Τhe Embodied, Proceptual and the Formal World in the Context of Functions

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Th e Em bodied Procep tua l and

Formal Worlds in the Context of

Funct ions

Co nstantinos Christou and Demetra Pitta-Pantazi

University of Cyprus

Alkeos Souyou l and Theodossios Zachariades

University of thens

Abstract In this study we use Tall et al.'s (2000) theory on mathematical concept development,

which describes three worlds of operations, the embodied, the proceptual and the formal (Tall,

2004; Tall , 2003; Wa tson , S pirou, & Tall, 2002). Th e purp ose of the study is threefold: first, to

identify mathematical tasks in the context of function that reflect the three worlds of operations;

second, to investigate whether studen ts ' thinking corresp onds to the embodied, the proceptual, and

the formal modes of thinking; and third, to reveal the structure of and relationships among the

three worlds of operations as these unfold through students' responses. The study was conducted

with first-year university students. The results suggested that mathematical tasks can be categorized

on the basis of Tall et al.'s (2000) theory and indicated that sttidents exhibit different kinds of

thinking, which reflect to a large extent the three worlds of operations. Three classes of students

were identified in terms of the difficulty level of the tasks: (a) the proceptuaL (b) the proceptiial-

emhodii d,

and (c) the

formal.

According to Tail 's theory, embodied, proceptual, and formal

thinking develop in sequence in an individual's life. This study indicates that freshmen university

students, who have had mathematics as a major in higher secondary school, are only able to deal

with the embodied tasks once they had been successful with the proceptual ones. The leap to formal

thinking could only be achieved when proceptual m anipu lations w ere enhanced with competence in

embodied tasks.

Sommaire exeeutif D ans eette etude, nou s no us servons de a theorie de Tall et de ses collegues sur

Telaboration des concepts mathematiques, laquelle distingue trois ordres d'operations mathema-

tiques

l 'inclusion physique, le proceptuel et le formel (Tall, sous presse; Tall,

2003;

Watson ,

Spirou. et Tall. 2002). Le but de Tetiide s'articule sur trois aspects: d'abord. identifier les taches

ma them atiqu es d ans le contexte des fonctions qui reflctent les trois ordres d'o per atio ns, ensuite

verifier a quels ordres d'operations correspondent les modes dc pensee des apprenants. et enfm,

faire emerger les structures et les relations qui existent entre les trois ordres d'operations au fur et a

mcsure que ceux-ci se manifestent dans les reponses des ctudiants.

L'etude a ete menee aupres d'etudiants de premiere annee a TUniversite de Chypre et a

I 'lJniversite d'Athe nes. do nt les math ema tiques avaient ete ia matiere pri nc ip al au deuxieme cycle

du secondaire. Un test, comprenant un certain nombre de taches qui refletent les caracteristiques

des trois ordres d'operations (Tall, sous presse; Tall. 2003), a ete administre a 236 etudiants.

L'analyse des facteurs de confirmation a ete appliquee afin d'evaluer les resultats de l 'etude.

On peut deduire de l 'analyse des donnees presentees que les taches mathematiques petivent etre

classees en fonction des criteres definis par Tall et ses collegues. La modelisation proposee montre



CJSMTE RCESMT 5:2 April 2005

Les resultats montrent egalement que la reussite dans la resolution d'un probieme par plus de

50

des etudian ts d'une categorie correspondait au mem e niveau de reussite par plus de 50 dans

les categories suivantes. La presence d'une tendance eonstante dans ie niveau de difticulte qui

caracterise les trois ordres d'operations temoigne en faveur de l'existence d'une tendance de

deveioppement specifique.

Selon la «theorie des trois ordres», les apprenants progressent de Pinclusion au proceptuel

pour enfin passer au mode de pensee formel. Cependant. des que l'un ou I'autre de ces trois ordres a

ete acquis. il peut tres bien constituer la base sur laquelle construi re les deux a utre s. N otr e etu de

met en evidence ce genre de differenciation dans la sequence inclusion - proceptuel - formel. Plus

preeisement, l 'analyse des donnees indique que les etudiants en premiere annee de mathematiques

ont plus de facilite a executer d'abord les taches proceptuelles et seulement plus tard les taches

d'inclusion. du moins pour cc qui est des fonetions. En elTet. les etudiants n'etaient en mesure

d'affronter Ics taches d'inclusion que lo rsqu'ils avaient deja m aitrise les taches proc eptuelles. II

semble que les objets mathematiques et leurs proprietes, par exemple les graphiques. ne peuvent

etre compris que lorsque les etudiants maitrisent les procedes mathematiques representes par les

symboles. On peut done aflirmer que. pour ies etudiants de premiere annee universitaire, la

comprehension des fonctions passe d'abord du proceptuel a l'inclusion, et en dernier lieu au formel.

