la objetivación en la enseñanza del teorema de pitágoras

The objectification of the right-angled trianglein the teaching of the Pythagorean Theorem:an empirical investigation

Andreas Moutsios-Rentzos & Panagiotis Spyrou & Alexandra Peteinara

Published online: 13 August 2013# Springer Science+Business Media Dordrecht 2013

Abstract In this paper, we present the design and the results of a teaching experimentcarried out to investigate the hypothesis that it is feasible to facilitate the students’ possibilityfor experiencing the reactivation of the objectification of the right-angled triangle. For thispurpose, a teaching design of the Pythagorean Theorem was developed and taught to anexperimental class of 14-year old students. The results of our teaching were compared with acontrol class with the employment of a questionnaire and semi-structured interviews. Thequantitative and qualitative analyses supported our hypothesis that the students of theexperimental class would develop qualitatively different understandings of the theorem thanthe control class, thus suggesting their possibility for experiencing the reactivation of theobjectification of the right-angled triangle.

Keywords Objectification . Geometry . Husserl . Phenomenology . Pythagorean Theorem .

Pythagorean Triples . Teaching experiment

1 Phenomenology and mathematics education

Mathematics educators have implemented phenomenological ideas in various projects(Brown & Haywood, 2011; Freudenthal, 1983; Radford, 2003). For example, the realisticmathematics perspective draws upon Freudenthal’s didactical phenomenology to proposeteachings in which the learners, guided by the teacher, actively organise a ‘real’ situation withmathematical tools, thus coming to the guided reinvention of a mathematical idea (Gravemeijer,1994). In this study, we concentrate on Husserl’s phenomenology to investigate its pedagogicalcontribution in the teaching of the Pythagorean Theorem.

Husserl (2001) addressed the question of the objectivity of knowledge, attempting toresolve the apparent contradiction between “the subjectivity of knowing and the objectivity

Educ Stud Math (2014) 85:29–51DOI 10.1007/s10649-013-9498-y

A. Moutsios-Rentzos (*)Department of Sciences of Pre-School Education and Educational Design, University of the Aegean,Room 308 (‘Kameiros’ building), 25 Martiou, 85100 Rhodes, Greecee-mail: [email protected]

P. Spyrou : A. PeteinaraDepartment of Mathematics, University of Athens, Athens, Greece

of the content known” (p. 2). Intentionality is central to his phenomenology, referring to “theconscious relationship we have to an object,” arguing that every “intending has its intendedobject” (Sokolowski, 2000, p. 8). The phenomenologically described given object (noema) isnot to be confused with the intentional activity (noesis) and the “objective and the subjective arecorrelative, but never reducible to one another” (Audi, 1999, p. 405). Through his discussionabout bodily and mental experience and communication, Husserl differentiated amongst thesubjective experiences of a phenomenon (signifying its subjectivity), the intersubjective expe-riences of the shared communicated (through language) meaning, and the transcendentalsubjectivity, which “is not merely understood as a possible singular but rather also as a possiblecommunicative subjectivity … through possible intersubjective acts of consciousness, it en-closes together into a possible allness a multiplicity of individual transcendental subjects”(Husserl, 1974, p. 31).

Husserl suggested his own method of phenomenological reduction (epoché). He differ-entiated the natural attitude, “our straightforward involvement of things and the world,”from the phenomenological attitude, “the reflective point of view from which we carry outphilosophical analysis of the intentions exercised in the natural attitude and the objectivecorrelates of these intentions” (Audi, 1999, p. 405). Moreover, explicit thinking was differ-entiated from “the passive, thoughtless repetition of words,” though it may “fall back intopassivity or become ‘sedimented’ as people take it for granted and go on to build furtherthinking upon it” (p. 406). Epoché involves bracketing out the natural attitude and movingtowards a phenomenological attitude by investigating the sedimented intentional history ofthe object. Thus, epoché can be viewed as the transcendental moment in which the naturalattitude is suspended and explicit thinking is activated, thus signifying the reactivation of thesedimented history into the intentional history.

During the objectification of an idea, through oral speech the ideality is freed from itsimmediate linguistic community, while through written language it obtains a status of an“absolute ideal Objectivity” (Derrida, 1989, p. 79). Husserl’s idealities differ from Platonic ideas:in having “a history, they must be related to, i.e., they must be primordially grounded in, theprotidealizations based on the substrate of an actually perceived real world” (p. 45). Instead ofviewing knowledge (new or already known) as an anamnesis of former states of our existence,new knowledge is actively constructed once within history and our subsequent knowing of thealready constructed knowledge requires a process of reactivating of its objectification.

Phenomenological ideas and the notion of objectification as viewed by mathematics educa-tors appear to differ from Husserl’s. For example, though Radford (2003) shares Husserl’s viewthat objectification occurs within the semiotic system employed to describe and to represent thesignified ideality, he explicitly differs in considering objectification process where idealitiesbecome disclosed to consciousness through a sensuous and affective encounter mediated bysocial practices (Radford, 2013). Notwithstanding the differences, mathematics educators havediscussed teaching principles that may facilitate the students’ experiencing the reactivation ofthe objectification of a mathematical idea; for example, the aforementioned notion of ‘guidedreinvention’ or Radford’s (2006) generalisation layers identified in the students’ process ofknowledge objectification. Husserl argues that in the phenomenological analyses “sedimentedthought must be reactivated and its meanings revived” (Audi, 1999, p. 406). Though epoché hasbeen argued to be essentially an ideal and that it is impossible to assume a view from nowhere(see Nagel, 1989; Ricoeur, 1986), we argue that epoché may be appropriately incorporated in adesign aiming to teach geometrical ideas embedded in the world that the students experience.Moreover, echoing ideas from the embodied mathematics research (Lakoff &Núñez, 2000), weposit that the way that the human body experiences the world can act as the foundation uponwhich such a teaching design may be built.

30 A. Moutsios-Rentzos et al.

Building upon previous efforts (Spyrou,Moutsios-Rentzos& Triantafyllou, 2009; Spyrou&Moutsios-Rentzos, 2011) and upon our phenomenological perspective on teaching geometry, inthis paper we discuss a teaching of the Pythagorean Theorem designed to facilitate the students’transformation of the subjective experience of perpendicularity into the objective mathematicalidea of the right-angled triangle:What are the effects of a phenomenologically derived teachingof the Pythagorean Theorem in the students’ possibility for experiencing the reactivation of theobjectification of the right-angled triangle?

