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Proposal for the DG22
Reasoning and Proof in a Spatial Geometry Teaching Situation
Denise Grenier,
Universit Joseph Fourier, Grenoble
denise.grenier@ujf-grenoble.frDenis Tanguay,
Universit du Qubec Montral (UQAM)
HYPERLINK "mailto:tanguay.denis@uqam.ca"tanguay.denis@uqam.ca
1. Introduction
1.1. The transition to post-secondary education
The effects of the transition from high school to the final years of high school (Lyce in France, CEGEPs in Quebec, and upper secondary school in English-speaking North America) and to the first years of university have, in recent years, been the subject of a number of studies by researchers in math education. One of the more acute problems to emerge in relation to these transitions is the increasing use of formalism, symbolism and proof in post-secondary mathematics. What stands out in what some researchers in math education refer to as the obstacle of formalism (for example Dorier, 1997 or Sierpinska et al., 1999) is the problem of meaning which teaching should bring to the classroom math experience: Formal writing does not in itself transmit the meaning of the laws it states or of the objects it brings into play (Bloch & al., 2006). As regards proof in particular, its a matter of resituating it within a process where it acquires full meaning, a process as complete as possible, similar to the practice of the true mathematician-researcher.
1.2. Establishing the truth
Deciding whether a statement is true or false is one of the raisons dtre of proof. In the classroom, this speculating about truth is, for the most part, absent: usually statements to be proven are declared to be true (Show that) or, as is often the case in plane geometry, are obvious from the figure, whether provided or created by the student.
Our hypothesis is that understanding the process of proof in its entirety requires that students regularly be placed in the situation of experimenting, defining, modelling, formulating conjecture and proving, with formal proof thus appearing as a requirement in establishing the truth of the proposed conjectures. For years, this educational hypothesis has been the basis for the work of the Maths--Modeler team at the Universit de Grenoble in France, a team which creates and studies the devolution and management of research situations for the classroom (Situations de recherche pour la classe, or SiRC) at different levels of schooling and at very diverse institutions. (Grenier & Payan, 1998; Grenier, 2001: Godot & Grenier, 2004; Grenier, 2006). This work is being continued in collaboration with researchers at the Universit du Qubec Montral (UQAM). The situation we present here fits within the framework of the collaboration between these teams. Spatial geometry, a field where basic properties are not obvious, strikes us as a source of problems in which the conditions mentioned above may be combined, particularly as they relate to the activity of definition (see 1.4) and to the necessity of resorting to proof. We can foresee putting this situation into effect at several levels of schooling, especially at the end of high school (ages 17-19) where it provides an opportunity to introduce students to graph theory, an area of mathematics which, along with discrete mathematics, is increasingly prevalent in end-of-secondary school curricula.
1.3. Experimenting
Experimenting is an essential task in any process of proving, a process concluded by writing a formal proof. By studying examples and researching possible counterexamples, it is possible to formulate a conjecture, to attempt to understand why it works and finally, to establish proof. Current programmes at the secondary level in France and Quebec place emphasis on experimentation in mathematics.
1.4. Defining
Defining is an important task in the study of concepts or in establishing ad hoc theories. This task is absent from current teaching in maths. For example, according to De Villiers (1998): The construction of definitions (defining) is a mathematical activity of no less importance than other processes such as solving problems, making conjectures, generalizing, specializing, proving, etc., and it is therefore strange that it has been neglected in most mathematics teaching; or else, the process of constructive defining described by Freudenthal (1973), which consists of modifying an initial definition, by exclusion, generalization or specialization, replacement or addition of properties. This constructive definition is similar to Lakatos proof-generated definition (1984), which occurs in the dialectic between proof and concept formation, and is of particular interest to us.
