Constructing the Continuity Concept

ANA VALONGO1

MIGUEL FELGUEIRAS 12

1ESTG, Polytechnic Institute of Leiria

2CEAUL Lisbon

PORTUGAL

Abstract: Key concepts in Mathematics, like derivative or integral, are connected with the notions of limit and

continuity. Therefore, the construction of these concepts is relevant and should be well understood by the stu-

dents. In this paper the construction of the continuity concept by the students is studied, without relying on the

limit for that construction. With this purpose in mind, we start by defining the theoretical framework based on

Abstraction in Context. After that, a continuity notion that does not depend on limit is presented. Finally, the

results of a qualitative study performed with Public Management Quantitative Methods students, from the School

of Technology and Management of the Polytechnic Institute of Leiria, is presented. The epistemic actions that

were developed by the students while building the continuity notion are presented and analysed.

Key-Words: - abstraction in context, neighbourhood, limit, continuity

Received: July 23, 2021. Revised: March 14, 2022. Accepted: April 17, 2022. Published: May 5, 2022.

1 Introduction

In Portugal, the notion of continuity is relatively well

known by the majority of students entering higher ed-

ucation with Sciences or Economics basic formation

(Mathematics A). Those students usually have a solid

background in Mathematics, and therefore are not the

target of the present study. However, nowadays many

students ingress in scientific higher education courses

(like engineering degrees) coming from professional

schools, humanities courses, abroad courses or adult

courses (Mathematics B or below). The majority of

these last students need a preliminary course in Cal-

culus before trying to assist to more advanced classes

on Analysis, Algebra, Physics or Statistics. Even if

the intuitive perceptions of limit and continuity at one

point are reasonably apprehended by all the students,

from a conceptual point of view, in the passage to

the formal definition several issues might be detected.

These difficulties are related with symbolism, formal-

ism, graphic representations and with the notion of

approximation.

Commonly, the pre-Calculus course for students

with Mathematics B or below begins with a brief re-

view of functions (what is a function, properties, lin-

ear function, quadratic function) and after that lim-

its are introduced, usually without much success be-

cause limits notion is not very easy to understand. In

fact, studies on the difficulty of the concept of limit

abound in the literature. In [5] it is indicated that most

students cannot understand the concept of limit, and

therefore cannot calculate them. Moreover, it is in-

dicated that most of the difficulties later encountered

in concepts such as continuity, differentiation and in-

tegration come from difficulties with the concept and

calculation of limits (still on this subject, see for ex-

ample [14, 20]). Even when the technology was used

to aid the calculation of limits, the results were unsat-

isfactory, which may be related to the lack of a unique

strategy to calculate limits [13]. This last author also

mentions that for most students limit is seen as a dy-

namic process and almost as “approaching without

reaching”, and not as a concrete value. In [4] it is indi-

cated that, regardless of the limits being related to the

continuity of functions or with sequences and series,

their calculation and understanding is always compli-

cated. Finally, [5] states that there are no works pre-

senting effective strategies that allow overcoming the

difficulties felt with the understanding and calculation

of limits.

In this study, we intend to eliminate the depen-

dency on the notion of continuity in relation to the

notion of limit, starting by introducing the concept

of continuity and only later the limit. The notion

of a point’s neighbourhood will be the basis of this

approach. This perspective has the advantage that

the concept of continuous function at one point or

throughout its domain does not require, as a presup-

position, the understanding of the limit concept, com-

monly regarded as more difficult. This way, the con-

struction of knowledge by students can benefit. We

do not intend to diminish the importance of the limit

notion, but we believe that knowledge construction

and academic success will benefit form this swap.

After that, we study the epistemic actions that

surge throughout this process in order to understand

how the notion of continuity is built by the students.

