (Section 3).
My contribution will be partly based on a long term experience of didactic innovation from grade VI to grade VIII (started more than twenty years ago in some 40 classes), which gradually became the educational environment ("Genoa Group Projects") where to deal with some intriguing research problems:
- how to define the educational aims and methodologies concerning the approach to theoretical knowledge in such a way that "large" success meet with "high" standards in students' performances?
- how to balance the needs of those students who will finish school in grade VIII with the needs of those who will continue?
- how to interpret the difficulties met by the slow learners, how to (possibly) overcome them?
I will consider also many contributions coming from other Italian research groups (particularly: the Alessandria group leaded by Pierluigi Ferrari-Section 2 largely depends on his contribution; the two Modena groups leaded by Mariolina Bartolini Bussi and Nicolina Malara; the Turin group leaded by Ferdinando Arzarello; the Pisa group leaded by Maria Alessandra Mariotti ), as well as many ideas taken from current literature.
2. NATURAL LANGUAGE AND THE APPROACH TO MATHEMATICAL THEORIES
Despite their different subjects and focus on different aspects of mathematics education, mastery of natural language in its logical and reflective functions emerges from many experimental studies in the field of mathematics education (see later for specific references) as one of the crucial conditions in order to approach more or less elementary theoretical and cultural aspects of mathematics. In particular, the reported teaching experiments show how producing, comparing, discussing conjectures, proofs, solutions for mathematical modeling problems, etc. needs a strong committment by teachers as concerns students' development of linguistic competencies. Indeed only if students reach a sufficient level of familiarity with the use of natural language in the proposed mathematical activities, they can perform in a satisfactory way and fully profit of these activities.
Theoretical positions and educational implementations concerning different functions of natural language in teaching and learning of mathematics in school are widely reported in current mathematics education literature. Especially communication in the mathematics classroom deserved much attention by mathematics educators in the last two decades (cf. Steinbring et al, 1996). These studies influenced our own theoretical and experimental research. The contributions brought by this section intend to join the stream of research about natural language in mathematics education by going in depth in the analysis of specific functions that natural language plays in relationship with the theoretical side of mathematical enculturation in school, and related educational implications.
2.1. The Language of Mathematics
By 'language of mathematics' we mean a wide range of registers that are commonly used in doing mathematics. According to Halliday (1974), quoted by Pimm (1991), a register is "a set of meanings that is appropriate to a particular function of language, together with the words and structures which express these meanings". The functions mathematical language may play include teaching, learning, popularizing, communicating, playing, ... both in oral and written form and are deeply affected by the age level and linguistic or mathematical competence of people involved (students, teachers, researchers, ...). Most mathematical registers are based on ordinary language, from which they widely borrow forms and structures, and may include a symbolic component and a visual one. In principle, large portions of mathematics could be expressed in a completely formalized language (for example, a first order one) with no word component, i.e. with no component borrowed from ordinary language. Languages of this kind have been built for highly specialized purposes, and are hardly used by any group of people (including advanced researchers) in doing or communicating mathematics. So we argue that ordinary language play a major function in all the registers significantly involved in doing, teaching and learning mathematics.
2.2. Ordinary Language in Mathematical Registers
Some difficulties generally arise from the differences in meanings and functions between the word component (i.e. the words and structures taken from ordinary language) of mathematical registers and the same words and structures as used in everyday life. The difference is hardly noteworthy in children mathematics, whose forms and meanings have been almost completely assimilated in ordinary language, but it grows more and more manifest in the transition to advanced mathematics, that needs of highly specialized languages, with characteristics that clearly distinguish them from the ordinary one. For example, in mathematical registers some words take a meaning that differs from the ordinary one, or add some new meaning to the standard ones. This is the case of words like 'power', 'root' or 'function'. The use or the interpretation of some connectives may change too, which implies that the meaning of some complex sentences (like conditionals) may significantly differ from the standard one. Ordinary language and mathematical one may differ too as far as purposes, relevance or implicatures of a statement are concerned. For example, in most ordinary registers a statement like "That shape is a rectangle" implicates that it is not a square as well, for if it were, the word 'square' would have been used as more appropriate for communication purposes. This additional information is called an implicature of the statement and is not conveyed by its content only but also by the fact that it has been uttered (or written) under given conditions. Also a statement like "2 is less or equal than 1000" is hardly acceptable in ordinary registers (as it is more complex than "2 is less than 1000" and conveys less information), whereas it may be quite appropriate in some reasoning process exploiting some properties of the 'less or equal' predicate. In general, a relevant source of trouble is the interpretation of verbal statements within mathematical registers according to conversational schemes (i.e. as it were within standard language). For more examples at this regard see Ferrari (1999).
These remarks suggest not only that some degree of competence in ordinary language is required in any mathematical register, but also that working with different mathematical registers may require something more on the side of metalinguistic awareness, in order to manage the transition between the different conventions. The idea of metalinguistic awareness has been applied to the exploration of the interplay between language proficiency and algebra learning by MacGregor and Price (1999). In their paper they focus on word awareness and syntax awareness, as the components having algebraic analogs. In our opinion it is necessary to consider a further component of metalinguistic awareness, namely the awareness that different registers and varieties of language have different purposes.
2.3. Ordinary Language and Algebraic Symbolisms
The relationships between the word and the symbolic component in mathematical registers are not only complex, but they have developed through the years as well. For many centuries suitable registers of ordinary language have been the main way of expressing fundamental algebraic relationships. The invention of algebraic symbolism has provided a powerful, appropriate tool for treating algebraic problems and for applying algebraic methods to other fields of mathematics and other scientific domains (physics, economics, etc.). A widespread idea amongst mathematics teachers is that algebraic symbolism, once learnt, is enough to treat a wide range of pure and applied algebra problems. Moreover, it is also common the belief that students' symbolic-reasoning skills develop first, with word-problem-solving ability developing later. For some evidence at this regard see Nathan and Koedinger (2000), who outline the SPM (Symbol Precedence Model) that induces a significant number of teachers to fail in predicting the behaviors of two groups of high school students.
We will argue that all the opinions that understimate the role of natural language in learning does not fit the actual processes of algebraic problem solving and will give both theoretical reasons and experimental evidence.
Mathematically speaking, algebraic symbolism can be regarded as a (part of a) formal system designed to fulfil specific purposes, among which we mention the opportunity of performing computations correctly and effectively.
Even if in the past algebraic symbolism has been introduced as a contraction of ordinary language, and in some cases (like "3+5=8" and "three plus five equals eight") it may be treated like that, there is plenty of evidence suggesting that the relationships between ordinary language and algebraic symbolism are more complex.
First of all, algebraic symbolism has a very small set of primitive predicates (in some cases, the equality predicate only); this requires the representation of almost all predicates in terms of a small set of primitives; for example, in the setting of elementary number theory, the predicate "x is odd" has not a symbolic counterpart and cannot be directly translated; its symbolic representation requires a deep reorganization resulting in an expression like "Ey(x=2y+1)", that, in addition to the equality predicate, involves a quantifier and a new variable that does not correspond to anything mentioned in the original expression. In other words, there are plenty of algebraic expressions that are not semantically congruent (in the sense of Duval, 1995) to the verbal expressions they translate. The lack of semantic congruence may induce a number of misbehaviors, like, for example, the well known reversal error (for a survey and references see Pawley and Cooper, 1997).
Another major source of trouble lies in the fact that ordinary language employs a lot of indexical expressions (like 'this number', 'the age of Maria', 'the triangle on the top', 'the number of Bob's marbles'), not available in algebraic symbolism, that are automatically updated according to the context. So, in a story telling that at the beginning Bob has 7 marbles and then he wins 5 more, the expression 'the number of Bob's marbles' automatically updates its reference from 7 to 12. The same would not happen for a mathematical variable: if at the beginning of the story one defines B='the number of Bob's marbles', then at the end Bob has B+5 (and not B) marbles.
These features of algebraic symbolism have been recognized as sources of analgebraic thinking (Bloedy-Vinner, 1996)
These peculiar characters of algebraic symbolism constitute its main reasons of strength (as they allow algebraic transformations, i. e. the possibility of transforming an algebraic expression in such a way to both preserve its meaning and produce a new expression easier to interpret or useful to suggest new meanings). But it shows also the intrinsic limitation of algebraic language in comparison with natural language: algebraic language cannot fulfil neither a reflective function nor a command function. The fact that symbolic translations of verbal expressions are not semantically congruent to them, even though they preserve reference or truth value, implies that they cannot be used neither to organize one's processes, nor to reflect on them, since the non-referential and non-truth-functional component of their meaning (like, for example, connotation, or the use of metaphors), which often are crucial in resoning, are lost. Also the use of indexicals that update their meanings according to the context is a fundamental tool for reflection and control, as it allows the subject to.