The link between definition and proof is also examined in Tanguay (2007), in which the emphasis on learning formal proof goes through a necessary refocusing of the student, from the truth of the contents to the validity of deductive sequences: the student must understand that in formal proof, it is no longer a matter of his producing true statements but rather valid steps of reasoning. Analysis of what is involved in this refocusing (op. cit., p. 4) makes it possible to explain why the student at van Hiele level 2 (see, for example, Crowley, 1987), who comprehends definitions as a litany of properties, has such difficulty understanding formal proof: as long as definitions and properties are glued together in his mind, as long as he isnt able to detach one from the other, as long as they are all either simultaneously true or simultaneously false, the student wont know how to organize these propositions in the proof, to say which intervenes before the others, which results from what and why.
2. The Situation
2.1. Stating the problem
The following three questions were given to students:
Question 1. Describe and define regular polyhedra.
Question 2. Produce them with specific materials.
Question 3. Prove that the previously established list is valid and complete.
2.2. Mathematical analysis of the problem
First, lets look at the definition. There are several possibilities, which from the start raises the question of their equivalence. According to the most standard definition, a regular polyhedron is a convex solid in space, defined by faces which are all congruent to the same regular polygon, and having the same number of adjacent faces at each vertex. This last property can be replaced by either one of properties b), c), and d) below. For a convex polyhedron whose faces are congruent to the same regular polygon, the four statements below are indeed equivalent:
The vertices have the same degree.
All the dihedron angles are congruent.
The polyhedron is inscribable in a sphere.
The group of direct symmetries of the polyhedron acts transitively on the vertices.
The first result one arrives at is that of types of faces possible, in relation to the degree of a vertex. As such, in a convex polyhedron, the sum of the digon angles on the same vertex must be less than 2( (see 2.3.4.). If the faces are the same regular polygon, it is easy to see that the only configurations possible consist of 3, 4, or 5 equilateral triangles, 3 squares or 3 pentagons sharing the same vertex. But even having established that, it is not easy to demonstrate the equivalence between properties a), b), c) and d) above, for a convex polyhedron whose faces are an identical regular polygon. For example, Hartshorne (2000, Chap. 8) constructs Platos five solids (each of the above-mentioned configurations at the vertices leads to one of the solids) and shows that they verify the four properties. He then shows that a convex polyhedron whose faces are the same regular polygon and which moreover verifies a), is necessarily similar to one of the constructed solids.
For solids whose vertices are of degree 3, proof remains relatively basic: the configuration three polygons that share the same vertex is sufficiently rigid for the three digon angles to uniquely determine the three dihedral angles (Hartshorne, 2000, lemma 44.5). As such, three equilateral triangles, three squares or three pentagons sharing the same vertex determine the dihedral angles of, respectively, the tetrahedron, the cube and the dodecahedron. Given that a convex polyhedron the faces of which are the same regular polygon and whose vertices are of degree 3, we map a vertex and the three faces which are adjacent to it onto one vertex and, accordingly, three adjacent faces of the tetrahedron, the cube or the dodecahedron. We then extend the mapping to the configurations sharing those vertices which are at the other end of the already placed edges, and then to the entire solid, going from one vertex to another.
However, for the degree 4 and 5 vertices, this rigidity is lost, which one soon realizes by placing four (respectively, five) equilateral cardboard triangles around the same vertex: one can vary the dihedral angles without changing either the length of the sides or the digon angles. Thus, the preceding argument no longer works. Hartshorne (2000, theorem 44.4) demonstrates that a convex polyhedron on which four (respectively, five) equilateral triangles share each vertex is necessarily similar to the octahedron (respectively to the icosahedron). In doing so, he draws on relatively complex technical findings, in particular Cauchys Rigidity Theorem (op. cit., theorem 45.5), which we wont state here.
2.3. Didactical a priori analysis
2.3.1. The definition
The condition that all faces must be identical is, without a doubt, the regularity property which spontaneously comes to the fore. The next question deals with the type of admissible faces, which refers to the concept of regularity in relation to polygons. There are thus basically three criteria to consider: the congruence of sides, the congruence of angles and convexity. The combined criteria of congruence of sides and convexity are not sufficient because one must take into consideration that a non-square rhombus, for example, isnt regular. But the first two criteria are sufficient. Generally, when taught, the fact that these first two criteria imply convexity (because there cannot then be a reflex angle) is not even mentioned.
