Boero P., Garuti R. (1994) Approaching rational geometry: from physical relationships to conditional statements. PME XVII Tsukuba, Japan.
	Abstract Reflections on some historical and epistemological aspects of the statements of theorems in geometry suggested a teaching experiment with students in grade VII, concerning the production of geometry statements and the comparison between the statements produced and the statements contained in the text-books. An analysis of the students' papers proves that through such activities, in an adequate educational context, they are able to approach geometry statements constructively.
© P. Boero & R. Garutti

1. Introduction Research concerning the didactical and cognitive problems about approaching theorems and proofs has expanded especially on the proofs (see References); particularly, few works concerning the construction and analysis of the statements of rational geometry have been published. This report concerns an exploratory study on the students' early approach to the statements of theorems in geometry through the production of statements during suitable tasks, and the comparison between the statements produced and the statements in the text-books. Our hypothesis is as follows: in an adequate educational context students from 11-12 years of age should be able to perform these activities and constructively approach "rational geometry" (i)). Our research developed from a reflection on some historical and epistemological aspects of the approach to geometry statements (synthetically mentioned in § 2); from this we derived the guidelines of a teaching experiment concerning the approach to geometry statements in grade VII (ii)) (see § 3). The aspects considered in §2 also suggested some keys suitable to analyze students' papers (see § 4). 2. Historical and epistemological aspects concerning geometry statements The statements of theorems in "rational geometry" are general (each statement expresses the properties of a class of figures - all rectangular triangles, all circles..., not of a particular figure), abstract (the statements concern geometrical figures and not concrete objects), conditional (statements are logically articulated in the form: "in the hypothesis that ...it is true that ..." ).We may also observe that (also depending on their content) statements may be expressed in a procedural way (especially when they concern how to perform a geometric construction or to work out a measure: see Heron's Theorem) or in a relational way (as many theorems in Euclid's "Elements"). From a historical point of view, Euclid's "Elements" have been the first systematic organization of a hypothetical-deductive kind, based on the "evident" properties (axioms) of the elementary geometrical entities, of geometrical knowledge expressed in the form of general, abstract and conditional statements. Historians consider it a controversial problem how in the Greek culture the complex transition was achieved from the particular methods used for particular concrete problems, to the "Thales statements" as general properties of abstract geometrical figures, and from these to the "Euclidean statements" as statements of a conditional kind which are true since they can be proved (in them the hypotheses are conditions under which a thesis can be proved). Indeed the oldest geometrical propositions of the times of Thales probably were statements on the truth (based on evidence) of some general properties of abstract geometrical figures not requiring any proof based on other more elementary properties (Szabo,1961). Surely at the time of Aristotle the conditional (as well as abstract and general ) nature of the geometry statements was known (Heath, 1956). Serres (1992) assumes that the investigation concerning peculiar contexts (such as that of sun shadows), in a suitable cultural environment, might have suggested the transition to abstract geometrical shapes. Szabo (1961) relates about different hypotheses interpreting the transition from Thales to Euclid, and points out their possibile complementarity. He refers to ideas of Kolmogorov (concerning the development of mathematical proof in a socio-cultural context which was very advanced for the time), of van der Waerden (concerning the need - felt by the Greeks - to build general statements in order to overcome the limitations and contradictions existing between different practical methods derived from the Egyptians or the Babylonians), of von Fritz (who analyses the relationships between the development of Aristotelian logic and the structure of geometry statements and mathematical proof). And to its own ideas (concerning the implications of the development of the early Greek philosophical thinking - especially the Eleatic School, on the development of geometrical thinking towards abstraction, generalization and logical deduction). Arsac (1961) discusses Szabo's position, thus showing possible interactions between internal motivations of the synchronic development of the structure of geometrical statements and mathematical proof (depending on crucial problems, such as that of incommensurability between the side and the diagonal of the square), and external cultural development (especially concerning philosophy, starting from the Eleatic School). Historical and epistemological research may offer interesting suggestions to plan didactical activities aimed at making the transition to statements of "rational geometry" accessible to students, starting from their working experience in particular and concrete geometrical situations. This is in our view an open didactical problem, not so far exhaustively and globally investigated in the research on mathematical education. Particularly, in the few experimental works to be found in literature on the costruction of geometrical statements by younger students, we can see that they are only concerned with the transition from particular cases to general properties (recently, by also exploring geometrical situations realized through suitable softwares, such as Cabri (Laborde & Laborde, 1992) or Geometric Supposer (Yerushalmy & Chazan, 1990)), or from concrete to abstract situations (Gerdes, 1988), while the problem of the construction of "conditional" statements is virtually overlooked. 