For further reflections about the relationships between natural language and algebraic language in the perspective of approaching algebra in school, see Arzarello (1996); Malara, (1999).
2.4. Educational Implications
Research perspectives and problems about the role of natural language in mathematical activities considered in the preceding subsections raise some important questions related to teachers' preparation and educational implications for classroom work. We will sketch some aspects of this "problematique" and some didactical implications.
2.4.1. Teachers' Preparation
The development of linguistic competencies in mathematical activities strongly depends on teacher's mediation. Concerning this issue, teachers' difficult must be taken into consideration. Here some obstacles come from the widespread idea that natural language is not an efficient tool in developing and communicating mathematical knowledge, due to its redundancy and lack of precision. Many reasons are advocated to support this idea: the supposed prominence of mathematics as a formal system, the need for a purely syntactic treatment of mathematical relations, the difficulty many students meet in managing and understanding natural language with a sufficient level of precision, and (last but not least) the theory of Piaget (who considers communication as the main function of natural language). Many mathematics teachers (encouraged by current textbooks) are tempted to reduce the relevance of natural language in classroom work: tasks are formulated with a large support of images and/or stereotiped linguistic expressions, non-verbal answers (diagrams, formulas, etc.) are allowed, and verbal explications are represented at the blackboard with a large support of non verbal tools (diagrams, schemas, algebraic expressions, etc.). Add the increasing number of foreign students in the classroom, with its obvious consequences: mathematics (and mathematical formalisms) is universal, so we should try to teach it by reducing the verbal side of the activities. Add also the fact that ill-paid teachers (a common situations in many countries of the world) can prefer avoiding the "waste of time" needed to correct students' homework and classroom problem solutions by a strong committment to students to use technical, synthetic formalisms. Finally, the quantity of mathematical content that can be presented to students by using these formalisms is much bigger than in the case of a verbal presentation.
All the reasons considered above make a different perspective (concerning the development of linguistic skills related to the use of natural language as a crucial issue in mathematics education) difficult to be accepted by both prospective and in-service teachers.
According to our experience it is not sufficient to produce good theoretical arguments against the dismission of natural language in mathematics class: the discussion of well chosen examples of students' behaviours seem to be necessary (cf. Boero, Dapueto and Parenti, 1996). Discussion should put into evidence the crucial function of natural language in mathematical activities, according to preceding considerations. Discussion should also focus on the quality of students' performances in relationship with their current mastery of natural language in mathematical activities.
2.4.2. Promoting Verbal Activities in Mathematics Classes
Once the idea of a relevant role of verbal language in mathematical activities is accepted by prospective and in-service teachers, still remains the educational problem related to the choice and management of suitable classroom activities. As concerns the management of classroom activities, we can consider the role of the teacher as an indirect mediator (when he selects and exploits students' linguistic productions), as a direct semiotic mediator (when he provides students with appropriate linguistic expressions in order to fit their thinking processes) and as a cultural mediator (when he provides students with important "voices" as linguistic models of theoretical behaviours in mathematics).
Let us consider the following example: students are asked to find the square of double surface of a given square. According to the didactical contract in the classroom, they produce a verbal report about their trials to solve the problem. Sometimes this verbal report follows the steps of the activity, sometimes it is written during the activity and reflects the ongoing reasoning. Here is an example (grade VII):
Daniele: "I think that I can double the side of the square. Then everything will be double, and the area will be double"
(a partial drawing is produced)
"No, it is not true that I get a double area by doubling the side of the square. It is evident that the area becomes four times larger. I must find a smaller increase. I could take one time and one half the length of the side.
(drawing with careful measurements and calculations)
"No, it does not work. It is bigger than I need. I should decrease once more. I could take 1,4 times larger. I can make the calculation without making the drawing: it i sufficient to multiply 1,4 by the length of the side, and then multiply by itself."
"Double area means 8. I am near, but I have not obtained 8"
The teacher selects some texts produced by students and for each of them invites students to identify analogies/differences with their own solutions, then opens the discussion about the chosen text. By this way the content and expression of the chosen text are put under scrutiny. Here there is an excerpt taken from the discussion about the preceding text:
(Sabrina) "Daniele has found a very good result, the side with length 1,4 times 2 (namely, 2,8) gives a surface which is very near to the area 8, that is the double of the initial area"
(Pietro) "But Daniele has not reached the solution. The solution means to find the square of double area"
(Elena) "And Daniele has put his numbers by chance, why one time and one half, and then 1,4, and not other numbers?"
(T) Daniele, try to explain why did you choose those numbers
(Daniele): "I would say to Elena that if you see that doubling the sides of the square gives a four times bigger area, then you must decrease the side, in order to decrease the area. I have chosen 1,5 because it is a number in the middle between 1 and 2 (which does not work).
As the reader can notice, teacher's interventions stimulate students to find better expressions and provide students with appropriate words.
The same activity is repeated for three texts (usually, starting from the worse of the chosen texts). Then the teacher presents to the students a long excerpt taken from Plato's "Menon" (the well known episode of the dialogue between Socrates and Menon's slave about the problem of doubling the square: cf. last section of this paper).
Analogies and differences are found with students' productions; Plato's text is discussed as a "model" of dialogic treatment of the identification and overcoming of a mathematical mistake. The three phases of the dialogue are put into evidence (production of the mistake and identification of it by counter-examples; trials to overcome it; finally, solution of the problem guided by the teacher). A discussion follows about the choice of a "common" mistake of students, and the nature of that mistake. Finally, students are asked to produce an 'echo' to the Plato's dialogue by writing a dialogic treatment of the chosen mistake. Here is an example of a high level individual production (the chosen mistake concerns the idea that "By dividing an integer number by another number, one always gets a number smaller than the dividend")
Socrates (SO): Tell me, my boy, what is the result of 15:3?
Slave (SL): Five.
SO: Is it smaller than 15?
SL: What a question! That is clear!
SO: And yet, how much is it 20:5?
SL: Obviously 4, Socrates
SO: Then is it smaller than 20?
SO: Then, how do you think that results of the divisions are?
SL: I think that they are always smaller than the dividend.
SO: Are you sure?
SL:(°) Yes, because "to divide" means "to break in equal parts."
SO: Now perform this division: 15:1.
SL: Uhm, ...it makes 15.
SO: But 15 is equal to the dividend.
SL: It is true.
SO: Why is it equal?
SL: Because dividing by one is how to give an amount to one person, it remains equal.
SO: So does your theory still work?
SL: Not completely. Now I see that in some cases it does not work.
SO: Are you still sure you are right?
SL: Yes... Perhaps... No... Perhaps there is one case in which the result is larger...or perhaps not... My Zeus, I understand nothing! (five minutes elapse). SO: What is the result of 2:0.5?
SL: These are difficult questions. I am no longer able to answer.
SO: Take this square (drawing)and divide it into small squares!
SL: This way?(the drawing is divided into 16 pieces by drawing 3 horizontal and 3 vertical lines, all equally spaced)
SO: Yes, good. Now the unit is the small square [drawing]. How much is 0,5 compared to 1?
SL: One half.
SO: Now make one half of the small square.
SO: Do the same for all the small squares.
SL: Just a moment...Done.
SO: How many halves?
SL: 1,2,3... 32, Socrates.
SO: How many unit squares, at the beginning?
SL: Sixteen, Socrates.
SO: Then you got a result greater than the starting number.
SL: Uhm... Of course.
SO: And how is one half written as a fraction?
SL: Uhm... perhaps 1/2.
SO: Good! Are you able now to divide a number by a fraction?
SL: Yes, surely!
SO: Then divide 2 by 1/2. How many times is 1/2 contained in 2?
SL: According to the preceding rule, I must invert the fraction and then multiply. OK, it makes 4.
SO: How can you represent this?
SL: I'll try... Two squares..[drawing] One half twice for each [drawing]. It works: 4.
SL: I understand: the division is not only "breaking into equal parts", but also seeing how many times a number is contanined in another!
SO: Make an example by yourself!