One might expect students to consider the criterion the same regular polygon for the faces to be sufficient. One possible reason is that non convex polyhedra or polyhedra whose vertices are not all of the same degree are not specifically studied as objects of learning at the primary and secondary levels. One can indeed easily construct convex polyhedra all faces of which are equilateral triangles and whose vertices are not all of the same degree: glue together along their pentagonal bases two pyramids, or even more simply, use two tetrahedra. The analogous construction with two square-based pyramids results in the octahedron, which verifies all the regularity criteria. Another family of counterexamples can be obtained from each of Platos solids by gluing a pyramid on each face. One can thus construct non-convex star polyhedra, all faces of which are congruent equilateral triangles. Finally, there are four non-convex polyhedra whose vertices have the same degree: these are known as Kepler-Poinsot solids (e.g. Bouvier et al., 2001). It is surprising that for certain textbook authors, in spite of this abundance of quasi-regular polyhedra, the convexity or equality of degrees is taken for granted and is implicit when referring to regularity.
2.3.2. The constructions
Question 2 deals with the construction of Platos five solids. Construction starting from vertices and edges seems to be the most effective at stimulating thinking about the relationships between the degree of the vertices, interior angles of the faces, dihedron angles and general organization of the constituent elements. Materials provided for this are plasticine and toothpicks, or magnetic balls and metal sticks. This stage should make it possible for students to produce several (all?) of the five regular polyhedra, to study their characteristics (numbers of faces, of vertices, etc.) and to formulate a conjecture about how many there are. The next step is to prove that the constructed polyhedra are truly valid (that is, they conform to the definition, without distortions), and that no others exist.
2.3.3. Validating the constructed solids
To validate whether a constructed polyhedron is truly regular, one can expect to make use of the definition in which the equality of degrees occurs at vertices. For polyhedra that can be constructed with triangular faces, confirmation of the accuracy of this definition may be convincing and accepted as (non formal) proof, because of the rigidity of the solid produced. However, for polyhedra with non-triangular faces, one might be led to think that other elements would be necessary, because one might question whether or not the faces produced are the result of distortion. One might even suppose that false regular polyhedra, whether imagined or actually constructed, would be accepted as valid. Construction of a physical object is not proof of its mathematical existence.
2.3.4. Proof that only five regular polyhedra exist
First, one needs to be convinced that there must be at least three faces sharing each vertex: with only two faces, there is no way to add to these two faces (unless you have more than two!) to create a solid. Next, one must establish that the sum of the digon angles sharing a vertex be less than 2(. If this sum equals 2( which would be the case if, for example, three regular hexagons shared each vertex , then the polygon used for the faces tessallates the plane and one can not enclose a volume. If this sum is superior to 2(, then the polyhedron opens towards the outside in the vicinity of the vertex and the polyhedron cannot be convex. These three statements constitute the kernel of proof: next, it is sufficient to consider the finite number of possible situations, having ascertained that by virtue of these three statements, one can not make a regular polyhedron with polygons having six or more sides, nor can more than five equilateral triangles (and even more so, more than five squares or pentagons) share a vertex.
The problem is that the arguments which we have just given to justify the first three statements could only convince someone who already has a good idea of the way in which faces are organized around a given vertex. There is a real risk of these arguments appearing to be accepted on faith, which a teacher might attempt to impose on students whose spatial sense is uncertain. For example, some individuals will respond that if the polyhedron opens towards the exterior at each vertex, it might eventually fold up on part of this exterior, at which point what we believed to be the exterior was in fact the interior! It is difficult to find arguments to counter such objections other than to say that it doesnt make sense! and we have difficulty seeing how to make a more formal justification for the three statements without conducting arguments on an entirely different level, such as the ones given in Hartshorne (2000), already cited.