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2022.19.11

Ana Valongo, Miguel Felgueiras

E-ISSN: 2224-3410

109

Volume 19, 2022

2 Abstraction in Context

Abstraction in Context (AiC) is a theoretical frame-

work for studying students’ processes of construct-

ing abstract mathematical knowledge as it occurs in

a context that includes specific mathematical and cur-

ricular components, as well as a particular learning

environment. Seminal works by [6, 9, 15], among

others, introduced this issue in the area of Mathe-

matics Didactic. Essentially, “Abstraction” is a the-

oretical activity that contemplates a set of tasks per-

formed by one or more individuals, motivated by a

given problem integrated in a “Context”, which en-

compasses the personal and social involvement of in-

dividuals. Currently, this methodology is used as a

theoretical support in works that seek to understand

the construction of mathematical knowledge, being

mentioned in several studies that seek to understand

how students reach concepts of various levels of diffi-

culty [10, 12]. In those studies, the teacher introduces

the subjects in various ways (practical problem, the-

oretical concept to deepen, notion to generalize) and

helps students while they build their own considera-

tions on the subject. It seeks to reflect the spirit of

the Bologna process, with the key aspect being the

autonomy of the student, accompanied by the con-

structive observations of the teacher. AiC as three

stages, namely necessity, emergence and consolida-

tion. The emergence process of new mathematical

knowledge is the central part of the AiC and it is usu-

ally analysed under the epistemic actions Recogniz-

ing (R-Action), Building-with (B-Action) and Con-

structing (C-Action). Together they form the RBC

model, to which we can join the Consolidation (Co).

These are the epistemic actions which compose the

RBC+Co model [9, 10, 12].

3 The Notion of Neighbourhood

We mentioned, in the Introduction, that there are ad-

vantages by introducing the notion of continuity be-

fore the notion of limit. It is now important to see how

this process can be implemented in practice. It should

be noted that this is not a new idea, even in the Por-

tuguese context. In [21], a support manual for mathe-

matics teachers of the 10.º grade, continuity is not in-

troduced with the limits. However, this perspective is

only available for the teachers and not for the students,

and is provided to teachers solely as “general culture”

information. Later, in the 12.º grade, students work on

continuity using the definitions of Cauchy and Heine

[22], but usually consider that a function is continuous

at a point abelonging to its domain when the limit in

that point is equal to the value of the function in that

point.

Many students justify a function continuity in a

given point referring only to the equality of the lat-

Figure 1: Continuity at x= 2 where lim

x→2f(x)6=

f(2) .

eral limits without mentioning the function value in

that point (see Figure 1).

But the main issue is the fact that in 12.º grade the

limit notion is based on accumulation points, mean-

ing that xcannot be equal to a, and therefore there is

no limit at isolated points (see Figure 2). Of course,

we can define the limit considering (or not) isolated

points, but this causes, in many cases, confusion

among the students.

Figure 2: Continuity at x= 4 where lim

x→4f(x)doesn’t

exist for limit notions based on accumulation points.

The following question arises: “How to introduce

the concept of continuity without using limits?”. Fol-

lowing [11] line of thought, we will base the concept

of continuity on the notion of a point’s neighbour-

hood. The notion of a point’s neighbourhood is easy

to understand because it can be seen as a symmetrical

range around a point (see Figure 3).

In a formal level, the notion of a point’s neigh-

bourhood will be defined as follows.

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2022.19.11

Ana Valongo, Miguel Felgueiras

E-ISSN: 2224-3410

110

Volume 19, 2022

Figure 3: Neighbourhood of point awith radius 

Definition 1 Neighbourhood at a point.

Let abe a real number and a positive real number.

The neighbourhood of ais the set of real numbers

whose distance to ais less then , that is,

Vε(a) = {x∈R:|x−a|< ε}=]a−ε;a+ε[(1)

Thus, students have to understand the behaviour of

a certain ffunction around the vicinity of a, which

seems to us a simpler and more elegant way of intro-

ducing the concept of continuity at a point, leading to

a set of advantages that will be tackled later.