SL: 1:1/4 [he performs and illustrates it]
In general, comparing the initial text with the final production we have observed how the different roles played by the teacher had left traces in the student's production:
- an increased precision in the linguistic expression (with the use of appropriate terms);
- the assimilation of a dialogic treatment of the mistake (according to the cultural model provided by Plato's dialogue).
2.4.3. The Problem of Teaching Mathematics in Multi-language Classes
This problem is not easy at all, and the tendency of most teachers is to bypass this kind of difficulties by using the universal, technical languages of mathematics (arithmetic and algebraic languages, but also diagrams, arrows, etc.) in order to create common tools of communication between students and with the teacher. But if natural language is necessary, due to its reflective and planning functions, in personal mathematical activities as well as in social interaction, the fact of privileging technical languages of mathematics can result in a damage for the development of mathematical knowledge. On the other part, in a multi-language class the problem of verbal communication and production cannot be avoided for other subjects (like sciences or history), and then an autharchic position by the mathematics teacher ("I can avoid the necessity of communication in naturla language") can result in a general damage for the whole cultural preparation of students.
The problem of teaching mathematics in multi-language classes has been dealt with in many studies in the last years (for a survey, see ....). An interesting emerging trend was to design teaching situations in which language diversity could help mathematical understanding: indeed, it happens rather frequently that different languages use expressions which are particularly appropriate for speaking about specific concepts and properties, or that some linguistic expressions allow to grasp specific aspects of the mathematical knowledge carried by those expression. In Boero & Randnai Szendrei (1998) an example is provided, concerning the manner of speaking about numbers in Hungarian and in other languages. In particular, we may consider the different situation in Italy and in Hungary, concerning learning of natural numbers: in the Italian language, the names of natural numbers ('one, two, three, four...') are also used to indicate the days of the month (only the first day is commonly named 'first of...') ; in the Hungarian language, all the days of the month are named with the ordinal adjective 'first..., second..., third...' ; so in Italy and in Hungary the relationships between 'cardinal' and 'ordinal' aspects of natural numbers are different in the first mathematical experiences of pupils. Classroom discussions about these differences may result in an improved flexibility in managing the different "meanings" of numbers (and these discussions are possible, for instance, with those Hungarian classes whose Italian and Hungarian students learn mathematics in Italian).
Another example concerns a comparison between French and English (or Italian) as concerns the name of numbers like 85 or 75: in French, the name of 85 is "quatre-vingt-cinq", the name of 75 is "soixante-quinze"; in the first approach to numbers this can create a difficulty in comparison with the English (or Italian, or other languages), where tenths follow tenths from ten to one hundred. Comparison with one of these languages can help French speaking beginners to grasp the regularity of the tenths sequence and learn basic facts in the transition from units, to tenths, to hundreds, like the fact that the same, basic sequence is repeated with values which are ten, one hundred, etc. times bigger. But immediately later French names can be very useful in the classroom for all students to point out and experience some basic facts concerning the additive and multiplicative relationships between numbers: quatre-vingt means that 80 can be obtained by repeating 20 four times, with positive effects on mental calculations. As it happens frequently in mathematics, one difficulty can result in an opportunity of developing knowledge. This example suggests the opportunity to perform cross-cultural investigations in order to detect specific, linguistic aspects of basic mathematics in different languages which can be exploited as an opportunity for developing mathematical skills.
3. SEARCHING FOR CONDITIONS FACILITATING THE EARLY APPROACH TO THEOREMS
The research project reported in this section went on analysing mental processes underlying the production and proof of conjectures in mathematics. We believed that such analysis could give us some hints on suitable problem situations and the best class-work management modality for an extensive involvement of students in the construction of conjectures and proofs.
Firstly we took into consideration the conditionality of the statements, to which the logical structure of the proving process is connected. We have tried to formulate some hypotheses concerning the production of conditional statements and related proving developments. In order to do this, reference has been made to preceding studies, which suggested: the importance of the exploratory activity during the production of conjectures (cf. Polya's "variational strategies"; see also Schoenfeld, 1985); the relevance of mental images (as "a pictorial anticipation of an action not yet performed", Piaget & Inhelder, 1967 - see Harel, 1995; cf Simon, 1996 - "transformational reasoning") in the anticipatory processes in geometry; the possibility of deriving the hypothetical structure "if...then..." from the dynamic exploration of a problem situation (cf Caron, 1979).
We therefore came to the following hypotheses referred to a didactic situation where students are requested to solve an open problem through the formulation and proof of a conjecture. The hypotheses concern the crucial role that can be taken on by the dynamic exploration of the problem situation both at the stage of conjecture production and during the proof. The hypotheses are organised as follows (see Boero et al., 1996; for further results and extensions, see Guala and Boero, 1999):
- as to the conjecture production,
A) the conditionality of the statement can be the product of a dynamic exploration of the problem situation during which the identification of a special regularity leads to a temporal section of the exploration process, that will be subsequently detached from it and then "crystallize" from a logic point of view ("if......, then.... " );
- as to the proof construction,
B) for a statement expressing a sufficient condition ("if...then..." ), proof can be the product of the dynamic exploration of the particular situation identified by the hypothesis;
C) for a statement expressing a sufficient and necessary condition ("...if and only if..." ), proving that the condition is necessary can be achieved by resuming the dynamic exploration of the problem situation beyond the conditions fixed by the hypothesis.
These hypotheses brought us to the choice of suitable "contexts" familiar to students (i.e. "experience fields" - Boero, 1989; Boero et al, 1995) in order to promote the dsynamical exploration of the problem situation (see 3.1.).
Another hypothesis stems from our previous research on the feasibility of a constructive approach to theorems by students. In particular, during a teaching experiment concerning arithmetic theorems students were engaged in the production and proof of conjectures. It was observed that students kept a keen coherence between the text of the statement produced by them and the proof constructed to LEFT it (see Garuti & al., 1995). This textual coherence brought forward the problem of a possible cognitive continuity between the statement production process and the proving process. A similar behaviour in a problem solving situation implying the necessity of formulating and LEFTing conjectures was observed by C. Maher in very young students (grade IV) (see Maher, 1995).
Hypothesis D ("cognitive unity of theorems" as a facilitator of the student approach to theorems): the majority of grade VIII students can produce theorems (conjectures and proofs) if they are placed in a condition so as to implement a process with the following characteristics:
- during the production of the conjecture, the student progressively works out his/her statement through an intense argumentative activity functionally intermingling with the justification of the plausibility of his/her choices;
- during the subsequent statement proving stage, the student links up with this process in a coherent way, organising some of the justifications ("arguments") produced during the construction of the statement according to a logical chain.
This hypothesis might have important didactic consequences as to the school approach to theorems, radically calling into question the teaching traditions (see Subsection 5).
3.1. Our First Teaching Experiment
I will report here the first teaching experiment developed as a consequence of the preceding hypotheses.
The task concerning the production and proof of a conjecture was contextualized in the "field of experience"(Boero & al, 1995) of sunshadows. Students had already performed about 80 hours of classroom work in this field of experience. In particular they had observed and carefully recorded the sunshadows phenomenon over the year (in different days) and over the morning of some days. They had approached geometrical modeling of sunshadows and solved problems concerning the height of inaccessible objects through their sunshadows.
The field of experience of sunshadows was chosen because it offers the possibility of producing, in open problem solving situations, conjectures which are meaningful from a space geometry point of view, not easy to be proved and without the possibility of substituting proof with the realization of drawings. The field of experience of sunshadows is a context in which students can naturally explore problem situations in different dynamical ways. (cf. hypotheses A, B, C). In order to study the relationships between sun, shadow and the object which produces the shadow, one can imagine (and, if necessary, perform a concrete simulation of) the movement of the sun, of the observer and of the objects which produce the shadows. In particular, students had already realized some activities which needed the imagination of different position of the sun and of the observer in order to produce hypotheses concerning the shape and the length of the shadows.
In the two classes the activities were organised according to the following stages (whole amount of time for classroom work: about 10 hours):
a) Setting the problem :
"In the past years we observed that the shadows of two vertical sticks on the horizontal ground are always parallel. What can be said of the parallelism of shadows in the case of a vertical stick and an oblique stick? Can shadows be parallel? At times? When? Always? Never? Formulate your conjecture as a general statement."