A relatively accessible formal proof exists, different than the one previously described; in particular, the sum of the measures of the digon angles at the vertex does not enter into it. This proof is supported by studying the so-called Schlegels diagrams (representations by planar graphs of convex polyhedra). The steps of the proof are the following. By using the property that any regular polyhedron is inscribable in a sphere, we establish that there is representation by planar graph which is unique up to isomorphism, produced by stereographic projection on a plane tangential to the sphere, with a tangent point that is none of the polyhedrons vertices. The vertices and edges of the graph thus correspond to those of the polyhedron (two vertices are connected by an edge in the graph if and only if they are connected by an edge in the polyhedron). The next step of the proof consists in establishing the Euler formula for planar graphs: F+VE=2 (where F is the number of faces, V the number of vertices and E the number of edges in the graph), and in deducing from it that any regular polyhedron verifies the following relationship: If p is the number of sides of each face and q the number of edges at each vertex, then 2p + 2q pq ( 0. Resolution of this inequation in the integers results in only five possible pairs (p,q), and these pairs give rise to the previously constructed regular polyhedra.
3. Experimentation
The situation was explored in an experiment with students in the third year of a four-year teacher-training programme at UQAM, who are studying to become high school math teachers, and with pre-service math teachers in their third year of the Licence de mathmatiques at the Universit de Grenoble. The situation was described in a document given to the students. The researchers presenting this article taught the courses which included these sessions. Both were present at the UQAM session, while in Grenoble, only the researcher who gives the course there was present. In Grenoble, the experiment took place in one three-hour session, while in Montreal it took place in two sessions, one of three-hours and the second, a one and a half hour session two days later. This provided an opportunity to introduce these students to didactical problems in spatial geometry, of which they had little previous experience. The students worked in teams of three or four. At UQAM, two teams were filmed. We were able to collect working notes from teams at both universities.
The three questions given above were specified to ensure good devolution. Thus, Question 1 (Characterize and define regular polyhedra) was transmitted in a manner that would lead to a process of definition in several stages. Question 2 (Produce them with specific materials) was broken down into sub-questions, to ensure analysis of the constructed objects and to draw the students attention to those characteristics which would apply in Question 3 (Prove that the established list is valid and complete).
4. Analysis of the productions and findings
4.1. Definitions
Since we had not prepared materials for this stage, to define regularity the students spontaneously tried to construct shapes, using their usual materials (rulers, pens, pieces of paper), their aim being to have their own representations, and to agree on the characteristics to be retained. One student who was filmed noted, We dont have anything to manipulate! and asked the teacher if materials would be supplied.
The general definition that resulted was very broad: a polyhedron is a closed figure in space (whose edge is( made of polygons, and it is regular if its faces are all the same regular polygon. The convexity of the polygonal faces appears to be included in the students concept of regular polygon, though one team had its doubts. A student on the team asked the teacher if, by definition, a regular polygon is necessarily convex. On a piece of paper, she drew an octagon on which she changed two sides, folding them towards the interior, thus creating two reflex angles. It appears that the team members lost sight of the fact that a regular polygon also has congruent angles. The teacher intervened, setting them straight.
Initially, the polyhedrons convexity, congruence of dihedral angles and the equal number of edges (or the number of faces) at each vertex were missing from the proposed definitions. The criterion of resemblance to the sphere was raised but was vague, and sometimes mistakenly associated with the number of sides of the polygonal face: The more sides it has, the more it looks like a ball. We observed among the members of more than one team and most clearly from the comments made by one of the filmed teams the spontaneously occurring idea that regular polyhedra constitute an infinite family, with one polyhedron per type of polygon (with 3, 4, 5, 6, 7, sides) for the faces, with the number of faces increasing simultaneously along with the number of sides of these faces, and a resulting polyhedron closer and closer to the sphere. This idea led the filmed team to look for a formula which would connect the number of edges to the number of faces and number of sides for these faces. They did this by extrapolating from the tetrahedron and the cube which, for the time being, were the only two regular polyhedra at their disposal. Indeed, the team found and wrote the following:
N of edges: (N of faces ( N of sides per face)/2.