2 A phenomenological perspective on teaching geometry

2.1 Objects and objectification in geometry

We consider the geometrical object to be both a construction and an object of discovery. Ahuman conception that, once defined and secured within an axiomatic system, contains all itsconsequences, ‘waiting’ for the mathematician to discover them. Ayer (1971) contended thata “being whose intellect was infinitely powerful would take no interest in logic andmathematics,” as (s)he “would be able to see at a glance everything that his [sic] definitionsimplied” (p. 82). Following these and our phenomenological perspective, we regard thegeometrical object as a transcendental description of space within an (axiomatic) structure.

Lappas and Spyrou (2006) identified two levels of objectification in Euclidean geometry.In the 1st level, the results that stem from the subjective experience are objectified throughtheir numerical representation. Considering the right-angled triangle, the knowledge thatthree wooden sticks form a right-angled triangle is objectified through the numericalexpression ‘a2=b2+c2’ (where a, b and c are the lengths of the sides of a specific right-angled triangle). In the 2nd level, the existing ‘archetypal’ results are incorporated within amathematical theory, through an axiomatic proof. In this level, ‘a2=b2+c2’ is an algebraicexpression that is settled within an axiomatic system as a proved ‘truth.’ Through the twolevels of objectification, the subjective–of experiential origins–conceptions of the right-angled triangle are gradually quantified to be incorporated within an axiomatic system, thusbecoming a non-arbitrary, omnitemporal object.

Duval (2006) notes that the mathematical object is signified by the relationships amongstits various representations, but it is not identified with any of them. Moreover, he stressesthat ‘seeing’ a geometrical object (our intentions when we think about it) enables us tomentally ‘reorganise’ the geometrical object and to choose amongst the variety of itspossible configurations and sub-configurations to deduce a plausible argument. For Duval,the different semiotic representations of a geometrical object (including the figure of a right-angled triangle or the basic expression of the Pythagorean Theorem) as ‘transient phenom-enological objects,’ which can obtain the status of a geometrical object, provided that“attention can focus on some invariant (the assumed represented relations) and not only[for example] on their visual data and their perceptual organization” (p. 129). With respect togeometrical symbolism, he notes that “a ‘geometrical figure’ always associates both discur-sive and visual representations, even if only one of them can be explicitly highlightedaccording to the mathematical activity that is required” (p. 108). Though both the symbolicand the visual register are expected to be equivalent with respect to the correspondingmathematical process, the generalisations that occur during the students’ attempt to obtain adeeper level understanding of the concept may not be concurrent for both registers. Forexample, regarding the visual register, Duval draws our attention to the incongruity betweenthe mathematicians’ (indexical) ‘seeing’ a figure and the (iconic) everyday life ‘seeing.’

The objectification of the right-angled triangle 31

In the following section, Lappas and Spyrou’s analysis and Duval’s views are utilised todiscuss the role of the geometrical figure in the objectification process of a geometrical idea.

2.2 A phenomenological perspective on signs and generalisations in geometry

The construction of a mathematical object requires some kind of generalisation (Harel & Tall,1991; Mitchelmore & White 2000; Morgan, 2006). Radford (1999) synthesised ideas fromVygotsky, Peirce and Husserl to propose a semiotic-cultural approach to investigate objectifi-cation processes. Radford (2003) studied the generalising activities of novice students thinkingabout patterns, differentiating between presymbolic generalisations (including factualgeneralisation and contextual generalisation) and symbolic generalisations. A factual general-isation refers to “a generalisation of actions in the form of an operational scheme,” applying “toobjects of the same concrete level” (p. 47). On the other hand, contextual generalisations are“performed on conceptual, spatial temporal situated objects” (p. 54), generalising “not only thenumerical actions but also the objects of the actions” and dealing with generic objects “within therealm of reasoned discourse” (p. 65). In contrast, in a symbolic generalisation the students are“deprived of indexical and deictic spatial temporal terms,” causing a shift “in the relationbetween the knowing participant and the object of knowledge” (p. 66).

Following our phenomenological perspective, Lappas and Spyrou’s analysis, Duval’sviews and Radford’s layers of generalisations, we propose an objectification scheme aboutthe right-angled triangle, structured around the functions and the generalisations linked withfigural and numerical (arithmetic or algebraic) geometrical signs. We theorise that the relation-ship of figural and numerical geometrical signs evolves through a series of generalisations thatgradually imposes the axiomatic structure of numbers on the figural representation, with thepurpose of constructing an autonomous axiomatic structure of geometry (freed from both thespatiotemporal, sociocultural reference and the imposed numerical axiomatic structure).

At the first level, the figure acts as an iconic sign of a perceptually experienced triangularlyshaped object, while the number acts as an indexical sign of an act of measuring the sides of thatobject. At this level, the figure includes the ‘gestalt’ relationships characterising the triangularshape, whereas the numerical (arithmetic) relationships are linked with the specific triangularshape (both the actual object and its figural representation). Both representations designate thesame physical object, but the numerical relationships require the figural representation to obtaina geometrical meaning. Hence, at this level all generalisations are factual.

At the second level, both the figure and the numerical relationships are indexical signs ofa generic–yet ‘concrete’ for the students’ ‘expanded’ perception (Radford, 2003)–geometricalobject. Hence, at this level, the generalisation functions are contextual.

At the third level, the algebraic relationships still require the figural representation toobtain their geometrical sign status. The algebraic structure is imposed on the figuralstructure, but it is not sufficient on its own to signify the geometrical object, since the lackof the figural representation prevents it from obtaining an unavoidable geometrical signstatus. We argue that the generalisations at the visual registry are symbolic, since the figureincludes the potential of signifying the whole class of the right-angled triangles. In contrast,the generalisations at the numerical registry are not symbolic, due to their reliance on thefigure to obtain their geometrical status. Such a pre-symbolic generalisation, in which asymbolic sign mediates another type of sign in order to necessarily signify a geometricalobject, is identified as mediated generalisation.

At the fourth level, the figural and the algebraic symbolism fuse into a mathematicalstructure (for example, algebraic geometry) that designates the geometrical object withoutrelying on a figural representation and, thus, all the generalisations involved are symbolic.


2.3 Teaching for the possibility for reactivation of objectification in geometry

The main phenomenological idea is summarised in the phrase “back to ‘the things them-selves’” (Husserl, 2001, p. 168). Within a pedagogical design, this does not imply an attemptto re-enact something that happened a long time ago in a radically different socioculturalsetting, but a design within which the ‘natural attitude’ can be suspended, establishing a kindof epoché, which would allow the possibility for the reactivation of objectification. Thus, weintegrate Husserlian ideas within a didactical framework structured around five didacticalprinciples: ‘appropriate generalisations,’ ‘prescientific materials,’ ‘bodily experience,’ ‘com-munication’ and ‘feasibility.’