Les resultats de cette etude indiquent egalement que la comprehension proceptuelle ne suflit pas a

elle seule a mener a la comprehension formelle. En effet. le passage au formel n'a pu se realiser que

lorsque les manipulations proceptuelles ont ete associees a i'acquisition des competences liees aux

taches d'inclusion.

Introduction

This study is based on Tail's

three-world theory

of mathematical concept developmen

(Tall, 2004; Gray & Tail, 1994; Tall et al., 2000; Gray & Tall. 2001; Tall, 2003; Watson &

Tall, 2002; Watson et al., 2002). This theory describes three worlds of operations—the

embodied

the

symbolic-proceptual

and the

formah

the theory constitutes an attempt to

encompass the whole spectrum of students' cognitive development in different mathematical

topics, from elementary to advanced mathematical thinking.

The present study provides an empirical verification of the three worlds of operations

and traces the different types of thinking projected by freshmen university students in the

context of functions. The paper is organized as follows; First, we briefly describe the theory

of the three worlds and the theoretical perspectives that relate to the concept of function. We

then present the methodology and the results of a relevant study conducted in Cyprus and

Greece. Finally, the conclusions of the study and their implications for educational planning

and teaching are discussed.

The oret ical b ackground: Th e three wo rld theory

Tall ct al. (2000), based on the assumption that individuals build up conceptual

structures by perception, action, and reflection, put forward the theory that these three

activities produce three fundamentally different worlds of mathematics, the embodied, the

proceptual. and the formal. The embodied world is the basis for all of our activities. It begins

with our perceptions and actions on the real world and, through the use of language, it builds

from perceptual representations to more abstract representations.

The proceptual world is the world of calculations in arithmetic and symbolic



Tbe E mbodied Proceptual and Formal Worlds in the Context of Functions

in embodied and proceptual mathematics leads to recognizing the properties of the concepts,

until we are able to use these properties as fundamental axioms and definitions in the

coherent formal structure of theorems and

proof

Tall et al. (2000) suggested that each of the three worlds focuses on different qualities:

The embodied world focuses on the objects and their properties, the proceptual world on

processes represented by symbols, and the formal on properties and relationships among

them. The three modes of operation develop in sequence as the individual progresses through

his or her school life. First, the student acts in the embodied world, which has its roots in the

kinaestheticjiconic

modes of thought. For example, students first understand the number 5

based o n th eir p ercep tions of an d action s to do with five conc rete item s (e.g., five fingers, five

manipulatives, five icons of countable objects); then they reach an understanding of the

number 5 in its abstract form. Later on, the proceptual world builds on embodied actions and

the stud ent c an view the sym bol 5 either as a collection of five objects o r the sum of

+ 2,

4 + 1 ,

and so on. Once an understanding of whole numbers is achieved, more sophisticated

concepts may follow, such as fractions, decimals, or algebra.

Later on. formal thinking develops. Students at this level are able to work on axiomatic

systems and definitions and deduce properties from the axioms and definitions in a sequence

of theorems. At this level, thinking is not based on familiar objects but on axioms, which are

fonnulated to define specific mathematical structures. Properties can then be deduced by

formal proof to build a number of theorems. For example, for the concept of natural

numbers A'— {0. 1, 2,

}

we accept axio m atically the existence of a set /Vo and a function

s\ NQ

-^ JV; with the following properties:

1. The function s is one-to-one.

2.

The re exists an element of NQ , which we represent with the symbol 0, so that

s{n

^ 0,

for each element n of A'o-

3 .

If A is a sub-set of

JVO,

it satisfies the properties:

a) 0 e ,4. and

b) if n

G

/I, then s{n e A for every element n of A'o, then A

—

N ^.