4 The Concept of Continuity

The notion of continuity that is usually taught, both

in secondary and in higher education, is based on the

concept of limit. However, historically the notion of

continuity pre-date limit’s notion - at least in the per-

spective of a function “without holes” [23]. This au-

thor considers that the term limit, in the most cur-

rent sense of the term, was only defined in the nine-

teenth century, while the concept of continuity was

already well established in the eighteenth century. In

1748, Euler published “Introductio in Analysin In-

finitorum”, where most of the mathematical concepts

used nowadays were defined (of course, with a very

different notation and presentation). In that work,

Euler indicates that a continuous function is formed

by a single analytical expression, while discontinu-

ous functions are formed by more than one analytical

expression [18]. Naturally, this notion of continuous

function lacks, in the light of current knowledge, from

mathematical rigorous, and denotes some flaws. A

simple counter-example is when a function is defined

by branches but still is continuous throughout the do-

main. Nevertheless, Euler’s work was pioneer for the

time and had a great impact on the later development

of mathematics.

Therefore, and as we mentioned in the Introduc-

tion, there seems to be pedagogical advantages in

introducing continuity first, and and only after that

limit. In a simpler and more intuitive notion, the con-

cept of continuity based on the notion of a point’s

neighbourhood can be defined as [11].

Definition 2 Continuity at one point.

Let fbe a real function defined in D⊂Rand a∈D

a point in the domain. The function fis continuous

in aif

∀δ > 0,∃ε > 0 : ∀x∈R,

(x∈D∩Vε(a)⇒f(x)∈Vδ(f(a))) .(2)

Note that in this definition we do not remove afrom

its own neighbourhood, and therefore the function is

continuous at isolated points.

5 The Pedagogical Proposal

As already stated, we changed the teaching order of

the concepts of limit and continuity. Therefore, we

seek to verify whether the cutting of dependence on

the notion of continuity in relation to the notion of

limit leads to a better understanding of continuity.

Also, we seek to identify the epistemic actions that

arise during the abstraction process and that are rele-

vant for the construction of mathematical knowledge,

as well as the difficulties felt by the students, specially

at the level of the application of symbolic language

and interconnection of concepts. The research ques-

tions associated with the pedagogical proposal are as

follows:

• what epistemic actions are possible to identify,

during students abstraction process, related with

the constructing (C-Action) of new mathematical

knowledge, namely:

–while developing an understanding of the

problems;

–identify the need to use other mathematical

concepts or previous constructions;

–implement intermediate strategies and solu-

tions;

–organize knowledge and ideas;

–construct the concept of continuity;

• how do these epistemic actions are sequenced

and related?

Throughout an experimental study, we tried to an-

swer the previews questions. The final goal is to

present a detailed description of the entire knowledge

building system. The experimental study was im-

plemented in a Quantitative Methods curricular unity

(QM). This course belongs to the Public Management

degree from School of Technology and Management,

Polytechnic Institute of Leiria (ESTG). The students

who attend to QM usually did not have Mathemat-

ics A in high school, and therefore their mathemati-

cal background is similar to engineering students that

must attend to a preliminary course in Calculus (see

Introduction). Even more, QM has similar contents to

the preliminary course in Calculus in ESTG. For these

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2022.19.11

Ana Valongo, Miguel Felgueiras

E-ISSN: 2224-3410

111

Volume 19, 2022

reasons, we believe that the results can be extended to

the less prepared engineering students attending pre-

liminary curricular unity in Calculus.

6 Research Methodology and Data

Collect

The methodology adopted in this work is qualitative

and interpretive. According to [7], qualitative re-

search is fundamentally concerned with processes and

dynamics, and is dependent on the researcher. Bog-

dan and Biklen [1], in one of the best-known works

on this subject, state that the multiple ways of inter-

preting experiences depend on the relationships be-

tween the various actors in the learning process. In the

Portuguese context, [16, 17] studied the methodology

to apply in Mathematics classes. Thus, we sought to

understand the process of knowledge construction for

the involved students, through the analysis of the evo-

lution of their productions in class (oral participation,

resolution of exercises, etc.). Later, some case stud-

ies that seemed more important for understanding the

process of knowledge construction were analysed in

detail. About 15 students participated in the study,

but not all of them performed all the tasks. Students

that were repeating the course had one of the weekly

lessons partially overlapped, which conditioned atten-

dance. The collected data (concerning continuity no-

tion) was produced by students frequenting QM, dur-

ing two classes of 120 minutes each. In general, stu-

dents seemed rather motivated with the aims of the

study and did their best to participate.