(Individual work or work in pairs, as chosen by the students)
Some thin, long sticks and three polystyrene platforms were handed, in order to support the dynamic exploration process of the problem situation.
b) Producing conjectures: many students started to work with the thin sticks or with pencils. They started to move the sticks or to move themselves to see what happened. Other students closed their eyes. The absence of sunlight or spotlight in the classroom hindered the experimental verification of conjectures they were formulating: it was the mind's eyes that were "looking". Students individually wrote down their conjectures.
c) Discussing conjectures: the conjectures were discussed, with the help of the teacher, until statements of correct conjectures were collectively obtained which reflected the different approaches to the problem by the students.
d) Arranging statements: through different discussions, under the guidance of the teacher, the following statements," cleaned" from metaphors and more precise from a linguistic point of view than those produced by students at the beginning, were collectively attained:
-" If sun rays belong to the vertical plane of the oblique stick, shadows are parallel."
- "If the oblique stick moves along a vertical plane containing sun rays, then shadows are parallel."
- "The shadows of the two sticks will be parallel only if the vertical plane of the oblique stick contains sun rays."
The first two statements stand for two different ways of approaching the problem on the part of the students: the movement of the Sun and the movement of the sticks; the third statement makes explicit the uniqueness of the situation in which shadows are parallel.
After further discussion the collective construction of the two statements below was attained:
- "If sun rays belong to the vertical plane of the oblique stick, shadows are parallel. Shadows are parallel only if sun rays belong to the vertical plane of the oblique stick "
- "If the oblique stick is on a vertical plane containing sun rays, shadows are parallel. Shadows are parallel only if the oblique stick is on a vertical plane containing sun rays"
In order to help the students in the proving stage it was preferred not to express the statement in its standard, compact mathematical form "if and only if..." (its meaning in common Italian cannot be distinguished from the meaning of "only if..." ) .
e) Preparing proof; the following activities were performed:
- individual search for analogies and differences between one's own initial conjecture and the three "cleaned" statements considered during the stage d);
- individual task: "What do you think about the possibility of testing our conjectures by experiment?"
- discussion concerning students' answers to the preceding question. During the discussion, gradually students realize that an experimental testing is "very difficult", because one should check what happens "in all the infinite positions of the sun and in all the infinite positions of the sticks".
This long stage of activity (about 3 hours) was planned in order to enhance students' critical detachment from statements, motivate them to proving and make clear that since then classroom work would have concerned the validity of the statement "in general".
f) Proving that the condition is sufficient (activity in pairs, followed by the individual wording of the proof text);
g) Proving that the condition is necessary (short discussion guided by the teacher, followed by the individual wording of the proof text).
h) Final discussion, followed by an individual report about the whole activity (at home).
(an example of student behaviour))
Underlining indicates traces of connections between conjecture production and proof construction.
Formulation of the conjecture with shifting of the stick (phase b):
(Beatrice) "I tried to put one stick straight and the other in many positions (right, left, back, front) and with a ruler I tried to create the parallel rays. I sketched the shadows on a sheet of paper and I saw that: if the stick moves right or left shadows are not parallel; if the stick is moved forward and back shadows are parallel. Shifting the stick along the vertical plane, forward and back, the two sticks are always on the same direction, that is to say they meet the rays in the same way, therefore shadows are parallel. Whereas shifting the stick right and left the two sticks are not on the same direction anymore and therefore do not meet the sun rays in the same way and shadows in this case are not parallel. Shadows are parallel if the oblique stick is moved forward and back in the direction of sunrays."
Proof (phase f): "Shadows are parallel because, as we already said, sun rays belong to the vertical plane of the oblique stick.
But all this does not explain to us why this is true. First of all, though the sticks stand one in an oblique and the other in a vertical position, they are aligned in the same way and if the oblique stick is moved along its vertical plane and is left in the point in which it becomes vertical itself we see that they are parallel and, as a consequence, their shadows must naturally be also parallel, and also parallel with the shadow of the oblique stick, which has the same direction of that produced by the imaginary, vertical stick."
We notice that in the cases of Beatrice, just as for the majority of students, the dynamic process that brought to the production of the statement (movement of the sun or movement of the stick) is found again in the proving process. Yet the dynamic exploration implemented during the construction of the proof, though it shows remarkable similarities with the one implemented during the production of the conjecture as to the type of movement, differs deeply as to the function assumed in the thinking process: from a support to the selection and the specification of the conjecture, to a support for the implementation of a logical connection between the property assumed as true ("vertical sticks produce parallel shadows") and the property to be validated. The movement of the stick keeps the direction of its shadow (since it happens in the vertical plane containing sun rays) and, therefore, opens the possibility to reason in a transitive way (e.g.: the real, vertical stick produces a shadow parallel to the one of the imaginary, vertical stick; the oblique stick produces a shadow aligned with that of the imaginary, vertical stick; therefore the oblique stick produces a shadow parallel to that of the real, vertical stick). It also seems interesting to underline the fact that the hypothesis fixes the vertical plane on which the movement takes place that allows to relate logically the property to be proved with the property assumed as known.
The teaching experiment suggests some interesting hints about the links between argumentative reasoning in the phase of the production of the conjecture and proof construction. Actually, as concerns the production of the statement, argumentative reasoning (cf Douek, 1998; 1999) fulfils a crucial function: it allows students to consciously explore different alternatives, to progressively specify the statement and to LEFT the plausibility of the produced conjecture . On the other hand, students that produced wrong conjectures later show the need of reconstructing the valid conjecture in order to produce the proof.The fact that poor argumentation during the production of the statement always corresponds to lack of arguments during the construction of the proof seem to confirm the close connection that exists between production of the conjecture and construction of the proof. Moreover, the consistency among personal arguments provided during the production of statements and the ways of reasoning developed during the proof seems to be confirmed:
- by the fact that the type of argumentative reasoning made during the production of the statement by one student is resumed by him/her (often also with similar linguistic expressions) in the justification of the statement subject to proof;
- by the fact that the kind of dynamic process (movement of the sun or the stick) recorded at the conjecture stage is almost always the same as the one used at the proof stage.
3.2. Facilitating Conditions Emerging from the Teaching Experiment
In our teaching experiment, the "dynamic" learning environment of sun shadows was chosen in order to enhance the dynamic exploration of the problem situation on the part of students (keeping into account their background related to the same field of experience: see 3.2.). The great majority of the students (29 out of 36) has productively taken part in the statement construction and subsequent proof. This fact raises the problem of searching for learning environments similar or even more effective than that of the sun shadows as well as the problem of the transfer to "static" mathematics situations.
As regards the problem of finding suitable learning environments to develop the conjectures processes (dynamic exploration of problem situations), there are many learning environments which can be usefully compared with that of sun shadows (in particular, in the perspective of the "dynamic geometry" indicated by Goldenberg & Cuoco, 1995): Cabri or Geometric Supposer or Geometer's Sketchpad, even the "mathematical machines", the "representation of the visible space" and "gears" (Bartolini Bussi and Boero, 1998; Bartolini Bussi et al, 1999). Comparisons like these could propose different potentials and limits for the different learning environments.
As mentioned at the beginning of Section 3, the hypothesis D) seems to have important didactic implications, since it calls into question the traditional school approach to theorems.In fact, usually in Italy and in other Countries the teacher asks the students to understand and repeat proofs of statements supplied by him, which appears one of the most difficult and selective tasks for grade IX-X students. Only as possible last stage (often reserved to the top level students or students choosing an advanced mathematical curriculum) the teacher asks the students to prove statements, generally not produced by students but suggested by the teacher. Even more seldom students are asked to produce conjectures themselves. If our hypothesis is valid, during this traditional path students' difficulties can at least partly depend on the fact that they should reconstruct the cognitive complexity of a process in which mental acts of different nature functionally intermingle starting from tasks that by their nature bring them to partial activities that are difficult to reassemble in a single whole. Our teaching experiment suggests an alternative didactic path, based on situations where the cognitive unity of theorems can work as a "facilitator" for proving (see Garuti et al, 1998).
3.3. Validating Hypotheses Concerning Physical Referents and Proving Theorems
All the quoted experiments (performed from Grade V to Grade VIII) deal with problem situations concerning "Sunshadows", "Gears", etc. Theredore we must consider in what sense the students of the reported experiment have performed a mathematical activity concerning theorems.
The object of the experiment is a hypothesis concerning the physical phenomenon of sunshadows; it has as a geometric counterpart, at the level of model, a statement of parallel projection geometry. Students produce their conjecture as a hypothesis concerning the phenomenon of sunshadows; when they verify their conjecture most of them seem to be aware of the fact that they must get the truth of the statement by reasoning, starting from true facts. Most of them produce a validation realized through a deductive reasoning. Actually their reasoning starts from properties considered as true ("two vertical sticks produce parallel shadows") and gets the truth of the statement in the "scenary" determined by the hypothesis.