3. Planning the teaching experiment Based on the analysis briefly explained in the previous paragraph, we have identified a preliminary didactic context and two subsequent tasks which seemed adequate to effectively and rapidly achieve a direct constructive approach of the students to the aspects of the statements of the theorems of rational geometry that we deem essential. The educational context has been the study of the phenomenon of the shadows of the sun (see Garuti & Boero,1992 for the motivations and the first part of the didactic itinerary); in this context the students of the two classes we will consider in this report had gradually come to solve problems concerning the determination of the heights of objects (inaccessible by direct measuring), by using the proportionality existing between the heights of the objects projecting the shadows, and the lengths of the projected shadows. To introduce the didactic activities forming the object of this report, two ancient versions (iii) of the "Thales and the height of the pyramids" anecdote were told, then compared and discussed. In this context, the notion of "triangle similarity" was introduced under the double aspect of "preservation of the ratios between the sides" and "preservation of the angles". The first task proposed was: "Imagine you are Thales, and write what he might have written down in his testament in order to explain his findings to posterity. If you can, write a general statement". From the point of view of motivation, this task could be expected to stimulate a number of students to reconstruct the meaning of the work done. It was also assumed that the task would stimulate some students to produce statements with the typical attributes of the statements of rational geometry, to be used in the following comparison and discussion activities. Indeed the students had initially faced the problem of the height of a lamp-post and, subsequently that of a tower; the Thales anecdote was concerned with the height of a pyramid ... It was therefore assumed that some students would feel uneasy (in writing down their testament) about the many different examples of "concrete" problems, and that the progress to generality and abstraction of the statements relating to geometrical figures might be for them an adequate and practicable answer to the need to overcome their embarassment. It was also assumed that in the transition to generality and abstraction of the statements relating to the geometrical figures, some students might feel the need to underline the parallelism of straight lines as a "hypothesis". In fact, in facing the "concrete" problems met in the "field of experience" of sun shadows many students had become convinced that the proportionality between the heights of the objects and the lengths of the shadows projected depended on the parallelism of the sun rays, and therefore might consider this parallelism as the condition of the ratio invariance. At this point, we deemed impossible a spontaneous evolution of all the students towards statements of rational geometry (due to a lack of reference models). This was the motivation for the second task, proposed a few days after the "testament" had been produced, which reads as follows: "Try to establish which of the following four statements resembles yours, and explain why". In this task, the students are offered four different formulations of Thales's statement (see Annex) taken from high school text-books.The statements have been selected according to different criteria: an analysis of the texts produced by the students, so that each one of them may identify with an "official" statement; statement content; significance in relation to the strategies adopted by the students during the work on proportional reasoning; diffusion in the text-books. Drawings were modified in comparison with those usually found in books so that the students may perceive analogies with the shadows phenomenon without however representing this phenomenon explicitly. As to the objectives of the second task, we can remark that the request to recognize analogies and differences between a student's own statement and one of the four proposed, calls upon the distinctive attributes of a geometry statement (see § 2) and can therefore represent an important step in the process of becoming aware of them. As for the methods of observation, in planning the teaching experiment we considered that the two classes (which had been working two years with the same Maths and Science teacher) were by then used to writing down their reasoning extensively and accurately. It could therefore be assumed that the verbal records produced during the two tasks would supply enough material for our study. 4. Analysis of the students' behaviour (see TABLE I, annexed) 4.1. Statement production The texts produced by the students can be analyzed according to different keys, linked to the analysis carried out in the second paragraph. I) Particular-General The text types identified are listed below: a) some students stick to a particular case (Pc), for instance the case of the equinox (iv) or to another numerical example; b) others generalize (G) by stating general properties of the sun rays and/or generally describe the process adopted for which no examples are given; c) others still have mixed behaviours (Pc/G) because they go from a particular case to a general statement or else state the general aspects first, and then apply them to a particular example. The G or Pc/G texts are prevailing perhaps because, beyond the explicit request in this sense, the production of a testament shifts the attention from the geometrical representation of a particular phenomenon to the construction of a text concerning general properties. However, we can see that it is not easy to make a distinction between the "particular" as an example to express a general property, and the "particular" as a failure to have a more general view of the property in question (see later on: examples concerning the "testaments" of students [1] and [27]. II) Concrete-abstract The behaviours are of a different kind: a) some students stick to the phenomenon they have observed, and the elements they describe are concrete objects (rays, sticks, pyramids, ...) (C); b)others explicitly state the properties of abstract geometric entities(parallel straight lines,triangles..)