At the end of the class that preceded the begin-

ning of the continuity study, the pedagogical pro-

posal was briefly presented to the QM students by

the researcher. Note that the researcher responsible

for the study implementation attended to the classes,

but was not the teacher, even if sometimes he helped

the teacher. This is a procedure already considered

has advantageous by some researchers [8], mainly be-

cause this way the researcher can try to collect the

“best” data and at the same time the teacher can strug-

gle for students’ success. After this introductory pro-

cedure, all students had to answer, individually and

in paper, to a first question. With the answers to that

question, we intended to identify the students’ prelim-

inary knowledge about the notion of continuity. In

subsequent classes the process was similar. The pro-

fessor distributed tasks, of increasing difficulty, about

the continuity of a function, both in a point or in its

domain. Each exercise was first projected onto the

board, without any major considerations being made

about its resolution. The students answered to the

questions on paper, and after that the answers were

discussed in class, being subject to correction after

the discussion phase. This iterative process sought

Table 1: Epistemic Actions for the RBC model

R-action

Interpret

Acquired structure

Neighbourhood

B-action

Strategies

Previous construction application

Intermediate solutions

Justification

C-action

Reorganization

Continuity at a pointLeft-continuity

Right-continuity

Continuity on the domain

Continuity on an interval

Communication

to contribute to the process of construction of mathe-

matical, abstract and advanced knowledge by the stu-

dents. After performing a set of tasks, the students

answered again to the initial question, trying to assess

whether or not they were able to understand the conti-

nuity notion. Moreover, we tried to understand if they

evolved in relation to their initial knowledge. Only

after this phase the formal notion of continuity was

debated, and later written in accordance with Defini-

tion 2.

Given that it would be very difficult to record all

the discussion that took place in class, when solv-

ing the proposed tasks, the lessons were recorded and

later viewed. All the appropriate authorizations were

obtained, and it was ensured that the recordings would

only be used for academic purposes. With this proce-

dure it was possible to collect a large amount of data,

since that information came with the written compo-

nent (individual answers to the questions) but also

with the oral component (video recording of the dis-

cussion of the problems).

Considering that it wouldn´t be possible to con-

sistently analyse the production of a large number of

respondents, two students were selected for detailed

analysis (AF and DV). These students were selected

considering that some students didn´t answer all the

proposed questions and, among those who answered

all the questions, some did it in an incipient way.

According with the RBC+Co model introduced in

previous questions, the categories and subcategories

concerning the epistemic model [19] can be seen in

Table 1. Note that Consolidation (Co) does not have

subcategories because it will only appear when apply-

ing continuity construction in other problems. Table 1

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2022.19.11

Ana Valongo, Miguel Felgueiras

E-ISSN: 2224-3410

112

Volume 19, 2022

Table 2: Epistemic actions for the R-action

Interpret

Piecewise function

Quadratic function

Acquired structure

Quadratic formula

Domain

Graphical representation

Neighbourhood

At a point

will be used as an artefact [2, 3, 24] by the researchers

since it is the primary tool to understand students pro-

duction. We will not analyse this table here, because

in the next section its utility will be highlighted. Note

that throughout the text references to the lines of Ta-

ble 1 will appear, but to avoid repetitions they will not

be explicit.

7 Results

In this section we focus our attention in one of the

questions that were presented during the classes. The

selected question is relevant since it deals with conti-

nuity in an isolated point but also in an interval, for a

function with different expressions in parts of its do-

main. Therefore, we will explain the epistemic ac-

tions that occur throughout the process.

Question Let fbe a real function of a real variable x

defined as

f(x) = 2, x = 3

x2−x−2, x ≤2.(3)

Verify if fis continuous on its domain.

In the remaining subsections, the transcriptions

from the students (AF and DV), professor (P) and re-

searcher (R) interventions are identified by letter and

number. For example, P3 refers to the third interven-

tion in the discussion, in this case performed by the

teacher.

7.1 R-action

The students began to Recognize that they were deal-

ing with a piecewise function, with a quadratic func-

tion in the second branch. The epistemic actions In-

terpret and Acquired Structure occur almost simulta-

neously, because students acknowledge the need of

graphical representation, that rely on quadratic for-

mula and domain. Clearly, there is a strong interlink

between these actions. Finally, the students needed to

recall Neighbourhood at a point in order to study con-

tinuity. In a nutshell, the epistemic actions and their

subcategories can be found in Table 2, with the class

discussion in Table 3.