Based on these axioms we define two operations, the sum and multiplication. All other

properties ofthis set and its operations are deduced from these axioms. However, it needs to

be stated that formal thinking is also partly based on students' abilities to observe the

properties ofthe objects ofthe embodied world and to carry out the processes associated with

and the manipulation of procepts.

he con cep t of function and th e aim s of the study

The function co ncept is a com plex, multifaceted idea whose power and richness perm eate

almost all areas of mathematics. In the school curriculum, function is an advanced topic,

which is typically not explored in detail until the secondary level. Due to the unifying role of

the function concept in mathematics and its ability to provide meaningful representations of

com plex, real-world situation s (He id. Ch oa te, Sheets, Zbie k. 1995), curre nt reform

recommendations call for an emphasis on functions to be integrated throughout the school

curriculum, beginning in the elementary grades (National Council of Teachers of

Mathematics [NCTM]. 2000). Given the importance of functions in mathematics and in




verbal, or symbolic form. Y erushalm y (1997) con cen trated on students abilities to deal with

these different kinds of representations and on translations among them. Sfard (1992)

discussed the difliculty students face in translating and relating algebraic and graphical

representations of function. Norman (1992) found that even mathematicians studying for a

ma ster's degree tend to use just one kind of repre senta tion, the graphical on e. Christou et al.

(2002) identified hierarchical levels among the graphical and symbolic representations of

mathematical functions and verified an association with students' ability to identify various

representations of the mathematical functions.

Generally, the views of different researchers seem to converge on the idea that most

teaching approaches do not make any reference to relationships and translations among

different kinds of rep rese ntatio ns of the function c once pt (Ye rush alm y. 1997). Th e different

representations and the difficulties that students face—the relationships among the

different kinds of thinking that students employ and the question of how these relate to

difl erent representations of functions—are not yet clear.

A characteristic of recent developments has been the focus of attention not only on the

mathematics to be taught but also on the mental processes by which the concept of function

is conceived and learnt (Tall. 2003). To understand students' mental processes, we need first

to realize that there is a spectrum of possible appr oac hes to the concept of function, from

examining real-world functions that can enact intuitions using visuo-spatial representations,

through looking at numeric, symbolic, and graphic representations, to employing formal

definitions (Tall. 2003). A student's ability to deal with graphs has the characteristics of visual

embodiment, whereas the algebraic manipulation of function symbolism has proceptual

characteristics (Tall, 2003). It is also possible that an individual may approach the function

concept through its definition. However, this approach is more likely to be adopted by

experts than by secondary school students, who often disregard definitions.

In this study, we hypothesize that the concept of function has distinct aspects that

represent TalPs three worlds, and we propose to investigate the relationship among these

three worlds as it unfolds through the responses of freshmen university students to function

tasks.

Specifically, the aims ofthe study were as follows:

1. to investigate wheth er different task s in the con text of functions can be categ orize d as

embodied, proceptual. or formal

2.

to trace grou ps of studen ts that reflect these three mod es of thinkin g

3.

to exam ine the structu re of and the relationship s am ong these three wo rlds of ope ra-

tions as they are projected through freshmen mathematics students' responses

etho oiogy

articipants and tasks

Data reported in this paper were collected in 2002 through questionnaires administered

to 236 first-year students who had had mathematics as a major subject in secondary school

and were studying at the University of Cyprus and the University of Athens. The tests were

administered during the students' calculus course, towards the beginning of their first

semester.



The Embodied, Proceptual, and Formal Worlds in the Context of Functions

Embodied Tasks

1 . Mark w ith the letter [F] the graphs that were genera ted by functions of the form y

f(x) and [O]

those thaf are impossible to be generated by functions of the form / = f(x). Explain your

responses (x is Ihe horizonfal axis in all fhe graphs).

(El)

E 2

Proceptual Tasks 1. Investigate whefher the following relationships define functions. Explain your answers.

2. Identify fhe letter indicating the independent variable and provide the largest field of values of

the relationships thaf define fundions. If there are any consfanf functions mark them as [C]-

(P1 ) .v ' -2 v = 0

P2) h

a -2a

P3) f(y}

e

Formal Tasks (F1) Give the definition of function.

(F2) Whe n is a funcfion f(i) equal lo another function g(t)7

(F3) For security purposes any of Ihe gym mem bers entering the gym leave fheir money

af fhe recepfionist. Is fhere a funcfion fhat expresses fhe relationship befween Ihe gym

members' names and fheir money? Explain your answer. (The assumption is ffiaf ffiere are no

gym members with the same name and fhat all members bring a certain amounf of money) .

(F4) Does the process of matching every funcfion that has a derivative and ils denva live

constilufe a funcfion? Explain your answer .