(A) c) finally, some students begin by mentioning concrete elements, but later in the discussion they gradually go on to talk of geometrical entities (C&A). According to this key we can see how the majority of the students are still tied to the physical phenomenon observed. This can be explained on the one hand, by the relevance of the geometrization work performed in the previous year, and on the other hand, by the fact that the transition to abstraction possibly requires a further didactical mediation to go from sun rays to parallel straight lines, from the cross section "drawing of shadows/objects/rays" to similar triangles. III) Conditional nature of the statements About the mechanism determining the transition from "parallelism of the sun rays as a cause of proportionality" to "parallelism of the straight lines as a condition of proportionality" it can be said, in all four cases ([1],[2],[3],[18]) where this transition spontaneously occurs (IF), that this might imply some degree of awareness that the straight lines (unlike the sun rays) may not be parallel, and the conviction that parallelism is the property of the rays granting proportionality between objects and their shadows. IV) Procedural-Relational The students' behaviours are essentially of three kinds: a) Procedural (P), when the students describe only the measuring process of an object of height unkown by using its shadow; b) Relational (R) when the students detect the parallelism as a cause (or a condition) for the ratio invariance and relate the elements of the phenomenon (parallel rays, same angular height,...) or of the geometrical figure without however caring to describe the process to work out the measure; c) Procedural-Relational (P&R): when the students describe both the process to reckon the height unknown and the "causes" (or "conditions") for the ratio invariance. According to this key, we can see that the students are almost equally distributed in the three groups: the procedural statements probably reflect a will to make explicit (in a more or less general way) the resolutive methods concerning the problem of inaccessible height constructed during the previous year's activities; while the relational statements probably correspond to an idea of mathematics developed by a number of students of this age, due to their school experience about mathematics (as a "study of the properties of mathematical entities"). We can see how the distinction between the two types of statements does not seem to be cognitively relevant here (both types of statements are proposed by students of any level). Let us now analyze some of the papers to see how the aspects we have so far observed have mixed in reality; examples such as these strengthen our conviction that, in the particular situation of statement production we are considering, the attributes "general" and "relational" or "abstract" do not indicate any superior cognitive performance. a) [27]: he only considers the equinox case and concrete objects (Pc-C-P): « I have considered a pyramid, I have measured its shadow and knowing that it was the equinox day, I have found the height of the pyramid. The ratio existing between the pole and the shadow is one » b) [ 24]: his text is characterized by a high degree of generalization both in the description of the procedure and in the drawing illustrating it . In this drawing we may detect an approach to abstraction (G-C&A-P&R). c) [ 33]: he builds the text starting from the case of the equinox and goes on to the general case by linking the ratio invariance with the parallelism of the sun rays. He therefore completes the picture by giving a numeric example where he makes the procedure explicit. In this general statement, he states a general property of the "parallel rays". This text is unique for its complexity and the dialectic existing among the various aspects (Pc&G-C-P&R). « I, Thales, [....] have discovered [...] : given that the sun rays are parallel, I have worked out the height of the pyramid [...] I have used another object, i.e. a short pole [...] In terms of proportions the stick was as long as its shadow, and the pyramid was as high as its shadow and so [...] I then saw that even if the shadows were not exactly as long as the objects [...] Therefore, if the stick is contained in its shadow a certain number of times, then another object is also contained in its shadow the same number of times ... [numerical ex.]. Since the rays are parallel, if a shadow is contained in another a certain number of times, hen a stick is also contained in another the same number of times ». d) [1]: his text opens with the equinox example, then goes on to a different numeric example. This student does not express any relation or generalization on these two cases. Afterwards, he "jumps" to a purely geometrical statement. In this text, the various aspects do not flow together at an explicit level, and there is a sudden progress to the "theorem" (Pc&G-C&A-IF-P). 