First, we adopted Radford’s (2003) view that the students’ making a series of appropriategeneralisations is crucial for their experiencing the reactivation of the objectification of aconcept. We argue that an appropriate didactical design, through a series of generalisations,may help the students’ possibility for experiencing the transformation of the subjectiveexperience into an objectified geometrical idea (through the quantified relationships com-pressed in that idea).

Secondly, Husserl claimed that “mathematical abstraction has roots in the prescientificworld” and that the “geometrical forms have their roots in the activity of measuring and inthe idealization of the volumes, surfaces, edges, and intersections we experience in the life-world” (Audi, 1999, p. 407). Though ‘replicating’ history in the classroom is probablyimpossible (Radford, 1997), an ancient idea may be adapted to be “compatible with moderncurricula” (p. 32; see also Arcavi & Isoda, 2007; Fried, 2001; Jankvist, 2009). The theorememerged in various civilisations and inter-cultural similarities can be found in its develop-ment (Maor, 2007). Though the standard teaching at Greek Gymnasio is based on theconcept of area (in line with Euclid’s Proposition I47; Heath, 1956), historically thePythagorean Triples usually appeared as an ‘introductory’ concept to the theorem and the‘area’ was employed as means for introducing the general case. Thus, we decided to utilisein our teaching the figurative numbers, which embody a two-fold representation (numericaland geometrical) of certain numbers. We posit that the embedded link between the tworepresentational systems of figurative numbers may facilitate the students’ focusing on themathematical object. Moreover, the figurative numbers allow the students’ generalisations tobe ‘reconstructed’ and to incorporate through the notion of area the rational (and later the real)numbers. For example, the 4th figurative number that has 42 dots may be later re-viewed as asquare having side 4 and area 42. The figurative numbers constitute a representation that mayfacilitate the convergence of the two registers involved, thus helping the students’ generalisingthe conceptual ‘equivalence’ between the right-angled triangles (with the length of its sidesbeing natural numbers) and the Triples to the wider set of real numbers.

Thirdly, gravity has affected the evolution of the humans’ sensory and neural systems thatidentify or model verticality to maintain their upright posture crucial for survival (Merfeld,Zupan & Peterka, 1999; Noback, Strominger, Demarest & Ruggiero, 2005). Radford (2003)stresses that the construction of knowledge involves nonsemiotic representations including“physical and sensual means of objectification … that give a corporeal and tangible form toknowledge as well” (p. 41). Following these and our assumption that the bodily experienceof gravity is invariant throughout history, we posit that a teaching that builds on thisexperience, linking ‘natural perpendicularity’ (that the horizon and gravity form) with theright-angled triangle, may help the students to physically experience the reactivation of theobjectification of the right-angled triangle (see also ‘embodied mathematics’; Lakoff &Núñez, 2000). Such a teaching reinstates the anthropological and non-arbitrary nature ofthe theorem and facilitates the students’ moving away from their ‘natural attitude’ (including


an environment where the right angle is deeply embedded in its structure), towards ‘goingback to the things themselves.’

Fourth, since objectification relies on the communication (oral or written) of subjectiveexperiences, we foster a learning environment that allows the students to reflect upon theiractions and thoughts, as well as to test, to communicate and to argue about them.

Finally, feasibility is crucial for any pedagogical intervention: our teaching complementsthe existing curriculum (utilising simple materials) to maximise the possibility of itsimplementation in the current school reality.

2.4 Identifying the level of the reactivation of the objectification

We argue that the objectification of the right-angled triangle occurs through a shift of theintentionality whilst experiencing the phenomena and the quantification of qualitativerelationships that identify it as such (including triangle, right angle, largest side oppositeto the right angle). Though the Triples are a way of partially quantifying these relationships,a rule that can be deductively generalised is required for the triangle to obtain higher levelsof objectification. The basic expression of the theorem quantifies the desired relationships onthe premise that is not just a mere arithmetic (or even algebraic) rule they learned; it is anumerical (or algebraic) relationship signifying geometrical relationships. Hence, theaforementioned five principles were incorporated in teaching design (see §3.2.2 below) toallow the students to re-view their experiences of the right-angled triangle and to communicatejudgements that suggest their (possibility for) capturing the whole complexity of the quantifiedgeometrical relationships, compressed in the expression ‘a2=b2+c2.’ Consequently, to identifythe level of the reactivation of the objectification of the right-angled triangle experienced by thestudents, we should investigate both whether their communications about the right-angledtriangle are based on the expression of the theorem or only on qualitative warrants (visual,authoritative or other) and whether this expression quantifies geometrical relationships (thewhole web or fragments of it) or is disjointed from a geometrical meaning.

3 Methods and procedures

3.1 Overview of the design: sample, methods and procedures

The study was conducted with the 2nd graders (14 years old) of two classes of a GreekGymnasio. The experimental class followed our design. The control class was taught oneadditional school hour exercises linked with the theorem, so that both classes would spendthe same time at the theorem. Both classes were taught by the same mathematician and theytook a school test she designed (two problems requiring the application of the PythagoreanTheorem and its inverse) usually given at the end of the standard teaching of the theorem.

The five-phase study was conducted from October 2011 to January 2012 (see Table 1).Phase A (Figurative numbers) included an introduction of the figurative numbers (20/10/2011,see §3.2.1 below). Phase B (Teaching Intervention) included our teaching intervention(06/12/2011, see §3.2.2 below), just before the theorem was introduced to the students(Phase C, Standard teaching, 07/12/2011; see §3.2.3 below). Phase D, Evaluation, included aquestionnaire (19/12/2012, see §3.3.1 below) administered in both classes to compare theresults of our teaching. In Phase E, we conducted Post-teaching interviews (10/01/2012, see§3.3.2 below): videotaped ‘semi-structured’ interviews with three pairs of students of eachclass, according to their School test score (‘high’ – ‘medium’ – ‘low’).