P = proceptual tasks, E - embodied tasks, and F = formal tasks

Figure 1:

The embodied, proceptual, and

formal

tasks

example, Christou et al. (2002) found that the most difficuh embodied tasks were the ones

that co uld not be generate d by functions of the form y = f(.v) (see tasks E l, E2, and E3 in

Figure 1). The und erlying assum ption was that stude nts wo uld have achieved the procep tual

and embodied worlds once they were in a position to respond correctly to these tasks.

The problem s reflecting the em bodied ope ration s investigated students abilities to

identify graphs that were generated by functions ofthe form

v

= f(-v) and gra ph s that could

not be gen erated by functions o f this form (see tasks E l , E2 , and E3 in Figure 1). Th e focus of

these tasks was on the grap h as an object and on the relative chang e of v with respe ct to .v and




define functions, to clearly identify the letter indicating the independent variable, and to

provide the largest field of values. Thus, the emphasis here was on the manipulation of

symbols.

Finally, four tasks were presented in ord er to investigate students ability to opera te in

the formal w orld. Two of these tasks (F l and F 2; see Figure 1) simply looked into stu den ts'

abilities to provide and use definitions of functions. More specifically, Task Fl was used to

determine wh ether there were any students w ho were able to do the proceptual and embodied

tasks without knowing the definition of function. Tasks F3 and F4 required students to

refiect on the definition of function in its abstract form and to identify the properties of

functions (see Figure 1). In responding to these tasks, students could not rely on symbolic

manipulations or extract information from a given graph.

Scoring and analys i s

Students* fully c orrect respo nses were mark ed 1 and incorrect resp onses 0. If a stud ent

gave a partly correct response—for example if he or she gave a correct answer but a wrong

justification —this again was marked with 0. The confirmatory factor analysis (CFA), which

is part of a more general class of approaches called structural equation modelling, was

applied in order to assess the results ofthe study. CFA is appropriate in situations where the

factors of a set of variables for a given population is already known because of previous

research. In the case ofthe present study, CFA was used to test hypotheses corresponding to

the theoretical notion of the three worlds. Specifically, our task was not to determine the

factors of a set of variables or to find the pattern ofthe factor loadings. Instead, our purpose

of using CFA was to investigate whether the established theory of the three worlds fits

our data.

One of the most widely used structural equation-modelling computer programs,

MPLUS {Muthen & Muthen, 1998), which is appropriate for discrete variables, was used to

test for model fit. In order to evaluate model fit. three fit indices were computed: the /~ to its

degree of freedom ratio {x^/df , the com parative fi t index (CF I), and the root m ean-squ are

error of appro xima tion (R M SF A) (M arcoulides & Schum acker, 1996). The observed values

of x^ldf were to be less than 2; the values for CFI, higher than 0.9; and the RMSEA values,

close to 0. MPLUS was also used to trace groups of students who could reflect the embodied,

the proceptual, and the formal worlds of thinking. To this end, latenl class analysis (LCA)

was used, which is a part of mixture growth analysis (Muthen & Muthen, 1998). LCA is a

statistical method for finding sub-types of related cases (latent classes) from multivariate

data. The results of LCA can also be used to classify cases into their most likely latent class.

Thus, suppose that, given a sample of subjects measured on several variables, one wishes to

know whether there is a small number of basic groups into which cases fall. Once the latent

class model is estimated, subjects can be classified into their most likely latent classes by

means of recruitment probabilities. A re cruitmen t prob ability is the probability th at, for a

randomly selected member of a given latent class, a given response pattern will be observed.

esults

The results are presented in relation to the aims of the study. First, we examined the

hypothesis implied by the first aim of the study: whether the embodied, proceptual, and



The E mbodied Proceptual and Formal Worlds in the Context of Functions

0.37

* numbers in parenthesis indicate the z value

Fig ure 2: Confirmatory factor analysis with the loadings of each item on each factor

and the correlation between factors

structure of the three worlds, which might indicate their developmental nature. These

hypotheses are interwoven and aimed at investigating whether the existing theory stated by

Tall et al. 2000) can emb race a hierarchy of tasks that may contr ibute to instruction in

complex mathematical ideas such as functions.