It is striking to note to what degree these students have difficulty conceptualizing the dihedral angle. We indeed observe a lingering confusion between the internal angles of the faces (the digon angles) and the dihedral angles. Evidence of this is the following exchange, which took place when one team was reviewing the criteria retained for writing up a definition. The three team-members seemed to vaguely sense the necessity of adding a condition concerning the edges, which in fact emerges when angles are considered. This will lead the students to look for a formula which would allow them to calculate the number of edges (see above), but which they wont retain as a part of the definition.
Student B
Is there anything else? (to add as elements to the definition: the three team-members selected closed, all faces are the same regular polygon and the more sides there are, the more it looks like a ball.(
Student A (summing up(
All the faces , no, no I mean, were talking about faces, lets not even discuss about angles because, because its, its the polygons that form the angles. (Authors emphasis. It seems then that Student C expresses her doubts, in a few words which we cant quite hear. A answers:( Well, the angles inside (Theres also( The edges
Student B
Thats what I was going to say, the edges, you have to check the number of edges.
Student A
Im not sure how, well, theres something there, ... to calculate
Student C
No, thats not a definition, thats not part of the definition (referring to the fact that a formula for calculating edges should not be part of the definition(.
In the filmed lesson, the question of knowing whether other criteria should be added would be raised again by the teacher-researchers, initially in the form of questions: Does this ensure it (the polyhedron( enough regularity? and Can I deform the cube without deforming the faces?). Then come the hints; for example, the teacher takes a piece of paper which represents two faces and varies the angle, saying, At first glance, the angle between the two faces seems independent of the (digon( angles. But for the students this is not sufficient to draw a clear distinction between the dihedral angles and the digon angles. Finally, the teacher-researchers give counterexamples: the star polyhedron formed by gluing square-based pyramids on the faces of a cube, the polyhedra formed by gluing two tetrahedra or more generally, two pyramids along their congruent bases. Mention of the star polyhedron elicited the following thoughts from the filmed team quoted earlier: I dont get the impression that the angles on all the faces are the same, The angle between the faces and the angle between the sides, its not all that connected, There are reflex angles, its not convex anymore. Then the team-members added the criterion No reflex angle to their definition. Finally, the definition that was discussed in the larger group and selected for the next stage would be the same as the one which we had hoped to arrive at, and the question of the dihedral angle would be discussed and resolved.
4.2. Constructions
4.2.1. A conjecture: the only faces that work are those that tessellate the plane
The reasoning according to which the sum of the digon angles sharing the same vertex cannot be equal to 2( would not only need time to gain acceptance by the students or even to be seen as convincing, once explained, but clearly what is most surprising is that the inverse conjecture was specifically formulated by several teams. One might suppose that this arose from the fact that the most familiar regular polyhedra (the cube, the tetrahedron and the octahedron) verify this inverse conjecture. As a result, the pentagon was rejected (by two groups, one in France and one in Quebec). The students in one group were so convinced that a regular polyhedron cannot be constructed with pentagons that they took a dodecahedron from a neighbouring group to find out whats wrong!
4.2.2. A polyhedron with hexagonal faces
To the contrary, in each of the experiments, the hexagon was the object of a repeated number of trials, even among teams which didnt put forward the previously mentioned conjecture (4.2.1). Here the conviction emerged that the hexagon is possible but would require many faces and that the resulting polyhedron would be close to a sphere (4.1). The observation that construction is difficult is attributed, rather, to the inadequacy of the materials, or to the supposedly (too) large size of the polyhedron being sought. Thus it appears that the transition, from 360 of angles covered by three hexagons sharing the same vertex, to the impossibility of constructing the polyhedron, is not at all self-evident. In fact, during the group discussion period, it takes a long time for the argument to gain acceptance. Eventually however, the question is resolved, though several students remain perplexed.
4.2.3. A non regular octahedron
The discussion regarding the necessity of adding equality of degrees at each vertex to the regularity criteria a discussion which took place following presentation of the family of counterexamples produced by gluing two congruent pyramids together is perhaps the source of the following observation, one that came as a total surprise to the researchers: several teams did in fact build the octahedron but then rejected it, claiming that it was not a regular polyhedron. Some students said that it had a square face as well as triangular faces (they hadnt noticed that it disappeared). What was suggested most often was that its dihedral angles are not congruent, a perception most likely created by the fact that all its vertices would not have the same status, and that the polyhedron would not be perfectly symmetrical, which might be suggested by the material construction.