Tab

le1

The

five-phase

stud

ydesign

Aim

Sam

ple

Metho

d

Class

Size

Instruments

Analysis

Teaching

Phase

AFam

iliarity

with

square

figurativ

enumbers

Experim

ental

NPhaseA=20

Per

desk:Worksheet

(§3.2.1)

Qualitative

Figurativenu

mbers

20/10/20

11

Phase

BFacilitatin

gthereactiv

ation

ofobjectificationof

the

right-angled

triangle

Experim

ental

NPhaseB=17

(3stud

entsabsent)

‘Plumbbo

b’3sticks

Glass

bowl

Qualitative

Video

recording

Teaching

Interventio

n(three

sections)

06/12/20

11Per

desk:10

Sticks

Worksheet(§3.2.2)

Phase

CAsdescribedby

the

curriculum

Experim

entalControl

NPhaseC=37

(17+

20)

Asdescribedby

the

curriculum

(§3.2.3)

–

Stand

ardTeaching

07/12/20

11

Evaluation

Phase

DEvaluationof

thelevelof

reactiv

ationof

objectification

inbo

thclasses

Experim

entalControl

NPhaseD=37

(17+

20)

Questionnaire

(§3.4.1)

Schoo

ltest

Quantitativ

e

Evaluation

19/12/20

11

Phase

EQualitativeinvestigation

oftheresults

ofPhase

DExperim

entalControl

NPhaseE=12

(3pairs+3pairs)

(Schooltestscoring

Groups:High,

Medium,Low

)

InterviewsPer

pair:

Worksheet,woo

den

sticks

(§3.4.2)

Qualitative

Video

recording

Post-teaching

interviews

10/01/20

12


3.2 The structure of the teaching design

Our teaching is designed to work with the existing curriculum of 2nd grade of theGreek Gymnasio, rather than being a ‘stand-alone’ intervention. Drawing uponour didactical framework (§2.3) and upon previous research (Spyrou et al. 2009;Spyrou & Moutsios-Rentzos, 2011), the three-phase teaching includes: a) A shortactivity called Figurative numbers (Phase A), b) Our Teaching intervention (PhaseB), and c) The teaching in school according to the curriculum (Standard teaching;Phase C).

3.2.1 Figurative numbers

‘Figurative numbers’ is a less than 10 min, single worksheet activity designed to comple-ment the revision of natural numbers occurring in the beginning of the 2nd Grade. Thus, thefigurative numbers would be introduced to the class community and would become a toolthat the students may choose to utilise in our teaching intervention (scheduled around6 weeks later). The worksheet begins with a short paragraph introducing the figurativenumbers as an alternative representation of some natural numbers. The students are given atable with the numbers ‘4’ and ‘9’ represented as figurative numbers and they are asked tofind the following two figurative numbers and to fill in the table (Fig. 1). The activity endswith the students completing the rule: “In order to find the amount of dots of a squarefigurative number we find the … of dots of its side and then we … .”

Fig. 1 The table included in the ‘Figurative numbers’ worksheet


3.2.2 The teaching intervention

The three-section teaching intervention (Table 2) was designed to last one teaching hour andto precede and complement the current teaching (see §3.2.3). Since the 2nd graders are notexpected to prove the theorem, our intervention was focussed on the students’ possibility forexperiencing the reactivation of the first two levels of our objectification scheme (see §2.2).The students work in groups to communicate their ideas. On each desk, there is a set of tenwooden sticks (coloured with a different colour every 3 cm, representing numbers 1 to 10)and a 4-page worksheet.

In the first section, Natural perpendicularity, we aim to facilitate the students’ re-viewingand re-considering the sedimented names ‘right angle’ and ‘right-angled triangle,’ they usein their everyday life. A glass bowl is placed next to a ‘plumb-bob’ hanging next to the classwall, for the students to notice the angle formed by the surface of the liquid and the string ofthe ‘plumb-bob,’ as an example of natural perpendicularity. Subsequently, we focus onmaking the students aware of the, independent from verticality, link between the Basic Tripleand perpendicularity. For this purpose, they are given three wooden sticks (90 cm, 120 cm,150 cm; coloured differently every 30 cm). Though the students are not told about the lengthof each stick, they are expected to deduce that the sticks consist of respectively 3, 4 and 5equal parts, thus embodying the Basic Triple. We ask two of the students to use the sticks toconstruct a triangle that visually fits with ‘natural perpendicularity’ to link verticality withthe right-angled triangle (3,4,5). Furthermore, they are asked to construct a right-angledtriangle on the floor, thus leading them to the Basic Triple. We move the sticks and we askthe students to deduce which side should be the longest stick for the triangle to fit that angle.

Table 2 The three-section teaching intervention


The rest of the students (in groups) explore and verify these with the sticks on their desks,noting at the same time their conclusions on the worksheets.

In the second section, Pythagorean Triples, the students are asked to suggest more triplesof numbers that may form a right-angled triangle and to investigate their hypotheses with thesticks they have on their desks by visually comparing the triangular shapes of sticks with‘natural perpendicularity’. Next, we suggest the triples (9,12,15) or (5,12,13) that are notimmediately represented by the given sticks, which may help the students to realise thenecessity for changing the representation system and to work with numbers on theworksheet. In the second page of the worksheet, there is a figure that allows the studentsto investigate whether the proposed triples represent the lengths of a right-angled triangle(Fig. 2, left). Moreover, care is taken so that one of the triples is not a multitude of the BasicTriple (the findings of the previous studies and the pilot study suggest that the studentspropose the multitudes of the Basic Triple as suitable candidates) to trigger the necessity forfinding a different, more general rule.

In the third section, Find the rule, we ask for the students to find this rule. We may‘guide’ their thinking by asking them to consider different ways of representing numbers,with the purpose of the students’ proposing the figurative numbers as a tool that may helpthem in organising this situation. In the third page of the worksheet, the students are asked tocomplete a table that is visually and in content linked with the ‘Figurative numbers’ activity(Fig. 2, right). The way the table is structured may help the students’ finding the rule a2=b2+c2 (a,b,c ∈ ℕ). Furthermore, the triple (6,8,10) was included because the findings of the pilotstudy supported our hypothesis that these numbers (their sum totals to 100) would help thestudents noticing the rule. Moreover, the table offers the students an overview of differentrepresentations of the lengths of the sides of a right-angled triangle (geometrical, numericaland figurative numbers), which may facilitate their linking the representations, thus movingtowards the construction of the mathematical object.