CJSMTE/RCESMT 5:2 April 2005

Each of the embodied and proceptual factors involved three tasks, while the formal factor

involved four tasks. Figure 2 represents the model that best describes Tail's theory of the

three worlds of operations. Confirmatory factor analysis (CFA) was used to evaluate the

construct validity of the model. CFA showed that each of the tasks employed in the present

study loaded adequately (i.e., they were statistically significant, since values were greater

than 1.96) on each factor, as shown in Figure 2. It also showed that the observed and

theoretical factor structures matched for the data set ofthe present study and determined the

'goodness of fit* of the factor model (CFI = 0.946. .v^ -85 .13 4, i / / ' - 59 , .v^/£//=1.4,

R M S E A — 0.04). indicating that the em bodie d, pro cep tual, and form al worlds can represe nt

three distinct operations of students' thinking.

lass es of students and the developm ental trend

The second aim ofthe study concerned the extent to which students in the sample varied

according to the answers they provided in the test. Specifically, we examined whether there

were different types of student in our sample who could reflect the embodied, the proceptual,

and the formal worlds of operations. Mixture-growth modelling was used to answer this

question (Muthen & Muthen, 1998) because it enables the specification of models in which

one model applies to one sub-set ofthe data and another to another set. The modelling here

used a stepwise me thod the model was tested under the assu m ption th at there were two ,

three, or four classes of subjects. The best fitting model with the smallest AIC (3,417.340) and

BIC (3,569.749) indices (see Muthen & Muthen, 1998) was the one involving three classes.

Taking into consideration the average class probabilities as shown in Table I, we may

conclude that classes were quite distinct, indicating that each class had its own characteristics.

The means and standard deviations of each of the three worlds of operations (embodied,

proceptual, and formal) across the three classes are shown in Table 2. Table 2 shows that

studen ts in class 3 (A' = 0.69) outperfo rme d stude nts in class 2 and class 1 (i = 0.48, A = 0. 2I

respectively) in formal tasks, while students in class 2 outperformed their counterparts in

class 1. Ho we ver, the p erce ntage of success of stude nts in class 1 in all task s was below 50%.,

showing that these students had difiiculties in conceptualizing functions. Class 2 students

Table : Averag e class probabilities by class

Class I Class 2 Class 3

Class t

Class 2

Class 3

0.973

0.007

0.000

0.027

0.993

0,000

0.000

0.000

1.000

Table

Class

2:

Means and

standard deviations

Formal

Of Students' performance

Proceptual

in each class

limbodied

Class 1 (N^66)

0.21* (0.17)

0.47 (0.41)

0.45 (0.35)



The Embodied, Proceptual, and Formal Worlds in the Context of Functions

Table : Problems solved by more than 50 of students in each class

( l a s s I: Proceptual Class 2: Rm bodied-Proceptual Class 3: Formal

P1, *

P3, E3

P1 ,

P3, E3

P2 , El, E2, F l

P I ,

P3, E3

P2 , El, E2, F l

F2 , F3, F4

had difficulties in the formal tasks, since their success percentage was lower than 50Vo.

These students were successful in most of the embodied tasks (71%) and solved correctly

6 6%

of the proc eptua l task s. Finally, class 3 stude nts seemed not only to understan d the

embodied and proceptual tasks but also to have the ability to think formally (69%).

From Table 3, which shows the problems solved by more than 50% of the students in

each class, it can be deduced that there is a developmental trend in students' abilities to

complete the assigned tasks: Success on any problem by more than 50% of the students in a

class was associated with success by more than 50% of the students in all subsequent

classes. Thus, class 2. which was the largest (A'— 109), can be considered as the embodied

and proceptuai class because this class included those students who successfully solved

the embodied and proceptual p roblems, class 1 students (,^=366) solved two thirds o ft h e

proceptual problems and one embodied problem; thus, we can consider class I as the

proceptual class. It seems that, for some students, the links between the proceptual and

embodied worlds of operations were not very evident.

As is shown in Tab le 3. stude nts belonging to class 3 couid solve function prob lem s using

formal thinking ; this was the top class of studen ts. S tuden ts in class 2 systematically solved all

em bod ied and pro ce ptu al p rob lem s; it is difficult to classify th em into either of these two

worlds. W hat distinguished these students from those in class 3 was their inability to think

formally. Students in class

seemed to solve mo re proc eptu a than embod ied tasks, but their

ability to operate in these two worlds was much less than that of students in the higher

Model I

0.49 0.42

CFI = .859; jc^= 126.777; d f= 59 ; x2 df= 2 .15 ; RM SEA = 0.09

Model 2



CJSMTE/RCESMT 5:2 April 2005

Table :

Model 1

Model 2

Fitting ind ices ot model

CFI

0 859

0 951

1 and model 2

^ •

7

1

df

15

07

RMSFA

0 0 7

0 04

classes, This result seems to provide evidence that students are better able to understand

function problems that are presented in proceptual form.

he structure of the developm ental trend

The presence of a consistent trend in level of difficulty across the proceptual, embodied,

and formal worlds supports the hypothesis ofthe existence of a specific developmental trend.