A graphical representation from the relations be-

tween the different epistemic actions and class pro-

ductions can be seen in Figure 4. Figure 4 and re-

maining figures are at the end of the document in a

bigger size so that they can be properly read.

Table 3: Class discussion for the R-action

P1: for this function, what have you been doing?

AF2: calculating the zeros.

P3: using?

AF4: quadratic formula.

....

P7: and the vertex?

DV8: −b

2a, f −b

2a.

....

P15: after that, what can we do?

AF16: the graph!

R17: the students represent graphically ffunction.

P18: quadratic function is just when?

AF19: x≤2.

DV20: x= 3 is missing!

[R: referring to the domain].

....

P31: remember, for non continuity, what happens?

DV32: neighbourhood “catch” different values.

7.2 B-action

In the Building-with phase, students initial Strat-

egy was the graphical representation of the quadratic

function, that they identify as a parabola (Previous

construction application). To obtain the roots and ver-

tex, quadratic formula and vertex formula were used

as a preamble for graphical representation (x≤2).

The presented calculus are the Justification. We be-

lieve that the achieved graphical representation can be

seen as an Intermediate solution, where we can also

verify the Previous construction application for the

isolated point. After that, and from the graphic anal-

ysis (here seen as a Justification), students obtained

the function domain, that also represents an Interme-

diate solution. The written domain is another Justi-

fication. The epistemic actions for the Building-with

phase are summed up in Table 4, while student’s dis-

cussion about this subject is presented in Table 5.

A graphical representation from the relations be-

tween the different epistemic actions and class pro-

ductions can be seen in Figure 5.

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2022.19.11

Ana Valongo, Miguel Felgueiras

E-ISSN: 2224-3410

113

Volume 19, 2022

Table 4: Epistemic actions for the B-action

Strategies

Calculation of function roots

Calculation of parabola vertex

Graphical representation

Previous construction application

Use of quadratic formula

Use of vertex formula

Use of isolated point

Intermediate solutions

Obtained graphical representation

Obtained function domain

Justification

Written calculus for the function roots and vertex

Graphical representation

Written domain

Table 5: Class discussion for the B-action

P1: for this function, what have you been doing?

AF2: calculating the zeros.

....

P5: results are x=−1or x= 2,right?

AF6: yes!

P7: and the vertex?

....

P13: Ok! The results are?!

DV14: 0.5 and -2.25 [R: Referring to xand y].

....

R17: the students represent graphically ffunction.

....

P23: so, the domain is?

AF24: ]− ∞; 2] ∪ {3}.

7.3 C-action

The Constructing phase starts with the Reorganiza-

tion of previously obtained Intermediate solutions,

mainly Analysing obtained domain and Analysing ob-

tained graphical representation. The main issue, as

expected, occurred when dealing with Constructing

the Continuity on the domain (see AF26 in Table 7).

Only after teacher’s intervention (see P31 and subse-

quent interventions in Table 7) it was possible for the

students to Construct Continuity at an isolated point,

after the teacher had remembered Neighbourhood no-

tion. By the other hand, the Construction of Continu-

ity on an interval was easily achieved by the students

(see P28 and DV29 in Table 7). Therefore, Continu-

ity at an isolated point and Continuity on an interval

preceded Continuity on the domain, and a little “push”

from the teacher was required in order for the students

to achieve the late construction. Finally, Communi-

cation is only expressed when the students answer to

question, that is, when they write the domain (Table

5, P23 and AF 24) and say where the function is con-

tinuous (Table 7, DV37).

Table 6: Epistemic actions for the C-action

Reorganization

Analysing obtained graphical representation

Analysing obtained domain

Continuity on the domain

Continuity at an isolated point

Continuity on an interval

Communication

Table 7: Class discussion for the C-action

R25: Although most students identify the domain,

most of them does not answer to the question.

AF26: Professor! It isn’t continuous, right? It stops!