4.2.4. Conjectures on the degree of the vertices
Two contradictory conjectures arose concerning the possible number of adjacent faces at each vertex. One group suggested that the more triangles you add to a vertex, the more it (the polyhedron( resembles a sphere, while another group, using the cube and the tetrahedron to back them up, was, on the contrary, convinced that you cant have more than three faces (at each vertex( because this is a 3 dimensional space. These two conjectures were quickly disproved by the same groups that had proposed them, the first by the argument that the sum of the angles of the adjacent polygonal faces on one vertex can not be greater than 360 degrees, the second by exhibiting a constructed regular polyhedron, different from the tetrahedron and the cube.
4.2.5. Results
The collective summing up which followed the construction phase made it possible to sort out the false conjectures and eliminate inappropriate polyhedra, particularly the one composed of hexagonal faces (according to the argument that regular hexagons tessallate the plane) and in the end, to keep five polyhedra: Platos polyhedra. However, there were lingering doubts concerning the validity of those polyhedra which could be deformed and the eventual existence of other regular polyhedra which werent found. For these reasons, of real interest is the following phase, which consists of establishing a more formal proof using Schlegels diagrams.
4.3. Proof using Schegels diagrams
In both experiments we still managed to finish the proof, in spite of there not being enough time allotted to this phase and the teacher having to explain the final results. The construction of Schlegels diagrams associated with Platos five polyhedra did not present any real difficulty, though there was some uncertainty that led to inaccurate drawings, which were quickly corrected. The groups representations of the same polyhedron were at times very different, a situation which led to clarification of the notion of graphs being isomorphic, and of the uniqueness of representation by planar graph.
One difficulty the students encountered was in accepting that Schlegels diagrams, which dont account for the angles and lengths of edges of represented polyhedra, can be useful in proving findings on regular polyhedra: how can a model which does not display the regularity characteristics of an object be pertinent in proving a finding which concerns its regularity? Furthermore, the resolution in ( of the inequation 2p + 2q pq ( 0 posed a problem, probably because it was unusual and not attached to a familiar procedure.
Finally, it seems that the complexity of the proofs procedure (constructing Schlegels diagrams, studying the Euler formula, looking for solutions of an inequation in the integers, characterizing a polyhedron by pair (p, q)) prevented certain students from recognizing that, as such, they really had proved the result in question: Ive lost sight of it, I no longer know what were trying to demonstrate, Why will what were doing now make it possible to demonstrate that we only have five regular polyhedra?. However, those students who were able to follow the logic of the reasoning right to its conclusion experienced real intellectual satisfaction when confronted with the novelty and power of this proof.
5. Conclusion
On the whole, this situation seems to be consistent as regards knowledge of regular polyhedra, but also in that it questions the relationships between spatial and planar objects, as well as diverse representations of these objects. However, the validation phase still needs clarification. In particular, the definition and construction phases clearly do not lead to the study of Schlegels diagrams; rather, this appears to be a new problem, where it is a priori not at all clear how it might resolve the question of validating Platos polyhedra. Nevertheless, this proof brings a theoretical perspective to regular polyhedra which we evaluate as fundamental to moving beyond the findings linked to their physical construction.
BIBLIOGRAPHY
Bloch, I., Kientega, G. & Tanguay, D. (2006). Synthse du Thme 6: Transition secondaire/post-secondaire et enseignement des mathmatiques dans le postsecondaire. To appear in Actes du Colloque EMF 2006. Universit de Sherbrooke.
Crowley, M. L. 1987. The Van Hiele Model of the Development of Geometric Thought. In Learning and Teaching Geometry, K-12, 1987 Yearbook of the National Council of Teachers of Mathematics (NCTM), pp. 1-16.