In this way, students produce neither a statement of geometry "strictu sensu", nor a formal proof: objects are not yet geometric entities, deduction is not yet formal derivation. But their deductive reasoning shares some crucial aspects with the construction of a mathematical proof. Moreover, the whole activity performed by students shares many aspects with mathematicians' work when they produce conjectures and proofs in some mathematics fields (e. g.: differential geometry): mental images of concrete models are frequently used during those activities. As to proof, mathematicians frequently come near to realize the ideal of the formal proof only during the final stage of proof writing. During the stage of proof construction, the search for "arguments" to be "set in chain" in a deductive way is frequently performed through heuristics, the reference to analogical models and keeping into account the semantics of considered propositions (cf Alibert & Thomas, 1991; Thurston, 1994).
For these reasons we think that the activity performed during our teaching experiment may represent an early approach to mathematics theorems which is correct and meaningful from the cultural point of view.
Other examples of early approach to mathematics theorems are reported in Bartolini Bussi (1996); Bartolini Bussi et al (1999). Despite some relevant differences concerning aims and research methodology, those teaching experiments share with ours the exploitation of everyday life situations and concrete referents (plane representations of 3D-situations; gears) in order to provide students with an early experience of proving "theorems".
What do we need as a further step?
Here the theoretical construct of "theorem" as "statement, proof and reference theory" by Mariotti (see Mariotti et al, 1997) helped us in planning further teaching: students need to enter gradually the culture of theorems, i.e. that peculiar culture which consists in developing knowledge by producing conjectures (written as statements according to a socially acceptable shape) and proving them with reference to a theory, i.e. enchaining arguments (legitimated by the reference theory) in a deductive way, up to a final product written according to a socially acceptable shape.
Some teaching experiments were performed; they are partially reported in forthcoming papers by Parenti and by Chiappini & Molinari.
4. THE TEACHER AS A CULTURAL MEDIATOR FOR THEORETICAL KNOWLEDGE: WHY AND HOW
The most common educational strategies (either traditional or not) to approach theoretical knowledge appear to be unproductive for most students, even in upper-secondary and tertiary education. In Italy as in other countries, mathematics and science theories are 'explained' by the teacher to students as from the 10th grade; the students' job is to understand them, to repeat them in verbal or written tests and to apply them in easy problem situations. The results are well known: for most students, theories are only tools for solving school exercises and do not influence their deep conceptions and ways of reasoning.
Constructivism too presents limits as regards the approach to theoretical knowledge: see Newman, Griffin & Cole (1989). We have noticed profound gaps which are difficult to bridge even with the teacher's help. These are between the expressive forms of students' everyday knowledge, and the expressive forms of theoretical knowledge; between the students' spontaneous way of getting knowledge through facts, and theoretical deduction; and between students' intuitions, and the counterintuitive content of some theories.
The ongoing research study, which is partially reported in this Section, aims to give useful elements for interpreting and overcoming the above difficulties in the approach to theoretical knowledge.
4.1. Why: A Vygotskian (and Bachtinian) Perspective.
The difficulties encountered in the traditional and constructivist approaches pose a series of questions. We shall try to describe the route we have taken to reach the definitions and hypotheses presented in Section 3.
What constitutes the gap between spontaneous and theoretical thinking? To address this issue we have considered the distinction proposed by Vygotskij, between everyday and scientific concepts (Vygotskij, 1992, chap. VI). It is common knowledge that this is one of the most controversial aspects of Vygotskij's work. It has often been considered outdated as it contains a systematic critique of the position taken by Piaget in the twenties, a position later revised by Piaget himself. On the other hand, the most significant examples Vygotskij uses to develop his arguments concern language and social sciences, with some generalisation to mathematics and natural sciences that are not always pertinent. In Vygotskij's school, Davydov himself has pointed out several weak and even contradictory points (Davydov, 1972). In addition, Vygotskij claims it is possible to 'teach' scientific concepts and theories to the point where they are 'internalised'; yet his hypothesis does not succeed in overcoming the learning paradox: 'How can a structure generate another structure more complex than itself?' and, more particularly, 'How does internalisation take place?' (see the discussion of Bereiter's paradox in Engeström, 1991). All the above objections have lead to underestimation of other aspects of Vygotskian analysis, such as the following: the systematic character of theoretical knowledge (versus the a-systematic nature of everyday knowledge); and the transition of scientific concepts from words to facts, versus the transition of everyday concepts from facts to word. Only recently have some researchers (e.g. John Steiner, 1995) called attention to the significance of these aspects of Vygotskian analysis.
In seeking to refine our framework about theoretical knowledge, we have found three different sources of inspiration. Although they belong to different cultural domains and orientations, they seem to offer coherent and useful hints about peculiar characteristics of theoretical knowledge.
The seminal work of Vygotskij (1990, chap. VI) suggests that: theoretical knowledge is systematic and coherent; it allows production of judgements (predictions, validations...) about the experience through intentional reasoning based on highly organized and culturally rooted linguistic patterns; it organizes the experience (both material and intellectual) in connection with a cultural tradition; it needs a particular mediation to be trasmitted to new generations.
Wittgenstein's theoretical costruct of 'language games' (Wittgenstein, 1953) can be exploited to describe how the potentialities of language (particularly the possibility of defining, of proving, of making the rules of inference explicit and, in general, of eliciting and discussing peculiar characteristics of a theory) allow theories to be constructed, described and discussed. Wittgenstein's analysis of common knowledge (1969) suggests that it offers the grounding and basic grammar for culture (and 'certainty') at any level, but it reduces to a set of (possibly incoherent) pragmatic tools if not systematized by a theoretical discourse.
Sfard's recent investigation (Sfard, 1997) suggests that "the discourse of mathematics may be viewed as an autopoietic system [...] which is continuously self-producing. According to this conception, the discourse and mathematical objects are mutually constitutive and are in a constant dialectic process of co-emergence".
Taking into account these references, we may try to point out some particular characteristics of theoretical knowledge in mathematics, by considering both the processes of theory production (especially as concerns the role of language) and the peculiarities of the produced theories:
* theoretical knowledge is organized according to explicit methodological requirements (like coherence, systematicity, etc.), which offer important (although not exhaustive) guidelines for constructing and evaluating theories; these methodological requirements belong to cultural tradition;
* definitions and proofs are key steps in the progressive extensions of a theory. They are produced through thinking strategies (general, like proving by contradiction; or particular, like 'epsilon-delta reasoning' in mathematical analysis) which exploit the potentialities of language and belong to cultural tradition;
* the speech genre of the language used to build up and communicate theoretical knowledge has specific language keys for a theory or a set of coordinated theories - for instance, the theory of limits and the theory of integration, in mathematical analysis. The speech genre too belongs to cultural tradition;
* as a coherent and systematic organization of experience, theoretical knowledge vehiculates specific 'manners of viewing' the 'objects' of a theory (in the field of mathematical modelling, we may consider deterministic or probabilistic modelling; in the field of geometry, the synthetic or analytic points of view; etc.)
We think that the approach to theoretical knowledge in a given mathematics domain must take these elements into account, with the aim of mediating them in suitable ways; indeed each of the listed peculiarities is beyond the reach of a purely constructive approach.
Why is constructivist approach unable to bridge the gap between everyday and theoretical knowledge? On the basis of his distinction between everyday and scientific concepts, Vygotskij hypothesises that, in children's intellectual growth, their everyday knowledge has to be developed towards theoretical knowledge by establishing links with theoretical knowledge and that theoretical knowledge has to be connected with facts by establishing links with children's everyday knowledge. Yet, according to Vygotskij, the development of everyday concepts is not spontaneous: the child cannot be left alone to pursue this process because theoretical knowledge has been socially constructed in the long term of cultural history and cannot be reconstructed in the short term of the individual learning process. In short, 'exposure' to theoretical knowledge is necessary, and must be provided together with explicit links to children's knowledge.
