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Boero P.,
Chiapini G., Garutti R. Sibilla. A. (1992)
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1.IntroductionThe issue of the approach to mathematical theorems is dealt with in papers dealing especially with geometry, and above all, concerning the proof of theorems (see Balacheff, 1987; Hanna & Winchester, 1990; Hanna & Jahnke, 1993). Proving geometry theorems prevails also in high school students' work (see Moore, 1994). In Boero & Garuti (1994), an analysis had been performed on how grade VI / VII students may realise, in a convenient educational context, a constructive approach to geometry statements. That report indicated some issues to be dealt with more in depth, concerning: - The role of cultural mediation that may/must be performed by the teacher. This report refers to a study having the following objectives: - Analyze the behaviour of VI/VII grade students when approaching statements in elementary arithmetic (this is a field that has not been widely considered in the literature dealing with the approach to theorems). Similarly to the approach to geometry theorems, historical and epistemological analysis helped us to identify some distinctive characteristics of arithmetic theorems and their proofs apt to be taken as reference points in the investigation of the cognitive behaviours of the students. This agrees with Vygotskij's theoretical framework: "A suitably organised teaching-learning process results in mental development... Each scholastic subject has a specific relation with the child's development course, ... this takes directly to a review of the issue ... of the significance of each single subject from the point of view of the overall mental development" (Vygotskij, 1978). 2. Historical and Epistemological AnalysisAs for the historical analysis, it is known that even before the IV Century BC, the Greek mathematicians/philosophers discovered and proved many arithmetic properties. Concerning this period, Szabò (1961) underlines how the proofs relative to the properties of natural numbers form the first historical example of " " (i.e. "science", "research"). In Euclid's Elements (Heath, 1956) we find arithmetic theorems that, from an epistemological point of view and as we'll see with some examples, show already all significant features of modern statements (conditionality, generality, relational or procedural formulation, etc.). The proofs instead are discursive and use a geometrical segment model to express "general numbers" (the algebraic language is not yet available). Like geometry's, also arithmetic's statements are conditional, that is, expressed with the formulation: "if.... then...". In Euclid's Elements we find statements with an explicit conditional formulation: "If two numbers be prime to any number, their product also will be prime to the same" (Tome VII, prop.24) together with others where the conditional form is implied: "Any prime number is prime to any number which it does not measure" (Tome VII, prop. 29). As for the generality of the statements, the arithmetic field allows us to formulate significative statements with different degrees of generality. For instance, concerning the set of prime numbers, we may formulate statements relative to properties of the set itself (as for instance: "Prime numbers are more than any assigned multitude of prime numbers", Tome IX, prop. 20) or properties relative to a generic element of the set (as for instance tome VII, prop. 29 mentioned above). Arithmetic statements may be expressed either in a procedural or a relational manner, i.e. highlighting the procedure that leads to the result to be validated by the proof, or the relation, or property, that depends on the hypothesis being formulated and that must be proven. In Euclid's Elements we find statements expressed in a relational form (such as: Tome IX, prop.20 mentioned above) as well as statements expressed in a procedural form (Tome IX, prop.22: "If as many odd numbers as we please be added together, and their multitude be even, the whole will be even" ). Nowadays for many arithmetic theorems we use algebraic formalism in its functions of "generalisation - synthesis" (to express the statement) and of "transformation" (to prove the theorem by means of an algebraic "calculation"). For what concerns the introduction of algebraic formalism in arithmetic, the need to express the general resolutive methods of arithmetic problems with a suitable formalism, syntactically different from common language and from Euclid's segments model, is manifested in the work of Diofanto (IV Century AD). One thousand year had to pass, however, before Viete put together a formalism adequate to meet this need. Still from an historical and epistemological point of view, concerning the use of algebraic formalism, we can see how the statements and proofs relative to arithmetic theorems have formed, from the end of the past century on, the preferred field for logicians, mathematicians and artificial intelligence researchers to try and reduce the proofs to calculations, exploring thus the issue of the "truth" in mathematics and developing also automatic proving programs. Concerning the significance of arithmetic statements, in the history of the theory of numbers we may trace back different significance criteria, often cohexistent at the same time: intellectual challenge (e.g.: the "Fermat's theorem"), relative to the difficulty of the proof; knowledge of the deep structure of the set of natural numbers (e.g.: infinity of the set of prime numbers); paradigmacity and importance for the construction of more general algebraic structures (e.g.: the theorem: "given two natural numbers a and b, with a>b, there exist two natural numbers q and r such that a=bq+r" produces in Algebra an axiom in the introduction of Euclidean rings). Let us note that, according to these last two criteria, a wide knowledge of arithmetic (if not of an even larger mathematical domain) is required to evaluate the significance of an arithmetic statement. These historical and epistemological considerations have been useful for us, as seen in the next paragraphs, to focus the issue of the role of the teacher, set up the teaching experiment and analyze the behaviour of the students. 