Fig. 2 The ‘investigation of triples’ figure (left) and the ‘Find the rule’ table (right)


3.2.3 The standard teaching

The third part of our design included the standard teaching of the theorem, according to theGreek curriculum. A 3-hour teaching introduces the theorem to the 2nd graders of the

Fig. 4 Sample items of the identification of reactivation of objectification questionnaire

Fig. 3 The two different configurations of the equal squares of the standard teaching


Gymnasio (14 years old). First, they calculate the area of two equal squares, consisting oftwo different configurations of shapes (Fig. 3). Considering that equal shapes have the same

Fig. 5 Activity 1 (left) and Activity 2 (right) of Phase D

Fig. 4 (continued)


area, the students produce the basic equation of the theorem, while the theorem and itsinverse are formulated.

3.3 Identifying the level of the reactivation of the objectification: instruments

3.3.1 The identification of reactivation of objectification questionnaire

The level of experiencing the reactivation of the objectification of the right-angledtriangle was identified through a questionnaire investigating aspects of the relationshipweb (§2.4). Building upon our previous studies and upon the pilot study, five sets of 13Likert items (5-point scale) were included (Fig. 4): 1) Characterisation of a trianglegiven the lengths of its sides (right-angled, obtuse, acute; items 1–3), 2) Characterisationof an angle of a triangle given the lengths of its sides (items 4–7), 3) Characterisationof the length of the third side of an oblique triangle given its two other sides and allthe sides of the corresponding right-angled triangle formed with the given sides of theoblique triangle (items 8–9), 4) Characterisation of the length of the side of an obliquetriangle with the other two sides of the same length as another triangle the lengths ofthe sides of which are a Pythagorean Triple (items 10–11), and 5) Choosing the mostpersuasive argument to characterise a right-angled triangle (items 12–13).

3.3.2 The identification of reactivation of objectification post-teaching interview

The purpose of the post-teaching interviews was to investigate the findings of the quantitativeanalyses of Phase D. Two activities similar to Item 10 and Item 2 of our questionnaire wereincluded (respectively Activity 1 and Activity 2; Fig. 5), as a result of the statistical analysis(§4.2). Three pairs of students were chosen from each class based on their School test score:Group A (high-attaining students), Group B (medium-attaining students) and Group C (low-attaining students). The students were given as much time as they needed for each activity.

3.4 Data analysis

In Phase A and Phase C, we minimised our intervention, including only informal conver-sations with the teacher (to investigate whether or not the two phases were conducted asplanned) and our collecting the ‘figurative numbers’ worksheets with the students’ work(Phase A).

The analysis of the teaching intervention (Phase B) and the post-teaching interview(Phase E) was conducted with the video recordings with Qualrus (Idea Works Inc.,Columbia, MO). A flexible coding systemwas adopted: starting from our theoretically foundedobjectification scheme (§2.2), allowing for the identification of new meaningful experiences ofthe phenomena. The video data were independently analysed and, subsequently, the resultswere discussed until consensus was reached. In both processes, a phenomenological attitudewas adopted to identify our own data analysis biases and intentions (a data analysis epoché; seealso Ahern, 1999; Langdridge, 2008).

In Phase D, non-parametric statistical analyses were conducted with SPSS 17 (SPSS, Inc.,Chicago, IL). The Mann–Whitney U test was employed to compare the scores of theexperimental class with those of the control class in both our Questionnaire and the schooltest, while the relationship between the two scores was investigated through Spearman’s rho.The responses for each Questionnaire item and the response patterns were compared withFisher’s exact test.


4 Results

4.1 Phase A

Phase A included the ‘Figurative numbers’ activity (see §3.2.1) with the purpose for thestudents of the experimental class to get familiarised with the square figurative numbers. Ourinformal discussion with the teacher confirmed that it was conducted as planned lasting lessthan 7 min.

4.2 Phase B

Phase B, our teaching intervention (see §3.2.2), was taught by one of the researchers andlasted around 39 min. 17 students attended the class (three were absent), working in 8groups: 7 pairs and 1 triplet. For the purposes of this paper, we focus on reporting events thatsignify the re-positioning of the students’ intentions about experiencing, making sense ofand communicating the phenomena. The data analysis revealed two poles in the students’communication spectrum: a) egocentric communications, in which the individual issemiotically present within the communication, suggesting a strong noetic-noematic corre-lation, and b) desubjectified communications, in which the communications focus on thephenomenon itself employing an impersonal language, thus alluding a weaker noetic-noematic correlation.

In Natural perpendicularity, the students were expected to visually match a form of thenatural perpendicularity with the triangle (3,4,5). First, with the help of two students, weformed a triangular shape with the large sticks and placed it by the ‘natural perpendicularity’,in order for the students to perceptually register the visual match between the angle of thetriangle and the natural perpendicularity. The rest of the class was asked to do the same withthe small sticks they had on their desks. We encouraged the students to challenge their‘natural attitude’, their sedimented ‘reflex’ to identify the triangle (3,4,5) and, instead, toemploy an explicit externally set (‘objective’) criterion in their judgment, putting the ‘name’under the scrutiny of a perceptual verification, thus showing elements of a phenomenolog-ical attitude. All the students actively participated and 6 groups cooperated with each other,suggesting that they were accustomed to similar classroom interactions, which agrees withthe purposes of our teaching.

Two students tried to construct a vertical triangle similar to the one they had just seen,while two pairs constructed two triangles that visually matched the angle of natural perpen-dicularity: (3,4,5) and (6,8,10). When the pair who constructed the (6,8,10) showed it to theclass, three more groups tried to make it and another pair tried to construct a triangle with the(6,7,8). The criterion of the validity of each claim was the students’ physically comparingthe constructed triangle with ‘natural perpendicularity’; 6 groups on their own and 2encouraged by the researcher.

Subsequently, the students were expected to perceptually experience that the largest sideof the triangle (3,4,5) is opposite to the right angle, that the acute angle is opposite to a sidesmaller than 5, and that an obtuse angle is opposite to a side larger than 5. With the help oftwo other students we changed the angle between the sticks 3 and 4, next to the ‘plumb bob,’in order for the students to perceptually experience the phenomena: the ‘5’ stick was eithertoo long (acute angle) or too short (obtuse angle). The rest of the class was asked to try thesame with the wooden sticks on their desks. Fourteen students participated in the activity,two chose only to observe, while one student remained focussed on the sticks, rather thanfollowing the rest of the class. Consequently, in this section, most students appeared to


realise the perceptual (visual and embodied), inductive verification of the quantification ofthe quality of perpendicularity (through the triples).