R27: AF refers to the “leap” from the point (2; 0) to

the point (3; 2), the isolated point.

P28: The main question is: this function is

continuous where?

DV29: From minus infinity until 2.

....

P31: remember, for non continuity, what happens?

DV32: neighbourhood ”catch” different values.

P33: right. When we calculate the neighbourhood,

we must obtain different images for non

continuity, ok? Here [R: referring to x= 2]

is there any continuity issue?

DV34: no.

P35: And here? [R: referring to x= 3]. Who

believes that it is continuous and who says

that it isn’t?

AF36: It is continuous! The neighbourhood is

empty and far from 3.

DV37: so it is continuous in the entire domain!

A graphical representation from the relations be-

tween the different epistemic actions and class pro-

ductions can be seen in Figure 6.

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2022.19.11

Ana Valongo, Miguel Felgueiras

E-ISSN: 2224-3410

114

Volume 19, 2022

8 Conclusion and Future Work

In this paper we studied the Construction of conti-

nuity concept from university students, using RBC

methodology as a theoretical framework. Consider-

ing the epistemic actions, we can conclude that the

abstraction process was always triggered by the R-

Action, when students recognized the need of using

previous constructions (quadratic and piecewise func-

tions, domain, neighbourhood, ...). After that, stu-

dents used the previous constructions in order to ob-

tain intermediate solutions (graphical representation

and domain). These epistemic actions belong to the

the B-action. Finally, the C-action was always ini-

tiated by the reorganization of the constructions de-

veloped in the B-action. Construction may not be

isolated since some constructions are related to each

other, in many cases promoting new constructions as-

sociated to the concept under study (that is, continu-

ity on an interval and continuity at an isolated point

promoted the construction on the domain). Because

traditional notion of continuity relies on the limit no-

tion, that is usually complicated to apprehend by the

students, we defined continuity in a coherent way us-

ing neighbourhood. In Figure 7 the reader can see,

in a schematic way, all the major relations between

R-action,B-action and C-action. Future work should

focus in consolidation (Co-Action). To perform that

task, different questions must be dealt to the students

to check if they can use continuity construction in an-

other context. Also, the bridge between limit and con-

tinuity using neighbourhood would also be an inter-

esting topic of study.

References:

[1] Bogdan, R; Biklen, S (1999). Qualitative re-

search in education. An introduction to theory

and methods. Porto: Porto Editora.

[2] Bussi, B; Corni, F; Mariani, C; Falcade, R (2012).

Semiotic Mediation in Mathematics and Physics

Classrooms: Artefacts and Signs after a Vygot-

skian Approach. The Eletronic Journal of Science

Education, 16(3), 1-28.

[3] Bussi, B; Mariotti, M (2008). Semiotic media-

tion in the mathematics classroom: Artefacts and

signs after a Vygotskian perspective. Handbook of

international research in mathematics education,

746–783. Nova York: Routledge.

[4] Cornu, B (1981). Quelques obstacles a l’appren-

tissage de la notion de limite. Recherches en Di-

dactique des Mathematiques, 4, 236-268.

[5] Cottrill, J; Dubinsky, E; Nichols, D; Schwingen-

dorf, K; Thomas, K; Vidakovic, D (1996). Un-

derstanding the Limit Concept: Beginning with a

Coordinated Process Schema. Journal of Mathe-

matical Behavior, 15, 167-192.

[6] Davydov, V (1990). Types of Generalization in

Instruction: Logical and Psychological Problems

in the Structuring of School. Soviet Studies in

Mathematics Education, 2 (Original work pub-

lished in 1972).

[7] Denzin, N (1990). Interpretive interactionism.

Newbury Park: Sage.

[8] Domingos, A (2003). Compreensão de Conceitos

MatemáticosAvançados – AMatemática no início

do Superior. PhD thesis in Mathematical Didatic.

Lisboa: UNL.

[9] Dreyfus, T (1991).Advanced mathematical think-

ing processes. Advanced Mathematical Thinking,

25-41. Dordrecht: Kluwer.

[10] Dreyfus, T (2012). Constructing Abstract Math-

ematical Knowledge in Context. 12th Interna-

tional Congress on Mathematical Education.