3.The teacher as a mediatorWhen approaching arithmetic statements and proof, there exist at first a wide gap between the knowledge of the teacher and the knowledge of the students. In particular, the teacher possesses knowledge and experiences unknown to the students in the areas of: - Linguistic formulation of the statements and their characteristics of generality and conditionality. The knowledge of the students may be made closer to the knowledge of the teacher: - In part through constructive activity required of the students, therefore through an indirect mediation on the part of the teacher implemented with the choice of suitable tasks promoting the generation by the students of significant and useable products for classroom work. All these forms of mediation must take into account the cognitive requirements of the students, manifested through their productions and the cultural and cognitive meaning that these products have. 4. Planning of the Teaching ExperimentThe educational context where our teaching experiment was located was that of the Genoa Group Project for the Comprehensive School. Relevant to the study reported in this paper are: - The practise of written verbal reporting on the part of the students, concerning both the resolution of the problems and the relative reasoning and reflections. The teaching experiment involved two classes in 1992/93 and two classes in 1994/95. Let's now go on to the description of the assignments (for further details, see Sibilla, 1994): By means of the individual assignment: "Suppose you have a certain set of numbers. Apply the transformation '+1' to all the elements of the set. What are the effects of the transformation?" and in general, by means of assignments of the type "what happens if...?" referred to a set of numbers selected by the students, followed by the comparison between the "effects" that had been identified, the experiment aimed to create, in the numbers "field of experience" (by now sufficiently familiar to an 11-year old student), an initial awareness of the fact that there exist "unvarying" properties for the change of the given set of numbers being considered. The students, in other words, were to be led to identify and express in a conditional format, properties having characteristics of generality. Another purpose was also to raise the issue of the justification, through reasoning, of properties which do not appear immediately true (until their proper proof). By means of the individual assignment: "You have a given set. What transformation do you have to apply to the set so that the transformed set is only formed by even numbers? (Help yourself with tables if you want)" and the assignment: "You have a given set. What transformation do you have to apply to the set so that the transformed set is only formed by odd numbers?" and by means of the subsequent discussion, the experiment wanted to "force", via the identification and the expression of a variable, the process of algebraic formalisation. By means of the assignment "What happens if you add together two consecutive odd numbers? Is there a regularity? And if so, why?" and the following discussion, the experiment wanted to stimulate an experience of exploration of numerical facts possibly leading to the identification of various properties: general because they do not vary with the particular pair of selected consecutive odd numbers, and (implicitly) conditional because they would depend on the conditions (odd and consecutive) of the numbers. Other assignments, such as: "what happens if you add two even consecutive numbers?" and "what happens if you add three odd consecutive numbers?" were used to evidence the conditional (as well as general) character of the statements relative to the properties of natural numbers, as well as develop a dexterity with algebraic formalism as a tool to explore and prove arithmetic theorems. 5. Analysis of the behaviour of students in the initial stage5.1. Production and comparison of statements in a classThe assignment: "Suppose you have a certain set of numbers. Apply the transformation '+1' to all the elements of the set. What are the effects of the transformation?" produced in the four classes a large variety of answer texts, due to its character, purposely "unleading". Some of the texts that were produced appear to be rather superficial, of little consequence from a mathematical point of view, and not very general in character: for instance: "If I have the set 2, 3, 4, it becomes the set 3, 4, 5" ; other texts contain statements which, although not very general, appear to be quite significant: "If the set contains numbers ending with 9, the transformation +1 transforms them in numbers ending with 0 and they have an extra digit". Other statements that have been produced are more general: "The set is transformed in another set with the same number of elements". Vis a vis with these products by the students, the teacher must select those which are more suitable, trying to bring out (or mediate) those aspects which are important from a cultural point of view (see point 3.).In a VII grade class, for instance, the following statements were compared: a) If I have 3, 4, 5, 6, 7, their sum is 25. When I add 1 to the numbers, their sum is 30. The character of generality of the statements may be negotiated, at least in part, with the students. In practise, in this class, at this stage of the teaching experiment, the negotiation took place through the comparison of statements presenting common elements