4.2. Comparison between produced and official statements: an outline of results. During the recognition process the students should interpret systems of different signs: text, figure and symbols. They try to detect the statement analogue to their own by recognizing analogies in the drawing (DA) or the text (TA), or (also depending on the content of their texts) through a more articulated process of interpretation; in this second case, some students (Row), stimulated by "official" statements, revise their texts making them more general and abstract as well as closer to an official one; others (Ro) transform one of the official statements to bring it closer to their own statement, which therefore becomes a consequence (or a realization) of the official statement. 5. Discussion As is shown in the examples and also in the table, many texts produced during the first and second task are characterized by remarkable variety and complexity, and by the dialectical presence of the different aspects: particular/general, concrete/abstract, procedural/rational; this allows the students, from time to time, to rest on one and/or the other aspect according to what they wish to communicate. The official statements proposed in the second task are not interpreted by a number of students as formal texts, but seem to "speak" to them about shadows, sun rays and objects projecting shadows, and to also evoke the conquest by the students of the proportionality model. In the official statements they thus tend to recognize their own geometrical work experience. The comparison between produced statements and official statements seems suited to fulfill the double function of maintaining the relation between the official statements and the students' geometric experience and introduce them to the cultural system of rational geometry. With these considerations in mind, the teaching experiment suggests a certain optimism as regards the feasibility of the objective to get grade VII students constructively involved in approaching "rational geometry", obviously in an adequate educational context. The teaching experiment also suggests some questions, concerning the teacher's didactical choices, we would like to point out very concisely: the opportunity or not of forcing (by modifying the first task) the production of relational statements; or especially the production of conditional statements; the most appropriate moments to introduce classroom discussions (which might be quite useful to foster the classification process of statements and the detection of the typical characteristics of the statements in rational geometry); the ways to accomplish the transition to proof once the "conditional" nature of the statements has been recognized. A further question concerns the usage of the history of mathematics in mathematics education, apart from the usage that may be made in planning teaching experiments by deriving some suggestions from the hypotheses of historians (as we did). We think it might be useful to clarify the role (that we deem important both on the affective and cognitive ground) explicit references to personalities and intellectual activities so far back in time might have for students. __________________________________ (i) we shall use the expression "rational geometry" as in the Italian syllabi for secondary education ("geometria razionale"), to signify an axiomatic-deductive theoretical organization of the geometrical knowledge, apart from the particular system of axioms chosen. (ii) in Italy, it is usually in grade VII that comprehensive school students get in touch with the first theorems of geometry (generally the theorem of Pythagoras, sometimes the theorem of Thales, more seldom the theorems of Euclid). (iii) "Thales measures the heights of the pyramids by measuring their shadows, after having observed the time when our shadow is the same as our height ..." [Diogenes Laertius] .    "I especially admire you, Thales of Miletus, because, by placing your stick at the end of the shadow of a pyramid, you formed two triangles with the sun rays, and proved that the height of the pyramid is to the length of the stick as the shadow of the pyramid is to the shadow of the stick" [Plutarch]. (iv) they had experienced that in the equinox day at noon (at our latitude) the length of a pole is equal to the length of the projected shadow References Arsac G. (1987) L'origine de la démonstration : essai d'épistémologie didactique. Recherches en didactique des mathématiques 8(3) 267-312. Balacheff N. (1987) Processus de preuves et situations de validation. Educational Studies in Mathematics. 18(2) 147-176. Balacheff N. (1988) Etude des processus de preuve chez des élèves de Collège. Thèse. Grenoble : Université Joseph Fourier. Boero, P., Chiappini, G.P. and Garuti, Rossella (1993) Towards proportional modelling: an exploratory study about context sensitivity (preliminary report), DI.MA., Univ. Genova. Duval R. (1991) Structure du raisonnement déductif et apprentissage de la démonstration. Educational Studies in Mathematics 22(3) 233-261. Garuti R., Boero P. (1992) A sequence of proportionality problems: an exploratory study. Proceedings of PME. XVI , Durhan Gerdes P. (1988) On Culture, Geometrical Thinking and Mathematics Education. Educational Studies in Mathematics Hanna G., Winchester I.(eds.) (1990) Creativity,Thought and Mathematical Proof. Toronto: OISE. Hanna G., Jahnke N. (eds.) (1993) Aspects of Proof. special issue of Educational studies in Mathematics 24(4). Heath T. (1956) The Thirteen Books of Euclid's Elements. New York: Dover. Laborde C., Laborde J.M. (1992) Problem Solving in Geometry. In: J.P.Ponte et al. (ads.) Mathematical Problem Solving and New Information Technologies. Berlin: Springer-Verlag. Mesquita A. L. (1989) Sur une situation d'éveil à la déduction en géométrie. Educational Studies in Mathematics 20, 55-57. Piaget J., Inhelder, B. (19**) De la logique de l'enfant à la logique de l'adolescent. Paris: P.U.F. Serres M. (1993) Les origines de la géométrie. Paris: Flammarion. Szabo 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 Yerushalmy M., Chazan D. (1990) Overcoming Visual Obstacles with the Aid of Supposer. Educational Studies in Mathematics