In Pythagorean Triples, the students produced more triples, focusing on their argumen-tation backing their examples. Anna suggested (2,3,4), because “as we said the hypotenuseis the largest, and they are consecutive [numbers].” Georgia shared Anna’s rationale,suggesting (6,7,8), because “as we said the hypotenuse will be the largest, so it is 8 andthe other 2 will form the right angle, since they are consecutive [numbers].” Notice that theyemploy egocentric descriptions for the geometrical relationships and desubjectified for thenumerical relationships. We posit that these can be regarded as evidence of a factualgeneralisation with contextual characteristics, which are also in line with the first level ofour objectification scheme.

Another group suggested the triples (6,8,10) and already started to verify their sugges-tions with the sticks on their desks. Thus, we encouraged the rest of the class to investigatethe validity of the two suggestions (2,3,4) and (6,8,10). Moreover, we asked the students toplace their small triangles next to the plumb bob, to visually verify the match of this triangle.The students, with the help of the sticks on their desks and employing perceptual verificationcriteria quickly realised that the triangle (6,8,10)–and not (2,3,4)–was a right-angled triangle,thus suggesting a re-positioning of ‘natural attitude’ towards the inclusion of a perceptuallyderived argument for their judgements. Maria drew upon this realisation to attempt ageneralisation: “for the (6,8,10) we are more sure, because from the numbers we had before[(3,4,5)], these are the double.” At the same time she uses her hands to show this magni-fication. Thus, Maria utters a general rule that is based on concrete manipulations andverifications, thus attempting a factual generalisation with embodied characteristics. Hercommunications crucially differ in her inclusion of an embodied argument supporting thenumerical argument ‘these are the double,’ implying a move towards a more desubjectifiedcommunication, which, importantly, cares for the geometrical compatibility of the relation-ships between numerical and non-numerical arguments.

Two pairs of students showed the same magnification by constructing the (3,4,5) trianglewithin the (6,8,10) in two different ways (see Fig. 5). We posit that this is a qualitativelydifferent generalisation, since the wooden sticks embody both the students’ product of theirgeneralisation and the rationale backing it, thus acting as the means for desubjectification oftheir argument, disjointed from the embodied, subjective, representation that Maria used.Hence, though on a concrete level, this implies the students’ desubjectified orchestration ofdifferent semiotic registers: the wooden sticks, the numerical and the embodied representa-tion of magnification.

At this point, the students suggested more triples that were multitudes of the Basic Triple,including (12,16,20), (9,12,15) and (1.5,2,2.5). Kostas, referring on the way that we canconstruct these triangles, asked “How do we do it? With the sticks?” Drawing upon thiscomment, a short discussion with the students revealed the fact that the sticks they had couldnot be used, since they represent only the natural numbers from 1 to 10. Thus, the studentsexperienced a situation the organisation of which required them to resort to the figural andthe numerical representations in the worksheet. All the students, but one, were convincedwith this discussion and started to work on the worksheet on the (9,12,15) triangle (Fig. 6).After drawing the triangle on the worksheet, they appeared to be convinced that themultitudes of the basic triple are right-angled triangles. Thus, we provided them with thenon-multitude (5,12,13), representing a triangle ‘under investigation.’ All the students(including the one who preferred to construct the triangle by combining the sticks 5 and10,2 and 9,4) realised the existence of triples, not multitudes of (3,4,5), that can be right-angled triangles, thus experiencing the necessity for finding a more general rule.


In the Find the rule section of our teaching, we expected the students to recall the‘Figurative numbers’ (see §3.2.1) and, with the help of the table included in the worksheet(Fig. 3), to produce the basic numerical expression of the theorem.

Researcher: Does anyone remember a different way of representing natural numbers?[no response] Something that you said with your teacher … with figures …Maria: Do you mean the square numbers?

Maria reminded her classmates of the ‘square numbers’ and with the participation of morestudents we wrote examples of such numbers on the blackboard. Once the students rememberedthe figurative numbers, they started to fill in the table in the worksheet (Fig. 7). The purpose of thisactivity was for the students to link the geometrical object with the numerical rule satisfied by thelength of its sides, thus realising the link between numerical and geometrical relationship. Thestudents first noted the pattern in the triangle (3,4,5), verified it for the other triangles included inthe table, verbally expressed the rule and wrote it down in the worksheet (see Fig. 6):

Researcher: Can you now find the link between the numbers 9, 16, and 25? Do you asee relationship amongst them?Giannis: If we add 9 and16 we have 25.Re: So you say that 9 plus 16 equals to 25. Right! Let’s write it down. Can we check itin the rest of the triangles without using the dots [of the figurative numbers]?We continuedwith the other triangles. All students actively participated, uttering the numbers.Re: Can anyone, what we found, say with words what we found so that we can write itdown in the following page of the worksheet?Katia: The amount of dots [pause]Re: [we encourage her] Yes?K: of the hypotenuse?Re: Yes!K: So … if we say that the perpendicular sides err squared give the amount of dots ofthe hypotenuse…Re: Yes. Can we express it better?Georgia: squared

Giannis made a numerical contextual generalisationwithout any reference to the geometricalmeaning. In contrast, Katia provides a desubjectified communication of her contextual gener-alisation explicitly linking numerical relationships with figural characteristics (in line with thesecond level of our objectification scheme).

The teaching concluded with our asking the students to express a rule with mathematicalsymbolism that would hold true for a ‘general’ triangle drawn on the blackboard (with lettersrather than numbers indicating the lengths of its sides). The students inductively producedthe basic expression of the theorem (within the set of natural numbers). Though not symbolic

Fig. 6 The two versions of the ‘triangle within the triangle’ construction and the worksheet


Fig. 7 A table filled by a pair of students and the rule they noted


(the letters are indexical signs of embodied actions and active argumentation in a variety ofrepresentational systems), their generalisation is in a form conceptually and linguisticallycompatible with higher levels of our objectification scheme (and with symbolicgeneralisation).

Researcher: [drawing at the same time] If I have a right-angled triangle and we write‘a’ for the hypotenuse and ‘b’ and ‘c’ for the two perpendicular sides, how can we saywhat we said before about the specific numbers?Kostas: a squaredRe: [writing at the same time what the students utters] Yes…K: equals with b squared plus c squared [The rest of the class agrees with this]

4.3 Phase C

Phase C included the three school hour ‘Standard teaching’ (§3.2.3). Our informal discussionwith the teacher confirmed that the teaching was in line with the curriculum descriptions.

4.4 Phase D

The purpose of this phase was to investigate whether or not our teaching experimentfacilitated the students’ experiencing the reactivation of the objectification of the right-angled triangle through a Questionnaire (see §3.3.1) and the School test.