Which aspects of theoretical knowledge are to be chosen? In our view, cultural meaning and student motivation are the most important criteria. Therefore, priority should be given to leaps forward in the cultural history of mankind, even if, for the abovementioned reasons, these are the most difficult areas for school study. The sorts of topics we are referring to include, for instance, the theory of the fall of bodies of Galilei and Newton, Mendel's probabilistic model of the transmission of hereditary traits, mathematical proof and algebraic language - all aspects with a counterintuitive character. These are 'scientific revolutions' related to historical figures from the history of science (Galilei, Newton, Mendel, Euclid, Viete). In many cases, scientific revolutions have been accomplished by overcoming epistemological obstacles (Bachelard, 1938) which were a crucial part of previous knowledge. The same obstacles are often found in individual history as well (Brousseau, 1983).
How are the leading ideas of scientific revolutions expressed? Bartolini Bussi (1995) has suggested referring to the Bachtinian construct of 'voice' to describe some crucial elements of the turning points in scientific thinking. Bachtin's seminal work centers on literature, but some researchers in general and mathematics education have found several interesting elements therein (Bosch, 1994; Seeger, 1991; Wertsch, 1991). As far as the approach to theories is concerned, we draw on some aspects of Bachtin's work:
- the idea that human experience does not speak by itself but needs original voices that interpret it; the voices are produced in a social situation and gradually recognised by society until they become the shared way of speaking of the human experience;
- the idea that such voices act as voices belonging to real people with whom an imaginary dialogue can be conducted beyond time and space. The voices are continuosly regenerated in response to changing situations (they are not mummified voices to be listened to passively, but living tools for interpreting changing human experience).
How can students be 'exposed' to the leading ideas of scientific revolutions? If we transpose these ideas to the fields of science and mathematics (intended as a 'field of experience': Boero & al. 1995) we gain a useful perspective for our purposes: teachers can become mediators of 'voices' (of 'historical voices' in particular), which embody those scientific revolutions whose sense is to be conveyed to new generations. This process must take place in a social situation where the voices are renewed in accordance with changing cultural perspectives.
4.2. How: The "Voices and echoes game" as a Possible Mediation Methodology
Retrospective analysis of some teaching experiments (performed several years ago in the Genoa Group classes) confirmed the idea that scientists' voices may be exploited to approach theoretical knowledge and provided us with hints for further operational activity. As an example, let us consider the teaching experiment reported in Boero & Garuti, 1994. Students were asked to produce a brief, general statement about the relationships between heights of objects and the length of sunshadows they cast; they were asked subsequently to compare their statements with official statements of the so-called 'Thales theorem'. Analysis of the students' texts revealed an interesting phenomenon: many students had tried to rephrase their statements in order to make it resemble to the official statement, or to rephrase the official statement in order to make it to resemble their own. This was a constructive effort of a quite different nature from the production of an original statement; in fact it was an effort to 'echo' proposed 'voices'! A similar phenomenon is reported in Bartolini Bussi (1996), where the 'voice' of Piero della Francesca is exploited during a primary school perspective drawing activity.
Taking into account these experiences and the reflections summarized in the preceding section, we have undertaken the construction of a theoretical framework for a new methodological approach to theoretical knowledge. We have defined the 'voices and echoes game' (see 4.2.1) and elaborated a general hypothesis concerning the effectiveness of this game in approaching theoretical knowledge (see 4.2.2.). Consequently, we have planned a set of teaching experiments, which were performed in 19 classes (from grade V to grade VIII): they concerned the theories of Aristotle and Galileo about "falling bodies" (see Boero et al, 1997; Garuti, 1997; Boero et al, 1998; Boero and Tizzani, 1998); Mendel's laws (Lladò and Boero, 1998); Plato's "Menon" (Garuti et al, 1999).
Analysis of the first teaching experiment allowed us to elaborate a language (see end of 4.2.1.) that we consider useful for describing, classifying and interpreting student behaviour during the 'voices and echoes' game, and which is also helpful in recognising and conveniently managing that behaviour.
4.2.1. The 'Voices and echoes game'
What is the VEG? Some verbal and non-verbal expressions (especially those produced by scientists of the past - but also contemporary expressions) represent in a dense and communicative way important leaps in the evolution of mathematics and science. Each of these expressions conveys a content, an organization of the discourse and the cultural horizon of the historical leap. Referring to Bachtin (1968) and Wertsch (1991), we called these expressions 'voices'. Performing suitable tasks proposed by the teacher, the student may try to make connections between the voice and his/her own interpretations, conceptions, experiences and personal senses (Leont'ev, 1978), and produce an 'echo', i.e. a link with the voice made explicit through a discourse. The 'echo' was an original idea intended to develop our new educational methodology. What we have called the VEG is a particular educational situation aimed at activating students to produce echoes through specific tasks: "How might.... have interpreted the fact that...?", or "Through what experiences might ... have supported his hypothesis?"; or :"What analogies and differences can you find between what your classmate said and what you read...?", etc. The echoes produced may become objects for classroom discussion.
Students' echoes: students may produce echoes of different types (depending on tasks and personal adaptation to them). We distinguished between individual echoes and collective echoes (these are produced during a classroom discussion which may start from an external voice or some of the individual echoes selected by the teacher as voices). In Boero et al. (1997), individual echoes were classified. In this report we will focus particularly on resonances, the situations of greatest interest. In this case the student appropriates the voice as a way of reconsidering and representing his/her experience; the distinctive sign of this situation is the ability to change linguistic register or level by seeking to select and go deep into pertinent elements ('deepening'), and finding examples, situations, etc. which actualize and multiply the voice appropriately ('multiplication'). The echoes which develop at the collective level may consist of series of individual echoes of the voice at the origin of discussion ('source voice') with a high level of connection between successive echoes. In particular, both the examples related to the 'source voice' (multiplication) and the expressions and expressive registers (deepening) may undergo rapid and intensive enrichment. We called this phenomenon 'multiple echo'.
4.2.2. Mediating what? Some Examples from Recent Teaching Experiments
We note that the object of the 'voices and echoes game' is not to construct a concept or an original solution to a problem, nor is it to validate a student product. Rather, the point is to perform an active imitation ("echo") of a "voice" proposed by the teacher and appropriated by students under his guidance. In this way the transition of students' thought to a theoretical level can be enhanced. Our general hypothesis on this issue is that the 'voices and echoes game' may allow the classroom's cultural horizon to embrace some elements which are difficult to construct in a constructivist approach to theoretical knowledge and difficult to mediate through a traditional approach:
- contents (especially, counter-intuitive conceptions) which are difficult to construct individually or socially;
- methods (for instance, mental experiments) far beyond the students' cultural horizon;
- kinds of organization of scientific discourse (for instance, scientific dialogue; argumentation structured into a deductive chain) which are not a natural part of students' speech.
In the case of important and counter-intuitive theories (such as Galilei's and Newton's theory of falling bodies, which was the object of our teaching experiment), we think that the transition towards the revolutionary theory should be made by overturning the contrasting theory that preceded it. Consequently, the 'voices and echoes game' should start with historical voices that give a theoretical representation of students' intuitions and interpretations. There are a number of different reason for this approach: cognitive and didactic reasons (students need to take on board epistemological obstacles - see Brousseau, 1983 and, from a different perspective, Fischbein,1994); historical and cultural reasons (important scientific changes do not happen in a cultural vacuum, but occur when new theories substitute old ones); reasons related to student transition to a theoretical dimension (a theoretical dimension may be more accessible if it initially concerns theories which resemble students' conceptions about natural phenomena or mathematical entities).
An example (reported in Boero et al, 1997; Garuti, 1997; Tizzani and Boero, 1998) concerns the methodological requirements of theoretical knowledge. The students met Aristotle's and Galileo's selected texts ("voices"), and were asked to echo them in tasks of different types: "How might Aristotle have explained the fact that... ?"; "How might Galileo have opposed the idea that...?"
Another important potential of the VEG concerns the possibility of intervening in aspects of the student's mastery of theoretical knowledge - those related to detecting conceptual mistakes and overcoming them by general explanation.
Again, the Vygotskian elaboration about consciousness as a condition for accessing scientific concepts, clearly pointed out by Vygotskij in his seminal work about "common concepts" and "scientific concepts", seems to be useful to frame this complex operation. According to Vygotskij (1990, chap. VI), consciousness is related to mastery of scientific concepts for different reasons: "scientific" concepts are not isolated (and consciousness is needed to control connections and inner coherence of the system); "scientific" concepts are explicit (and consciouness is needed to manage explicitation and especially the relationship between mediating signs and meaning); "scientific" concepts are in dialectic relationship with common ones (and consciousness is needed to be aware of the borders between them). During an activity in which students participate effectively in examining their conceptual mistakes, all these aspects where consciousness intervenes can come into play: contradictions with known properties are frequently a motive the teacher advocates for helping students recognize a conceptual mistake; explicitation of some concepts is needed in order to point out ambiguities that may be the root of mistakes; in many cases the teacher must point out that common intuition is a possible source of mistakes.