and asking the students to establish what happened to some properties that they had identified if the reference set was changed, and to compare statements which (like c) and d)) had common elements. Through a teacher-led discussion, the students were able to make significant observations concerning the character of generality of the statements, in particular discovering that the first is valid only for that specific set. The second statement created some perplexity in the way it is formulated:"It almost appears that you can get odd or even numbers from any set of numbers. It does not talk of the initial situation. If the beginning set is formed by even numbers only, the statement is no longer true". The third and fourth statements were instead considered general (they are valid for all sets on which the "+1" transformation is carried out, but express a different degree of generality: "In the fourth, one is considered as odd, but it could also be 3 or 5"). Many students observed also that the fourth statement goes beyond the level of generality requested by the assignment. The conditional character of the statements, that was to be considered at several stages during the course of the teaching experiment, appeared in this class during the discussion of the statements above. In particular, the teacher led the class to the discovery that in the third statement the conditionality is explicit, while in the fourth, it is implied. In this class, the problem of the significance had been introduced by comparing the statement a) with the other three: For the students, at this stage of the work, the triviality of the statement was clear ("in a) it is like saying that all the numbers increase of one unit" ) without going further than this level of reflection. In our opinion, the capacity of the students to autonomously express a judgement relative to the significance may be the result only of an extended activity aiming to bring out as "significant" those properties which are not immediately apparent (intellectual challenge) and/or that contribute to a deeper insight in the numeric field. This part of the teaching experiment confirms (in the field of arithmetic) the hypothesis of feasibility of the objective to get grade VII students constructively involved in approaching statements of theorems in an adequate educational context, strongly depending on the role of the teacher , stated in Boero & Garuti (1994) for geometry theorems. 5.2 Procedural and relational statements: Approach to the proof in two classes.In two grade VII classes (39 students), with the same teacher, after the first comparisons, the attention veered on the following property, offered by a girl: "If I add 1 the divisor changes. For instance, 365 is divisible by 5, 366 is not". Rather than a statement, this is the observation of a property that, even in its rather poor formulation, the teacher was able to judge as being very significant, not being immediately apparent and suitable to illustrate some structural characteristics of the set of the natural numbers. The property was then submitted for discussion to the two classes. The students seemed little convinced that it was true and, moreover, they were perplexed concerning its formulation. At this point the teacher asked the students to ask questions in order to formulate the statement more precisely. These were the questions of the students: - "Is it true for all numbers? How can you be sure?" As it may be seen by the questions, the problem of the greater precision in the formulation of the statement was intimately weaved with the problem of the verification of its validity, even if gradually, through numerical examples, they started to realise that the property might be true. Later the students were asked to rewrite the statement. In substance two types of formulations emerged (considered by the students as being equivalent: "They say the same property") : I) A number and the number immediately after have no common divisors except for the number 1 (relational statement) At this point, it was interesting to determine how much the formulation of the statement could condition the proving process, and in what measure the proving process could be autonomously managed by the students in a situation where the students were strongly motivated to prove the property which they had discovered. For this reason, the students were asked to: "try and verify if the property expressed in the two statements is true and why". All the attempts made by the students of the two classes seem influenced by the formulation of the statement that they considered. Most of the students (32) refer to the relational statement: they find all the divisors of a number and its next and verify that there are no common divisors, except for 1. This procedure does not help them in the justification of the statement (none of them was able to attain to a real proof), but helps them in substance to get an opinion on its validity. Different behaviours, however, were observed amongst the students proceeding in this way: - Some try on "large" number, looking for an empirical verification of the validity of the statement. Some of the 7 students referring to the procedural statement consider the divisors of a number and try to establish if the same divisors are also the divisors of the following number, realising that the unit, added in the transformation, constitutes the remainder of the division. This is the way that a student comes to the following proof: "This statement is true because between a number and the number immediately after, you add 1, so that the divisor of the first number is not right for the other because the 1 that was added forms a remainder. (...) For instance 15:3=5 and 16:3=5 remainder 1." Another proof that was produced is: "When skip-counting, except when you skip-count by 1, it is not possible that there are consecutive numbers