The two classes did not significantly differ in their overall attainment in the School test(MdnExper=15,MdnContr=15, MExper=12.39,MContr=14.16; U=135, p>0.05, r=0.14) or in ourQuestionnaire (MdnExper=7, MdnContr=6, MExper=7.05, MContr=6.59; U=129, p>0.05,r=0.13). Focusing on the Questionnaire items (Fig. 4), the control class scored statisticallysignificant higher in Item 2 (NCorrectExper=6, NCorrectContr=15; p<0.001, phi=0.57) and in Item10 (NCorrectExper=7, NCorrectContr=13; p<0.05, phi=0.40) and lower in Item 12(NCorrectExper=9, NCorrectContr=2; p<0.05, phi=−0.39). Furthermore, statistically significantdifferent response patterns were found for Item 2 (p<0.001, phi=0.64), Item 10 (p<0.05,phi=0.50) and Item 11 (p<0.05, phi=0.51). Finally, the students’ score in our Questionnairedid not significantly correlate with their School test score for either the experimental class(p>0.05, rs=−0.19) or the control class (p>0.05, rs=0.18), suggesting their measuringqualitatively different aspects of the students’ knowledge.

At first, our teaching design may seem ineffective, since the control class were successfulin more items than the experimental class. However, the quantitative superiority of thecontrol class was not identified in all the items hypothesised to require the same way ofthinking. Moreover, the experimental class statistically significantly scored higher in Item12, indicating an understanding linked with the reactivation of objectification. For example,the experimental class may have developed the desired qualitative links between the basicexpression of the theorem and the right-angled triangle, which, nevertheless, may have actedas a source of a cognitive obstacle in an item such as Item 2 (which poses a conflict betweenthe visual and the numerical information of the figure). These qualitatively complex resultscalled for further qualitative investigation (Phase E).

4.5 Phase E

The mean duration of the post-teaching interviews was 11.7 min. In Activity 1 (Fig. 5), theexperimental class appeared to be able to choose the correct answer, though their strategies


appeared to differ depending on the attaining group. The high-attaining group (Group A) easilyrecognised the Pythagorean Triple (5,12,13). Eleni said “I remember it from the teaching inclass” and she continued by saying “but we could do the operations.” Subsequently, she arguedthat “since we have next to it a right-angled triangle, where the angle is 90°, and we have thehypotenuse 13, then since the angle is acute that is less than 90°, then the opposite side will beless than the initial, that is the 13.” The Group B pair (medium-attaining students) did notemploy the theorem to investigate whether or not the triangle is right-angled and they focussedon discussing the angle K. Sophia argued that “since K is acute, err less than 90°, it is more say‘inside’ [she uses her hands to show this], soΛΜ should be less than 13.”Note however that theargument and the words that she used are similar with Eleni’s. Anna andGrigoris fromGroup C(low-attaining students) also used the same argument and in similar wording with Group B.Overall, it is argued that all the students of the experimental group have developed thequalitative understanding required (the link between angle and length of opposite side) andcould appropriately quantify it to choose the correct answer. Nevertheless, only the high-attainers explicitly argued about the right angle of the given triangle utilising desubjectifieddescriptions, suggesting that they were the only ones that based all their judgements on thequantification of the qualitative relationships.

In the same activity, the control class appeared not to be as successful. The high-attainingpair struggled in the beginning. Kalia employed similar arguments to those of Group B andC from the experimental class, while Mary disagreed saying that “if the angle is acute, thenthis is the hypotenuse, therefore it would be bigger than 13.” When asked about whether ornot the angle A is right, they both resorted to the theorem, but they admitted that their initialjudgements upon which they relied to answer the question were visual. Akis argued that “ifthe triangle ABΓ is right-angled and the hypotenuse is 13, this [the unknown side] it isreasonable to be less than 13” and Georgia agreed. Nevertheless, no further explanation wasgiven, even when prompted, in any form (including gestures, numbers etc.). The studentsassigned to Group C looked at both figures and quickly decided that “ΛΜ equals to 13,”since “the figure there is just larger … but it is the same … it has to be the same.” Theyappeared not to be able to realise that different angles imply different lengths of the oppositesides. Consequently, we argue that the students of the control group appeared to havedeveloped a more fragmented understanding than the students in the experimental groupwhich was magnified with the medium and low attaining students.

In Activity 2 (Fig. 5), the students of the experimental class were all troubled by thefigure. The students of Group A argued that “when I saw the numbers I did not believe it waslike this [the figure]” (Leo), but they both quickly decided that the angle is acute. Leo arguedthat “because the side under the height is larger than the hypotenuse … that is why thetriangle is not right-angled and therefore Β acute.” Leo’s argument is based in the link‘hypotenuse’–‘right-angled triangle’–‘opposite angle is right.’ Note that though they bothquantified this link drawing upon the lengths of the sides, they did not employ the theorem intheir judgements. Sophia from Group B applied the theorem based on the figure, realisedthat it didn’t hold true and decided that the angle B is acute. Nevertheless, Dora applied thetheorem focusing only on the numbers (without looking at the figure) and deduced that thetriangle is right-angled, therefore B is right. Sophia’s initial argument was overwhelmed byDora’s operations with numbers and changed her mind, thus resulting in their choosing thewrong answer. Both students of Group C based their judgements on the visual inspection ofthe figure and quickly answered that B is right. Considering that the lower attaining inmathematics students may be reluctant to choose numerical warrants, we decided to promptthose students to notice the numbers on the sides of the triangle. Anna quickly respondedthat “the numbers must be wrong,” because “it cannot be the largest to be 6 and the relative


smaller to be 10, therefore it is.” Grigoris interrupted her and said “therefore it is acute” andAnna agreed with him. Grigoris, when asked about his rationale, argued that “if it is smallerlike this, the figure would lean.” Though the experimental class seemed to struggle with thisactivity and employed a mixture of egocentrical and desubjectified communications, five outof six based their communications on appropriate quantifications (partial or not) of theexpected relationships. The fact that they were puzzled by the conflict between the figuraland the numerical representation indicates the existence of a geometrical meaning in therelationship of the representations.