But how can productive classroom activities concerning students' conceptual mistakes be organised? According to Bachelard (1977) many conceptual mistakes come from ancient knowledge that is appropriate in earlier situations but which is no longer suitable. The teacher must take the responsibility for selecting and proposing appropriate tasks (those which lead to crisis of the ancient knowledge) and for helping the student to overcome his/her mistakes. The teacher's role is central for other conceptual mistakes as well: for instance, those related to misunderstandings or ambiguities. What's more, the student must be aware of the role played by the teacher and his own role as a condition for being able to reproduce by himself, in the future, the sequence of actions needed to detect and overcome conceptual mistakes (cf. Brousseau, 1997).
Our working hypothesis was that the VEG could intervene as an appropriate educational methodology for attaining both the aims pointed out in preceding analyses: to develop students' consciousness about the functioning of theoretical knowledge when conceptual mistakes come into play; and to promote awareness of the teacher's and student's roles during classroom activities concening conceptual mistakes. Indeed, detecting and overcoming conceptual mistakes plays a crucial role in the evolution of mathematics and science. It is therefore natural that the history of mathematics and, more generally, the history of culture should offer "dialogical voices" that speak about this issue (exchange of letters, imaginery debates, etc). The production of "echoes" of well chosen "dialogical voices" during suitable tasks could lead students to participate consciously in the process of detecting and overcoming conceptual mistakes as a preliminary step towards interiorization.
The teaching experiment reported in Garuti et al (1999) was planned and performed in order to test and develop our working hypothesis. The object of the experiment was a well known piece of the Plato's "Meno", that concerning the problem of doubling the area of a given square by constructing a suitable square (this means overcoming the mistake which consists of doubling the side length). Fifth and seventh grade students were asked to produce echoes (i.e. "socratic dialogues") concerning a common mistake for them. For further details, see end of Subsection 2.4.2.
Alibert, D. & Thomas, M.: 1991, 'Research on mathematical proof'. In D. Tall (Ed.), Advanced Mathematical Thinking, Kluwer Ac. Pub.
Artigue, M.: 1992, 'Didactic engineering', Recherches en Didactique des Mathématiques, 12, (special issue), 41-65
Arzarello, F.: 1996, 'The Role of Natural Language in Prealgebraic and Algebraic Thinking', in Steinbring, H.; Bartolini Bussi, M. G.; Sierpinska, A. (Eds): 1996, Language and Communication in the Mathematics Classroom, NCTM, Reston, VA, pp. 249-261.
Bachtin, M.: Dostoevskij, poetica e stilistica, Einaudi, Torino
Bachelard, G.: 1938, La formation de l'esprit scientifique, Vrin, Paris
Bartolini Bussi, M.: 1995, 'Voci della storia dell'algebra', Actes du Seminaire Franco-Italien de Didactique de l'Algèbre, Genova
Bartolini Bussi, M.G.: 1996, 'Mathematical Discussion and Perspective Drawing in Primary School', Educational Studies in Mathematics, 31, 11-41
Bartolini Bussi, M.G.; Boero, P.: 1998, 'Teaching and Learning Geometry in Contexts', in C. Mammana & V. Villani (Eds), 'Perspectives on the Teaching of Geometry for the 21st Century', Kluwer A.P., pp. 52-62.
Bartolini Bussi, M. G.; Boni, M.; Ferri, F. & Garuti, R.: 'Early Approach to Theoretical Thinking: Gears in Primary School', Educational Studies in Mathematics, 39, 67-87
Bloedy-Vinner, H.: 1996. 'The analgebraic mode of thinking and other errors in word problem solving', in Gutierrez, A. & L. Puig (eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education, vol.2, 105-112.
Boero, P: 1989, 'Mathematical Literacy for All: Experiences and Problems', Proceedings of PME-XIII, vol. 1, pp. 62-76, Paris
Boero, P.; Garuti, R.: 1994, 'Approaching Rational Geometry: from Physical Relationships to Conditional Statements', Proceedings of PME-XVIII, Lisboa
Boero, P.; Chiappini, G.; Garuti, R.; Sibilla, A.: 1995, 'Towards Statements and Proofs in Elementary Arithmetic: An Exploratory Study About the Role of Teachers and the Bahaviour of Students', Proceedings of PME-XIX, Recife, Brazil, Universitade Federal de Pernambuco, Recife, 1995, vol. 3, pp. 129-136.
Boero, P.; Dapueto, C.; Ferrari, P.; Ferrero, E.; Garuti, R.; Lemut, E.; Parenti, L.; Scali, E.: 1995, 'Aspects of the Mathematics-Culture Relationship in Mathematics Teaching-Learning in Compulsory School', Proceedings of PME-XIX, Recife, Brazil, Universitade Federal de Pernambuco, Recife, 1995, vol. 1, pp. 151-166.
Boero, P.; Garuti, R.; Mariotti, M.A.: 1996, 'Some dynamic mental processes underlying producing and proving conjectures', Proceedings of PME-XX, Univ. de Valencia, vol. 2, pp. 121-128.
Boero, P.; Garuti, R.; Lemut, E. & Mariotti, M.A.: 1996, 'Challenging the traditional school approach to theorems: a hypothesis about the cognitive unity of theorems', Proceedings of PME-XX, vol. 2, pp. 113-120, Univ. de Valencia, vol. 2, pp. 113-120.
Boero, P.; Dapueto, C.; Parenti, L.: 1996, 'Research in Mathematics Education and Teacher Training', in Bishop A. (ed), International Handbook of Mathematics Education, Kluwer Ac. Pub., pp. 1097-1122.
Boero, P.; Pedemonte, B. & Robotti, E.: 1997, 'Approaching Theoretical Knowledge Through Voices and Echoes: a Vygotskian Perspective', Proc. of PME-XXI, Lahti, vol. 2, pp. 81-88
Boero, P.Chiappini, G.; Pedemonte, B.; Robotti, E.: 1998, 'The voices and echoes game and the interiorization of crucial aspects of theoretical knowledge in a vygotskian perspective: ongoing research', Proc. of PME-XXII, Stellenbosch, vol. 2, 120-127.
Boero, P. & Tizzani, P.: 1998, 'La chute des corps de Aristote à Galilée: voix de l'histoire et echos dans la classe pour l'approche au savoir théorique, Proceedings of CIEAEM-49, Setubal, pp. 369-376.
Boero, P. & Radnai Szendrei, J.: 1998, 'Research and Results in Mathematics Education: Some Contradictory Aspects, in A. Sierpinska & J. Kilpatrick (Eds.), Mathematics Education as a Research Domain, Kluwer A. P., Dordrecht, pp. 197-212.
Boero, P.; Garuti, R. & Lemut, E.: 1999, 'About the Generation of Conditionality of Statements and its Links with Proving', Proceedings of PME-XXIII, Haifa, vol. 2, pp. 137-144.
Boero, P. & Scali, E.: 1999, 'Objets d'usage courant dans la vie sociale exploités en classe', Actes de la X-ème Ecole d'Eté de Didactique des Mathematiques, Tome I, pp. 245-252.
Bosch, M.: 1994, La dimension ostensiva en la actividad matematica. El caso de la proporcionalidad, Thèse, Universidad Autonoma, Barcelona
Brousseau, G.: 1997, Théorie des situations didactiques, La pensée sauvage, Grenoble
Brousseau, G.: 1983, 'Les obstacles épistémologiques et les problèmes en mathématiques', Recherches en Didactique des Mathématiques, 4, 165-198
Brown, T.: 1997, Mathematics Education and Language, Kluwer Acad. Pub., Dordrecht
Caron, J.: 1979, 'La comprehension d'un connecteur polysemique: la conjonction "si"', Bulletin de Psychologie
Cocking, Rodney R. & Jose P. Mestre (eds.):1988, Linguistics and Cultural Influences on Learning Mathematics, Lawrence Erlbaum Associates
Davydov, V.V.: 1972, Types of Generalization in Instruction, Pedagogika, Moscow
Davydov, V.V.: 1988, 'Learning Activity...', Multidisciplinary Newsletter, 1/2, 29-36
Douek, N.: 1998, 'Some Remarks about Argumentation and Mathematical Proof and their Educational Implications', Proceedings of the CERME-I Conference, Osnabrueck, vol. I, pp. 128-142
Douek, N.:1999, 'Argumentative Aspects of Proving: Analysis of Some Undergraduate Mathematics Students' Performances', Proceedings of PME-XXIII, Haifa, vol. 2, pp. 273-280.