Dorier, J.-L., Harel, G., Hillel, J., Rogalski, M., Robinet, J., Robert, A. & Sierpinska, A. 1997. Lenseignement de lalgbre linaire en question. Coord. by J.-L. Dorier. La Pense Sauvage. Grenoble, France.
Godot, K. & Grenier, D. 2004. Research Situations for teaching: a modelization proposal and examples, Proceedings of the 10th International Congress for Mathematics Education, ICME 10, Copenhague.
Grenier D. 2006. Des problmes de recherche pour l'apprentissage de la modlisation et de la preuve en mathmatique. To appear in Actes du colloque de l'Association Mathmatique du Qubec (AMQ), Sherbrooke, june 2006.
Grenier D. 2001. Learning proof and modeling. Inventory of fixtures and new problems. Proceedings of the 9th International Congress for Mathematics Education, ICME 9, Tokyo.
Grenier, D. & Payan, C. 1998. Spcificits de la preuve et de la modlisation en mathmatiques discrtes. Recherches en didactiques des mathmatiques, Vol. 18, n1, pp.59-99.
Hartshorne, R. 2000. Geometry: Euclid and beyond. Coll. Undergraduate Texts in Mathematics. Springer, New-York.
Lakatos, I. 1984. Preuves et rfutations. Trad. N. Balacheff et J.-M. Laborde. ditions Hermann, coll. Actualits scientifiques et industrielles. Paris.
Sierpinska, A., Dreyfus, T. & Hillel, J. 1999. Evaluation of a Teaching Design in Linear Algebra: the Case of Linear Transformations. Recherches en didactiques des mathmatiques, Vol. 19, n1, pp. 7-40.
Tanguay, D. 2007. Learning Proof: from Truth towards Validity. Proceedings of the Xth Conference on Research in Undergraduate Mathematics Education (RUME), San Diego State University, San Diego, Californie. On the Web, 15 pages.
Tanguay, D. 2004. La formule dEuler. Revue Envol. First part in n129, october-november-december 2004, pp. 11-18. Second part in n130, january-february-march 2005, pp. 11-14.
Ouvrier-Buffet, C. 2006. Exploring Mathematical Definition Construction Processes. Educational Studies in Mathematics, Vol. 63, n3, pp. 259-282.
The increasing use of dynamic geometry software in the classroom can lead students to question the necessity of proof, at least in the field of 2D and 3D geometry.
For example, the result If a straight line d is orthogonal to two secant lines included in a plane P, then d is orthogonal to any other straight line in P is given as the fundamental theorem in classe de seconde (France, aged 16-17), but is not even so much as demonstrated.
Quoted in Ouvrier-Buffet, 2006.
Typically, when asked to provide a definition, a student at van Hiele level 2 tends to name all the properties he knows pertaining to the object to be defined. For example, he will say that an isosceles triangle is a triangle with two congruent sides and two congruent angles, an axis of symmetry, a median, angle bisector, altitude and perpendicular bisector joined on this axis of symmetry.
The number of faces (or equivalently the number of edges) adjacent at a given vertex is called the degree of this vertex.
Two adjacent faces form a dihedral angle, which is the angle whose vertex is on the common edge and whose sides are perpendicular to this edge, and are included respectively in the plane of each of the adjacent faces.
By digon angle, we mean the angle formed by two adjacent edges included in a face. This terminology is not standard.
Except of course for the cube which, although not rigid, will be accepted as a regular polyhedron without any hesitation whatsoever.
The pair (p, q) is referred to as the Schlfli symbol of the regular polyhedron.
The students use the word closed but it doesnt have as precise a meaning for them as it has in topology. Here, they use closed in the sense of which encloses a finite volume.
According to our observations, none of the teams seemed to question whether the word polyhedron designated a solid in space or its boundary, of dimension 2. In this regard, there is a vagueness in their speech with which they seem quite at ease.
Indeed, regular polygons are included in the material covered in their geometry courses.
Nevertheless, all the students we observed had already taken a course in Linear Algebra, in which the dihedral angle between two secant planes is defined and calculated from the plane equations.
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