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Boero P.,
Rossella Garuti, Mariotti M. A. (1996)
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Abstract |
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1. IntroductionTwo previous reports had tried to focalize (through historic and epistemological analyses) the main cultural aspects of theorems in geometry (Boero & Garuti, 1994) and arithmetic (Garuti & al., 1995) in order to plan and analyse teaching experiments aimed at: - verifying the possibility of productively involving students in the approach to theorems; The teaching experiments carried out showed how the students consciously took over the conditionality and generality of the statements (under the proper guide and mediation of the teacher) and then tried to prove them. However, the activity of most of the students greatly depended on teacher's interventions and their acquisitions were mainly based on the constructive proposals of a small number of their schoolmates. This research project 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. In particular, 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) 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 hypothesis referred to a didactic situation where students are requested to solve an open problem through the formulation and proof of a conjecture.The hypothesis concerns 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 hypothesis is organised as follows: - as to the conjecture production, At this point we had to work out, put into practice and analyse a teaching experiment which let us explore the plausibility of the hypothesis, supplied the relative supporting elements and paved the way for further in-depth studies. In order to do this, reference has been made to: - our previous observations made about the behaviour of students struggling with the formulation of hypotheses and conjectures in both mathematical and non-mathematical fields (cf Boero & al., 1995). Those observations stressed the importance of the choice of the context ("field of experience") as a crucial factor in order to activate mental processes of dynamic exploration of the problem situation; The teaching experiment is described in § 2. The analysis and conclusions of the teaching experiment are shown in § 3. The discussion (§4) contains some reflections on our findings and indicates some of the developments suggested by our research. 2. Teaching ExperimentWe tried to identify a suitable learning environment and proper tasks for the development of a production process of meaningful conjectures in classes with a suitable background. In addition, we tried to construct a teaching experiment which favoured (through the teaching activity succession and the observation system) the emergence and recording of the processes which our hypothesis refers to. 2.1. Learning environmentWe have chosen the field of experience of "Sunshadows" as learning environment for our teaching experiment. The field of experience of sunshadows is a context in which students can naturally explore problem situations in different dynamical ways. 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. 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. 2.2. Classes and students' backgroundThe teaching experiment was carried out in two grade VIII classes of 20 and 16 students, at the beginning of the third school year with the same teacher. Students had already interiorized the habit of producing argumented hypotheses in different domains (mathematical and non-mathematical), writing down their reasoning. Students had already experienced situations of statements production in arithmetic and geometry; they had approached proof production in the arithmetic field (see Boero & Garuti, 1994 and Garuti & al, 1995). Concerning "Sunshadows", students had already performed about 80 hours of classroom work in this field of experience. 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. 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. 2.3. Classroom activitiesIn 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 the sun rays belong to the vertical plane of the oblique stick, shadows are parallel." 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 " 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 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). 2.4. Collected materialsThe following materials were collected: videotapes of the initial stages (a and b); tape-records of discussions and teacher-students interactions; all the students' individual, written texts.The data which we are about to consider mainly concern stages b), f) and g). 3. Some findingsThe teaching experiment analysis seems to confirm the validity of our hypothesis, as proved by the behaviour of the great majority of the students of the two classes. All students actively took part in the production of the initial conjecture. 29 students (over 36) were able to follow the activities (from c to h ) in a productive way. The elements found which confirm our hypothesis can be summarized up as follows: 3.1. As regards A) (relevance of the dynamic exploration on the problem situation during the conjecture production stage), the analysis of the videotape shows that at least one half of students (in the reality, probably more) performs the dynamic exploration of the problem situation in different ways: indicating with their hands the imagined movement of the sun, or moving themselves, or moving the oblique stick, or moving the platform supporting the sticks, etc. On the other hand, in 14 individual texts (out of 36) there is explicit evidence of the passage from the imagined (and/or simulated) dynamic exploration of the problem situation to focusing on a temporal section, with successive transition to the formulation of a statement "crystallized" from a logic point of view: (Simone)"If we took into consideration two sticks, of which one vertical, the shadows shall be parallel when the two sticks are viewed parallel by the sun. If we suppose that the person is looking in the position of the Sun, by going round the sticks we can observe that the sticks are parallel in a certain position and the shadows are also parallel since the difference in position of the two sticks cannot be seen from that position. Thinking about the shadow space we can say that the non-vertical stick seems to be within the shadow space. Let's imagine an imaginary vertical stick representing the oblique one, in line with the sun rays and the same stick, the oblique one cannot be seen so it seems to be vertical, forming parallel shadows. The shadows can be parallel if the sun is situated along the direction of the oblique stick [with a gesture he indicates the vertical plane of the oblique stick]" During the subsequent discussion, Simone explains how he produced this conjecture. Simone's gestures show that he moves the polystyrene plane supporting the sticks "at random" (notice should also be paid to the generality of his reasoning) after identifying himself with the sun. Then, he places a new stick (which he calls "imaginary stick" ) in the same position he described in the written text, making the polystyrene plane rotate until the non-vertical stick is completely hidden by the "imaginary" vertical one. At this point he says "well, now in this position the shadows are parallel because...". Finally, it is interesting to analyse the way in which certain initially wrong conjectures are overcome: at the beginning of stage b) some students hypothesize that shadows are always parallel, on the basis of a kind of "principle" (according to their previous school experience): "the sun rays are parallel, so they give parallel shadows". This conjecture is overcome by imagining and/or simulating the movements of the sticks or the sun. Those movements allow students to explore new alternatives. Here follows an example: (Lucia)"I think shadows are parallel because the oblique stick functions like a normal object perpendicular to the ground, so if the rays are equal for all the objects, the shadows will be parallel. 