Duval, R.: 1991, 'Structure du raisonnement deductif et apprentissage de la démonstration', Educ. St. in Math.
Duval, Raymond: 1995, Sémiosis et pensée humaine, Peter Lang
Durkin, Kevin & Beatrice Shire (eds.): 1991, Language in Mathematical Education, Open University Press
Elliot, P. C. & Kenney, M. J. (eds.): 1996, Communication inMathematics - K-12 and Beyond, National Council of Teachers of Mathematics
English, Lyn D (ed.).:1997, Mathematical Reasoning, Lawrence Erlbaum Associates
Engeström, Y.: 1991, 'Activity Theory and Individual and Social Transformation', Multidisciplinary Newsletter for Activity Theory, 7-8, 6-17
Fauvel, J.: 1991, 'Using History in Mathematics Education', For the Learning of Mathematics,3-6
Ferrari, P.L.: 1996. 'On some factors affecting advanced algebraic problem solving', in Gutierrez, A. & L.Puig (eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education, vol.2, 345-352, Valencia (E).
Ferrari, P.L.: 1999, 'Cooperative principles and linguistic obstacles in advanced mathematics learning' in Inge Schwank (ed.), Proceedings of the First Conference of the European Society for Research in Mathematics Education, Vol.2, http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html
Fischbein, E.: 1994, 'The irrational numbers and the corresponding epistemological obstacles', Proceedings of PME-XVIII, vol. 2, pp. 352-359, Lisboa
Garuti, R.; Boero, P.; Chiappini, G.P.; Sibilla, A.: 1995, 'Towards Statements and Proofs in Elementary Arithmetic: .....', Proceedings of PME-XIX, Recife, 1995
Garuti, R.: 1997, 'A Classroom Discussion and a Historical Dialogue: a Case Study, Proc. of PME-XXI, Lahti, vol. 2, pp. 297-304
Garuti, R.; Boero, P. & Lemut, E.: 1998, 'Cognitive Unity of Theorems and Difficulties of Proof', Proceedings of PME-XXII, vol. 2, pp. 345-352
Garuti, R.; Boero, P. & Chiappini, G.: 1999, 'Bringing the Voice of Plato in the Classroom to Detect and Overcome Conceptual Mistakes', Proceedings of PME-XXIII, Haifa, vol. 3, pp. 9-16.
Goldenberg, E.P. & Cuoco, A.: 1995, 'What is Dynamic Geometry?', paper presented at the International Conference "LEFTing and Proving in School Mathematics", Institute of Education, London
Guala, E. & Boero, P.: 1999, 'Time Complexity and Learning', Annals of the New York Academy of Sciences, 879, 164-167.
Granger, G. G.: 1992, La vérification, Editions Odile Jacob, Paris
Halliday, M.A.K.: 1974, 'Some aspects of sociolinguistics', Interactions between Linguistics and Mathematical Education Symposium, Paris, UNESCO.
Hanna, G.: 1989, 'More than formal proof', For the Learning of Mathematics, 9, 20-23
Hanna,G. & Jahnke, N. (Eds.):1993, Aspects of Proof, special issue of Educ. St. in Math.
Harel, G. and Sowder, L.: 1998, 'Students' Proof Schemes', in A. Schoenfeld et al (Eds.), Research on Collegiate Mathematics, Vol. 3, M.A.A. and A.M.S. pp234-283.
John Steiner, V.: 1995, 'Spontaneous and Scientific Concepts in Mathematics: A Vygotskian Perspective', Proceedings of PME-XIX, vol.1, pp. 30-44, Recife
Lakatos, I.: 1976, Proofs and refutations, Cambridge University Press
Lladò, C. & Boero, P.: 1998, Les intéractions sociales dans la classe et le role médiateur de l'enseignant dans la modélisation mathématique des phénomènes naturels: le cas de la génétique, Proceedings of CIEAEM-49, Setubal, 171-179.
Leont'ev, A. N.: 1978, Activity, Consciousness and Personality, Prentice-Hall, Englewood Cliffs
MacGregor, M. and Price, E.: 1999, 'An Exploration of Aspects of Language Proficiency and Algebra Learning', Journal for Research in Mathematics Education, 30-4, 449-467
Maher, C.A.: 1995, 'Children's Explorations Leading to Proof', paper presented at the International Conference "LEFTing and Proving in School Mathematics", Institute of Education, London.
Malara, N. A.: 1999, 'Theory and Practice: The case of a Fruitful Relationship for the Renewal of the Teaching and Learning of Algebra', Proceedings of CIEAEM-50, Neuchatel, pp. 38-54.
Mariotti, M.A.; Bartolini Bussi, M.; Boero,P.; Ferri, F.; Garuti, R.: 1997, 'Approaching geometry theorems in contexts', Proceedings of PME-XXI, Lahti, vol.1, pp. 180-195
Mestre, Jose P.: 1988, 'The role of Language Comprehension in Mathematics and Problem Solving', in Cocking, Rodney R. & Jose P. Mestre (eds.): 1988, Linguistics and Cultural Influences on Learning Mathematics, Lawrence Erlbaum Associates.
Morgan, C.: 1998. Writing Mathematically. The Discourse of Investigation, London, Falmer Press.
Nathan, M.J. and Koedinger, K.R.: 2000, 'Teachers' and Researchers' Beliefs About the Development of Algebraic Reasoning', Journal for Research in Mathematics Education, 31/2, 168-190.
Newman, D.; Griffin, P. & Cole, M.: 1989, The Construction Zone, Cambridge U.P., Cambridge
Pawley, D. and Cooper, M.: 1997, 'What can be done to overcome the multiplicative reversal error?', Proceedings of the 21th Conference of the International Group for the Psychology of Mathematics Education, vol.3, 320-327, Lahti (Finland).
Piaget, J. & Inhelder, B.: 1967, The child's conception of space, W.W. Norton , New York
Pimm, D.: 1987, Speaking Mathematically: Communication in Mathematics Classrooms, London: Routledge Kegan and Paul.
Pimm, D.: 1991, 'Communicating mathematically', in Durkin, K.& B.Shire, Language in mathematical education, Milton Keynes, OpenUniversity Press, 17-23
Schoenfeld, A.: 1985, Mathematical Problem Solving, Academic Press
Seeger, F.: 1991, Teaching and Learning as Socially Distributed Activity, paper presented at the Int. Conf. 'Psychology of Learning and Mathematics Learning', C.I.R.M., Trento
Sfard, A.: 1997,'Framing in mathematical discourse',Proc. of PME-XXI, Lahti, vol.IV,pp.144-151
Simon, M.: 1996, 'Beyond Inductive and Deductive Reasoning: The Search for a Sense of Knowing', Educational Studies in Mathematics, 30, 197-210
Steinbring, H.; Bartolini Bussi, M. G.; Sierpinska, A. (Eds): 1996, Language and Communication in the Mathematics Classroom, NCTM, Reston, VA.
Tate, William F.: 1997, 'Race-Ethnicity, SES, Gender, and Language Proficiency Trends in Mathematics Achievement: An Update', Journal for Research in Mathematics Education, 28-6, 652-679.
Thurston, W.P: 1994, 'On Proof and Progress in Mathematics', Bulletin of the A.M.S., 30, 161-177
Usiskin, Z.: 1996, 'Mathematics as a Language', in Elliot, P.C. & M.J.Kenney (eds.), Communication inMathematics - K-12 and Beyond, 1996 Yearbook, Reston, NCTM,231-243
Vygotskij, L. S.: 1978, Mind in Society: The Development of Higher Psychological Processes, Harvard University Press, Cambridge, MA
Vygotskij, L. S.: 1992, Pensiero e linguaggio, Edizione critica di L. Mecacci, Laterza, Bari
Wertsch, J. V.: 1991, Voices of the Mind: A Sociocultural Approach to Mediated Action, Wheatsheaf, Harvester
Whitehead, A.N.: 1925, Science and the Modern World, Cambridge University Press
Wittgenstein, L.: 1953, Philosophical Investigations, Basil Blackwell, Oxford
Wittgenstein, L.: 1969, On Certainty, Basil Blackwell, Oxford