 

TABLE I : this table summarizes the main characteristics of students' behaviours

Stud.

I

II

III

IV

RC

Stud.

I

II

III

IV

RC

1

PC&D

C&A

IF

P

TA

19

G

C&A

P&R

Row

2

G

A

IF

P

Ro

20

G

C

R

Ro

3

Pc&G

C

IF

P

P

21

Pc&G

C

R

Ro

4

*

*

*

*

*

22

G

C

R

TA

5

Pc&G

C

R

DA

23

G

A

P

TA

6

*

*

*

*

*

24

G

C&A

P&R

*

7

G

C

P

TA

25

Pc&G

C

P&R

Row

8

G

C&A

R

Ro

26

Pc&G

C

P

TA

9

G

C

R

Ro

27

Pc

C

P

DA

10

Pc&G

C

P&R

Row

28

Pc&G

C&A

R

Row

11

G

C

P&R

Row

29

G

C

P

*

12

Pc&G

C

P&R

Row

30

Pc

C

P

Ro

13

*

*

*

*

*

31

Pc&G

C

P&R

DA

14

Pc&G

C&A

R

Ro

32

G

C&A

R

Ro

15

G

C

R

*

33

Pc&G

C

P&R

Ro

16

G

C

P&R

DA

34

Pc&G

C

R

TA

17

*

*

*

*

*

35

Pc&G

C&A

R

Row

18

G

C

IF

P&R

DA

36

Pc&G

C

R

Ro

37

Pc

C&A

P

Ro

Pc=Particular; G=General; C=Concrete; A=Abstract; P=Procedural; R=Relational; IF=Conditional; Rc= Recognition; TA=Text Analogy; DA= Drawing Analogy; Ro= Revision official text; Row=Revision own text; *=absent.

 

ANNEX : official texts of the Theorem of Thales