In the same activity, the high-attainers of the control class again disagreed in their judge-ments. Mary immediately employed the theorem and deduced that the triangle is right-angledand therefore B is right, suggesting that her partial quantification of the right-angled trianglethrough the theorem lacked the quantification of the link ‘hypotenuse’–‘opposite angle is right.’Kalia, on the other hand, argued that “if it were right-angled, then A should be right” and after awhile concluded that B should be acute. FollowingMary’s rationale, Akis (Group B) concludedthat B is right, but Georgia wondered “Since the hypotenuse is always the largest, why is this10?” She applied the theorem based on the figure and realised that it didn’t hold true. After adiscussion, they decided that the angle must be obtuse, since the sides are larger (they may haveconfused the rule about the larger side linked with obtuse angle). Finally, the students of GroupC based their arguments on the visual inspection of the figure, concluding that B is right. Evenwhen prompted to think about the numbers they continued to argue that “it is evident from thefigure” and that “every triangle has a right angle,” which justifies their answer that B is acute.Overall, the students of the control class appear to have a fragmented understanding of the expectedlinks with the low-attaining students showing no evidence of developing even the desired qualitativerelationships. Though the control class appeared to employ more desubjectified (numerical) com-munications, this was not in line with a higher level of objectification, due to the weak (or the lack ofa) signification of the expected geometrical relationships.

5 Discussion and concluding remarks

In this study, we discussed of a multi-phased teaching design (incorporating the standardschool teaching) of the Pythagorean Theorem, with the purpose of felicitating the students’possibility for experiencing the reactivation of the objectification of the right-angled triangle.

First, the analysis of our teaching intervention suggested the relative success of the firstphase of our design (‘Figurative numbers’), since the students managed to introduce and tosuccessfully utilise the figurative numbers representation in our teaching.

Moreover, the students’ interaction with our teaching suggested their moving: a) towardsmore desubjectified communications, and b) towards a stronger link amongst numericalrelationships, non-numerical (figural, visual or embodied) relationships and the right-angledtriangle. Desubjectified communications are conceptually linked with Radford’s (2003) discus-sion about students’ movement from pre-symbolic to symbolic generalisations. The strongerlinks amongst numerical and non-numerical relationships and the geometrical object are in linewith our theoretical discussion about the two registers involved in the signification of thegeometrical object and the development of their inter-relationships. Furthermore, our teachingintervention concluded with the students producing the basic expression of the theorem: acontextual generalisation, which was conceptually and semiotically compatible with higherlevels of objectification.

The comparative quantitative analyses of the post-teaching Questionnaire and the Schooltest scores revealed complex results: a) not significantly different success for both the School


test and the Questionnaire, and b) varied performance and response patterns per Questionnairetask.

These results were further pursued through semi-structured interviews. The data analysisappeared to support the hypothesis that the students of the experimental class experienced ahigher level of the reactivation of the objectification of the right-angled triangle. Notably, weidentified a stronger link of a geometrical reference to the communicated relationships(numerical and non-numerical). Though they appear to be reluctant (in comparison withthe control group) to employ the theorem, most of their communications were based on bothnumerical and non-numerical representations. Their ways of experiencing of the right-angled triangle and the theorem made apparent to them the discrepancy between the visualand the numerical information of the figure in Item 2 of our Questionnaire (see Fig. 4), thuspreventing them from choosing the correct answer, but when given the chance to reasonabout their ideas (in Activity 1 of the interviews) they seemed to be able to choose thecorrect answer. On the other hand, the students of the control class appeared to be morecompetent in applying the basic expression of the theorem to determine whether or not atriangle is right-angled, but the post-teaching interviews suggested that this competence wasnot appropriately geometrically founded. When asked to determine the right angle of thetriangle they appear to rely solely on the figure, without linking their answer with thenumerical expression they just used. This disjunction from the figure quantification mayexplain their higher score in our Questionnaire, since for these students the figure and thenumbers were linked with different phenomena and, thus, there was no discrepancy betweenthe two experiences.

Consequently, the students of the experimental class showed evidence of experiencing thelinks amongst figural, numerical and the qualitative relationships orchestrated in the signi-fication of the right-angled triangle. Though it appears that such an experience restrains thelower mathematically attaining from providing a ‘seemingly’ right answer (by just applyingthe ‘rule’), we argue that it is mathematically more important that: regardless their attainmentthe students of the experimental class showed elements of appropriately linking the rela-tionships involved. On the one hand, the students identified the perceptually constructedqualities on the complex web of relationships within each representational system. On theother hand, through measurement within non-numeric representational systems, the quali-tative relationships are quantified and are mapped on the numeric (and later algebraic)representational system.

The mathematical object gathered Husserl’s interest, as an ideal object “already reducedto its phenomenal sense, and its being is, from the outset, to be an object [être-objet] for apure consciousness” (Derrida, 1989, p. 27). We theorised that Husserl’s complex programmeand its cornerstone notions are compatible with the learning of geometry allowing for boththe construction of a geometrical idea through the satisfaction of perceptual, psychologicaland sociocultural needs and the securing of this idea through a proving process within anaxiomatic system.

The findings of this study are in line with these conjectures, by identifying different waysof experiencing and communicating the relationships amongst figural signs, numerical signsand the geometrical object. The various semiotic registers are experienced as havingdifferent functions and relationships with the signified object and with themselves. Thereported findings support our hypothesis that the numerical representations are experiencedby the school students as geometrical signs through the mediation of a figural sign; thestrength and nature of this mediation characterises the experience of the theorem withrespect to the right-angled triangle. Without such a mediated relationship the basic expres-sion of the theorem remains a geometrically empty numerical relationship signifying a class


of numbers, rather than a geometrical object. Hence, the gradual substitution of egocentriccommunications with desubjectified communications is not sufficient for appropriate geo-metrical generalisations (as it may be in the case of algebra).

Overall, the findings of this study support the pedagogical contribution of adapting andincorporating phenomenological ideas in designing a teaching of the Pythagorean Theorem.Our didactical framework allowed the students to move away from their natural attitude,towards a weaker noetic-noematic correlation and to viewing ‘the things themselves.’ Thestudents managed to unfold aspects of the sedimented web of relationships of the right-angled triangle, thus intentionally, selectively and appropriately mapping the mathematicallyequivalent regions of different representations. Based on the embodied experiences ofgravity and perceptually derived activities, the students re-viewed important geometricalrelationships, reactivating intentions compatible with the possibility for experiencing theanthropological, non-arbitrary nature of the right-angle triangle, which arises through therelationships of different representational systems. Our phenomenological perspectivehighlighted that the ease of employing the basic expression of the theorem (usually accom-panied with higher school test scores) does not necessarily imply a quantified geometricallymeaningful relationship and, importantly, this perspective provided a theoretically foundedpedagogy for taking appropriate actions towards addressing such issues.

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