The data implied that students grasp calculus concepts first by perceiving them in the

proceptual world and then by applying the ideas prescribed in the embodied world. The

formal world is grasped only after embodied a nd pro cep tual rep resen tations have been

conceptualized. It appears that, in the context of functions, students do not follow the

sequence, emb odied, proceptua , formal, but the sequence, proceptual, em bodied, formal.

To further examine this sequence, we tested two models for specifying the nature of the

developmental trend of students ' understanding of function problems. The first model

assumes that students first understand the concept of function through the embodied world

and then are able to use proceptual and formal ideas (model 1; see Figure 3). The second

model, which results from the data ofthe present study, assumes that students comprehend

the concept of function proceptually and then they are able to conceive the embodied and

formal representations (model 2; see Figure 3). In both models, formal thinking seems to be

the result of students' experience with both the embodied and proceptual worlds.

Structural confirmatory analysis was used to examine the model that best fits the

empirical data. From Table 4 and Figure 3, we can deduce that the best model is model 2,

since it has the best fitting indices (CFI:=0.951.

.v^

= 69.85.

df=4S.

. Y ^ / # - 1 . 4 6 ,

RM SE A —0.06). M odel I does not fit the data , and no fitting indices are adeq ua te to

provide evidence that supports the structure implied in it (CFI = 0.859, x^ 126.777, df= 59,

x^ df—2.\5, RM SE A = 0.09). These results reaffirm the deve lopm ental trend as described

above and indicate that students are first more fluent in doing the proceptual tasks and then in

doing the embodied ones.

Conclusions

The first aim of this study was to examine whether it is possible to distinguish among

different types of tasks that refleet the three worlds of operations. It can be deduced from the

data presented in this study that mathematical tasks can be categorized, based on Tall et al. 's

(2000) theory, as embodied, proceptual, or formal. The second aim concerned the extent

to which students in the sample might vary according to the tasks provided in the test.

The mixture-growth modelling illustrated that three different classes of students could be

identified, classes which, to a large extent, reflected the 'three worlds of operations." Class I



The Embo died Proceptual and Formal Worlds in the Context of Functions

consistent trend in level of difficulty across the embodied, proceptual, and formal worlds

supports the existence of a specific developmental trend.

The analysis indicated that students are more successful first in doing the proceptual

tasks and later on in doing the embodied ones. Students were only able to deal with the

embodied tasks once they had been successful with the proceptual ones. It seems that the

mathematical objects and their properties, such as graphs, can only be understood once

students have mastered the mathematical processes represented by symbols. Thus, it ean be

argued that, for freshmen university students who have had mathematics as a major in higher

secondary school, understanding of functions progresses from proceptual, to embodied, to

formal. Possible reasons for this may be the emphasis in mathematics classrooms or the

nature of the specific mathematical topic. The fact that students can only understand

graphical representations of functions once manipulation of symbols has been achieved does

not suggest that embodied understanding is of

ess

imp ortance . ' On the contrary , the results

ofthis study suggest that proceptual understanding was not sufficient, on its own, to lead to

formal understanding. The leap to formal thinking could be achieved only when proceptual

manipulations were enhanced with competence in embodied tasks.

A number of teaching implications arise from these findings. Within the theoretical

framework of this study, it can be argued that learning in mathematics begins with embodied

thinking. As students progress to higher secondary education a 'disembodiment ' s tarts taking

place so that proceptual thinking overshadows embodied thinking. This disembodiment may

be due to teaching approaches in the classroom that emphasize proceptual ways of thinking

at the expense ofthe embodied ones (Vinner, 1992; Yerushalmy, 1997). However, in order to

achieve formal thinking, a re-embodiment is necessary, since proceptual thinking is not

sufficient on its own to lead to formal thinking. Students who systematically use only

procedures and mathematical symbolism distance themselves from the manipulation of

embodied representations, and this inhibits their transition to the formal way of thinking.

cknowledgements

We thank David O. Tall and Elena Nardi for commenting on an earlier draft of this

manuscript, as well as the three reviewers for their valuable comments. The research

presented in this paper was funded by the University of Athens (EAKE) and the University

of Cyprus .

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G ray , E.M .. Tall, D. O. (1994). Du ality, am biguity and flexibility: A proc eptua l view of

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Τhe Embodied, Proceptual and the Formal World in the Context of Functions

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