since the multiples of a number derive from the beginning number that is added all along, so that it is impossible that two multiples are also two consecutive numbers, for instance 2-4-6-8... or 3-6-9-12...". In this case also the proof appears to be influenced by the procedural formulation of the statement. The other five students, although not reaching to a complete proof, make some steps in the first of the two indicated directions. The discussion and the comparisons relative to these two proofs allow (under the guidance and with the mediation of the teacher) to bring out deficiencies, analogies and differences with the other texts that had been produced. The work of these two classes appears to be satisfactory as a whole, both for the quality of the two proofs that were produced, and for the critical capacities demonstrated by most of the students towards the unsuccessful attempts.In our opinion, it also provides some elements for further studies, especially the hypothesis that in the case of a statement produced (or assumed) by a student, its proving process may naturally evolve from it as a textual "development" of the statement itself. 6.The problem of the approach to the proof as an algebraic calculationThe overall positive result of the "arguing" approach to the proof, illustrated at point 5.2, raises the issue of the opportunity of a fast transition to the proof as a calculation carried out on the formula expressing the elements on which the property to be proved is to be verified. It is apparent that for properties such as those considered in the last two stages of our teaching experiment, the algebraic formalism makes easily accessible to the students proofs that would otherwise present many difficulties, while the proof of the property considered at point 5.2 would not be made any easier by the availability of algebraic formalism. There may be imagined therefore both a development of classroom activities such as that hypothesised in our teaching experiment (where the choice of the statements from stage 2 on lends itself to the best use of algebraic formalisms), and a development based on other types of statements (like some contained in Euclid's Elements) that may be proved also without resorting to algebraic formalism. On the basis of our experience, it would appear to us that the conquer of algebraic formalism and its use for proving, would involve the students in discussions and considerations regarding conventions, transformation rules, etc. that may distract them from the logical mechanics of the proof, without, on the other hand, producing extended learning results (as far as the ability of autonomously using algebraic formalism for proving). It would appear to us moreover, that the influence on the algebraic formalism on the development of the proof is very strong and may give place to a development of the proof linked to the transformation mechanism and not the analysis of the property to be proven. We would like to mention, in this respect, an episode occurred in a class where, by the end of the teaching experiment described at point 4. only one student manages to autonomously use algebraic formalism for proving. During the teaching experiment, the student must prove that the sum of two consecutive odd numbers is divisible by 4. He gives a proof in words, writing that "the sum of two consecutive odd numbers is like taking an even number, take away one from it, and sum it to the same even number and add one. So it is the sum of two even numbers and therefore is divisible by four ". At the end of the teaching experiment he was asked to prove the same property using algebraic formalism; he writes: " 2n + 1 + 2n + 3 = 2n + 2n+ 4 = 2n + 2n + 2 + 2 = (2n+2)x2" . On the other hand, it is through the very use of the algebraic formalism as a tool for the expression of properties and as a calculation tool when proving arithmetic properties that the students may grasp the powerfulness of the tool that has been introduced and may make the first significant experiences of its "transformation function". A compromise solution for this problem appears then that of using different classroom activities to approach the two objectives: development of demonstrative reasoning as an arguing and logical experience; introduction to the algebraic formalism and "exemplary" use in carrying out proofs as "algebraic calculations" ReferencesArsac,G.: 1987, 'L'origine de la démonstration:...', Recherches en didactique des mathématiques Balacheff, N.:1987,'Processus de preuve et situation de validation', Educational Studies in Math. Balacheff,N.: 1988, Une étude des processus de preuve en mathématiques, thèse d'état, Grenoble Boero P.: 1989, 'Mathematical literacy for all: ..' , Proceedings PME XIII, Paris Boero, P.; Garuti, R.,1994: 'Approaching rational geometry...', Proceedings PME-XVIII, Lisboa Duval, R.: 1991, 'Structure du raisonnement deductif ...', Educational Studies in Mathematics Hanna,G. & Winchester, I.(Eds.): 1990, Creativity,Thought and Mathematical Proof, OISE, Toronto Hanna,G. & Jahnke, N. (Eds.):1993, Aspects of Proof, special issue of Educational Studies in Math. Heath, T.: 1956, The Thirteen Books of Euclid's Elements , Dover, New York Mesquita,A.L.:1989,'Sur une situation d'eveil à la déduction ...',Educational Studies in Mathematics Moore, R. C.: 1994, 'Making the Transition to Formal Proof', Educational Studies in Mathematics Sibilla, A., 1994: 'Approccio alla costruzione degli enunciati, alle dimostrazioni e al formalismo algebrico...', in Numeri e proprietà, Dipartimento Matematica Università di Parma Szabò, A.: 1961,'The transformation of mathematics into deductive science and the beginning of its foundation on definitions and axioms',I/II, Scripta mathematica, vol. XXVII Vygotski, L. S.: 1978, Mind in Society, Harvard University Press, Cambridge, Mass.
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