3.2. As regards B) (relevance of the dynamic exploration of the situation determined by the hypothesis during the construction of the proof that the condition is sufficient) , the following texts well represent the individual texts produced by most students: (Giovanni) "The sun "moves". At a given moment it "sees" the two parallel sticks and relative shadows. As the sun is far away it "sees" the two shadows parallel, so it imagines the oblique stick to be vertical (imaginary stick) [introduced by Simone during the discussion phase]. But if the imaginary stick were real its shadow would cover that of the oblique stick, that is they are on the same line.Well, now we know that the shadow of the two vertical sticks are parallel and at this moment it is as if we saw two parallel shadows because that of the oblique stick is "under" that of the imaginary one. Now, if we removed the imaginary stick, the shadow of the oblique stick would appear again since it was "under" the parallel shadow of the imaginary stick, so the shadow of the oblique stick is also parallel to that of the vertical one". It seems to us that from these texts clearly comes out the fact that the dynamic exploration of the situation singled out by the hypothesis fulfils an important function in order to promote the logical connection between the property accepted as true (parallel sticks produce parallel shadows) and the property to be confirmed (shadows are parallel): 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. In this sense 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, differs deeply as to the function assumed during 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. 3.3. As regards C) (the dynamical exploration of the problem situation is resumed during the construction of the proof that the condition is necessary), we observe that: - in some cases the sun or its rays are moved: (Stefania): "If the sun rays do not longer belong to the vertical plane of the oblique stick, the sun would "see" three sticks: one vertical, one oblique and an imaginary vertical one that casts shadow. Taking for granted that the shadows of the two vertical sticks are always parallel independently from the position of the sun or its rays, then, the sun would cast three shadows, of which two parallel and one oblique with respect to the other two. And if this shadow of the oblique stick were not aligned with that of the imaginary stick, it will be neither parallel with the shadow of the vertical stick, so the shadows would not be parallel and the hypothesis would not be true" - in other cases students moved the stick (beyond the vertical plane identified by the hypothesis): (Sandra)"In order to prove the second part of the statement [the shadows are parallel only if the stick moves along a vertical plane containing sun rays]we can move and place the oblique stick in another vertical plane so as to obtain two vertical planes, that of the oblique stick and that of the imaginary vertical stick. With this operation the two shadows are no longer situated in the same line so the shadow of the oblique stick and that of the vertical stick are no longer parallel. In this way, I've denied the previous statement so the shadows will be parallel only if the oblique stick is placed again along the vertical plane of the sun rays". 4. DiscussionWhat relationship does it exist between our teaching experiment and producing and proving mathematical conjectures? 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 many 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). 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 2.2.). 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 transfert 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" and the "representation of the visible space" (Bartolini Bussi, 1995). Comparisons like these could propose different potentials and limits for the different learning environments. With regard to the problem of the transfert from strongly contextualized theorems in a dynamic environment as that of the sun shadows geometry to the theorems of "context-free" mathematics, a number of confirmations derive from the observations that followed the teaching experiment in the two classes during activities with traditional geometry theorems. In particular, many students (of both high and average levels) could imagine the dynamic exploration of the geometric figures proposed for the formulation of conjectures and proofs. In the future, it will be necessary to make more systematic observations and establish comparisons with classes which have not carried out the activity described in this report (but have performed all previous activities). A delicate matter concerns the variety of possible approaches to the conditionality of statements (and related connections with proving process). In fact, in Boero & Garuti (1994), a report dealing with the "Thales Theorem" and concerning the same learning environment of "Sunshadows", the following type of reasoning was identified in 3 students out of 34: "The length of the shadows is proportional to the height of the sticks due to the parallelism of the sun shadows .... If the lines are parallel, the lengths of the segments cut on another two lines shall be proportional" .The process appears to be very different from that considered in our hypothesis, since in this case the student passes from a recognition of causal dependency between parallelism and proportionality in the physical phenomenon, to the conditional statement that takes into account the possibility that lines cannot be parallel. This process asks therefore detaching from the physical phenomenon (that on the contrary can be deferred in the case of the approach to conditionality studied in this report). It is for this reason that we have formulated our hypothesis A) by emphasizing the possibility ("can") that the conditionality of statements were originated in the dynamic exploration of the problem situation without excluding other possibilities. Further research will certainly supply interesting indications in this field, especially with respect to conjecture production processes more accessible to students in the approach to theorems. Finally, we deem it important to examine deeply the implications of what surfaced during the teaching experiment as to the continuity that seems to exist between the way in which the dynamic exploration of the problem situation is attained and expressed and the way in which, during the proof construction, the dynamic exploration of the situation singled out by the hypothesis is in its turn attained and expressed. This continuity prompts the reflection on the holistic character that in an opportune educational context can be taken on by the process of theorems (statement and proof) production, apparently contrasting with the deep difference that exists between the argumentative reasoning needed to construct and make plausible the conjecture and the deductive reasoning to validate it (see Duval, 1991). Acknowledgements. Carlo Dapueto, Pier Luigi Ferrari and Enrica Lemut helped us to clarify and develop some ideas of this paper. We thank them very much. ReferencesAlibert, D. & Thomas, M.: 1991, 'Research on mathematical proof'. In D. Tall (Ed.), Advanced Mathematical Thinking, Kluwer Ac. Pub. 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