Seoul, Korea.

[11] Ferreira, J (2005). Introduction to Mathematical

Analysis. Lisbon: Gulbenkian Foundation.

[12] Hershkowitz, R; Dreyfus, T; Tabach, M (2012).

Exponential Growth - Constructing Knowledge

in the Classroom. 12th International Congress on

Mathematical Education. Seoul, Korea.

[13] Monaghan, J; Sun, S; Tall, D (2006). Construc-

tion of the Limit Concept with a Computer Al-

gebra System. Proceedings of the 3rd Interna-

tional Conference on the Teaching of Mathemat-

ics. Turkish Mathematical Society.

[14] Orton, A (1983). Students’ understanding of in-

tegration. Educational Studies in Mathematics,

14, 1-18.

[15] Piaget, J (1990). Genetic Epistemology. New

York: Columbia University Press.

[16] Pimenta, C (2016). A Construção do Conheci-

mento no Desenvolvimento do Pensamento Al-

gébrico. PhD thesis in Mathematical Didatic.

Covilhã: UBI.

[17] Ponte, J (2008). A investigação em educação

matemática em Portugal: Realizações e perspec-

tivas. Investigación en educación matemática

XII, 55-78.

[18] Santana, G; Otte, M (2010). The Conceptions of

Euler and Cauchy for the concept of continuous

function from the perspective of Pierre Bourtoux.

X National Meeting of Mathematics Education,

Salvador, Brazil.

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2022.19.11

Ana Valongo, Miguel Felgueiras

E-ISSN: 2224-3410

115

Volume 19, 2022

[19] Santos, A (2018). Construindo os conceitos de

Continuidade e de Limite – Uma experiência de

ensino com estudantes do ensino superior politéc-

nico. PhD thesis in Mathematical Didatic. Cov-

ilhã: UBI.

[20] Tall, D (1992). The transition to advanced math-

ematical thinking: Functions, limits, infinity

and proof. Handbook of research on mathemat-

ics teaching and learning, 495-511. New York:

Macmillan.

[21] Teixeira, P; Precatado, A; Albuquerque, C.; An-

tunes, C; Nápoles, S (1997). Funções, 10.º ano de

escolaridade. Ministério da Educação.

[22] Teixeira, P; Precatado, A; Albuquerque, C; An-

tunes, C; Nápoles, S (1999). Funções, 12.º ano de

escolaridade. Ministério da Educação.

[23] Thomas, G; Finney, R; Weir, M; Giordano, F

(2002). Thomas’Calculus. Boston: Addison Wes-

ley.

[24] Vygotsky, L (1978). Mind in Society. Cam-

bridge: Harvard University Press.

Contribution of individual authors to

the creation of a scientific article

(ghostwriting policy)

Ana Santos was the main researcher and conducted

the pedagogical experience.

Miguel Felgueiras was the class teacher and devel-

oped the class questions.

Ana Santos designed the figures presented in this text.

Miguel Felgueiras was the main writer of this article.

Both authors read the manuscript.

Sources of funding for research

presented in a scientific article or

scientific article itself

This work is partially financed by national funds

through FCT: Fundação para a Ciência e a Tecnolo-

gia under the project UIDB/00006/2020.

Creative Commons Attribution

License 4.0 (Attribution 4.0

International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2022.19.11

Ana Valongo, Miguel Felgueiras

E-ISSN: 2224-3410

116

Volume 19, 2022

Figure 4: R-Action

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2022.19.11

Ana Valongo, Miguel Felgueiras

E-ISSN: 2224-3410

117

Volume 19, 2022

Figure 5: B-Action

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2022.19.11

Ana Valongo, Miguel Felgueiras

E-ISSN: 2224-3410

118

Volume 19, 2022

Figure 6: C-Action

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2022.19.11

Ana Valongo, Miguel Felgueiras

E-ISSN: 2224-3410

119

Volume 19, 2022

Figure 7: Relations between the epistemic actions.

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2022.19.11

Ana Valongo, Miguel Felgueiras

E-ISSN: 2224-3410

120

Volume 19, 2022