1) The document discusses a study that examines secondary students' understanding of the concept of function across three dimensions: definition, representation (recognition and interpretation), and problem solving. 2) Confirmatory factor analysis confirmed these four dimensions comprise the conceptual understanding of functions. The ability to define the concept was found to influence performance on the other dimensions. 3) The study aims to verify the proposed theoretical structure of conceptual understanding of functions and examine how the dimensions are interrelated, as well as identify differences in student performance at different grade levels.

1. A Structural Model Related to the Understanding
of the Concept of Function: Definition and Problem
Solving
Areti Panaoura1
& Paraskevi Michael-Chrysanthou2
&
Athanasios Gagatsis2
& Iliada Elia2
& Andreas Philippou2
Received: 15 July 2015 /Accepted: 24 January 2016
# Ministry of Science and Technology, Taiwan 2016
Abstract This article focuses on exploring students’ understanding of the concept
of function concerning three main aspects: secondary students’ ability to (1) define
the concept of function and present examples of functions, (2) solve tasks which
asked them to recognize and interpret the concept of function presented in different
forms of representation, and (3) solve function problems. Confirmatory factor
analysis verified 4 dimensions comprising the conceptual understanding of func-
tions: definition, recognition, interpretation, and problem solving. Furthermore, the
important role of the ability to define the concept on the rest abilities was revealed,
leading to important didactic implications.
Keywords Definition . Functions . Interpretation . Problem-solving . Recognition
Introduction
For more than 20 years, the concept of function has been internationally considered
as a unifying theme in mathematics curricula (Steele, Hillen & Smith, 2013), while
students face many instructional obstacles when developing understanding of
functions (Sajka, 2003; Sierpinska, 1992). Kieran (1992) questions whether stu-
dents’ inability of conceptually understanding functions is related to its teaching or
is due to students’ inappropriate way of approaching function tasks. Accepting the
main constructive value that everything can be learnt by using the appropriate
teaching methods in specific ages, the present study concentrated on the students’
Int J of Sci and Math Educ
DOI 10.1007/s10763-016-9714-1
* Areti Panaoura
pre.pm@frederick.ac.cy
1
Frederick University, Nicosia, Cyprus
2
University of Cyprus, Nicosia, Cyprus

2. understanding of functions at two different educational levels. Sajka (2003) indicates
that students’ abilities in solving tasks involving functions are influenced by the typical
nature of school tasks, leading to the use of standard procedures. Students experience
difficulties in dealing with the concept of function when using textual, algebraic, and
graphical representations involving daily life situation in problem solving (Okur, 2013)
as, according to Thomas (2008), representational versatility consists of realizing
relationships between representations of the same concept and performing conceptual
and procedural interactions with each representation.
In relation to the above, our study examines students’ conceptions of function, as it
is one of the most important topics of the curriculum (Sanchez & Llinares, 2003). In
defining conceptions, we take Thompson’s (1992) definition, who invoked conceptions
as a more general mental structure, encompassing beliefs, meanings, concepts, prop-
ositions, rules, mental images, preferences, and the like^ (p. 130). Research indicates
representational challenges evidenced by students when dealing with functions (Cho &
Moore-Russo, 2014). Thus, teachers need to understand the images which have been
created in students’ mind about the specific concept as internal representations of the
concept and they need to know how mathematical ideas can be represented in order to
facilitate students’ understanding.
Aiming to define the different aspects of the understanding of functions, the focus of
this study is to examine secondary students’ understanding of the concept, in relation to
the use of multiple representations. In fact, our research is organized according to the
following research objectives:
1. Confirm the theoretical structure of the conceptual understanding of function (in
gymnasium and lyceum) which is proposed at the present study.
2. Examine how the different dimensions which are used in the structure of the
conceptual understanding of function are interrelated.
3. Identify the differences in students’ performance at the third, fourth, and fifth
grades of secondary education.
The rationality of the study is related with the examination of different theoretical
dimensions of the understanding of the concept in two different education levels
(gymnasium and lyceum) in order to be able to pose suggestions about the teaching
of the concept: (a) definition, as a verbal description of the concept image (Vinner &
Dreyfus, 1989) and (b) conversion of representations (recognition and interpretation) as
a fundamental process leading to the understanding (Duval, 2002) and the problem
solving as the implementation of the ideas into practice.
Theoretical Framework
The Concept of Function in Mathematics Education Research
Our longstanding work with teacher education and the difficulties students face at
the beginning of their studies has stimulated our curiosity on how students in
secondary education develop concepts in mathematics. Students’ great challenges
when entering university mathematics are reported by national and international
A. Panaoura et al.

3. studies (i.e. Jayakody & Sedaghatjou, 2014). Recently (in August 2015), these
challenges and difficulties were the topic of a session in EARLI 2015
Conference, which was dedicated to study prerequisites and learning activities in
the transition to mathematics university study.
The US Common Core State Standards for Mathematics (2010) states that high
school students should be able to (a) create functions that model relationships between
two quantities, (b) analyze and employ functions using different representations, and (c)
interpret functions for applications in terms of the context of the situation. In Cyprus,
the concept of function is one of the mathematical concepts which are first introduced
at the second grade of secondary education and has a central role at the curriculum for
the following years. However, the understanding of functions does not appear to be
easy (Tall, 1991) while students face many obstacles trying to understand the specific
concept (Sajka, 2003).
Doorman, Drijvers, Gravemeljer, Boon and Reed (2012) claim that functions
have different faces and making students perceive them as faces of the same
mathematical concept is a pedagogical and research challenge. At the same time,
we have to be focused on the extent to which students are able to succeed in
utilizing their knowledge and skills in situations other than the context in which
they were taught (Van Streun, Harskamp & Suhre, 2000), in order to talk about the
conceptual understanding of the concept.
The Role of Mathematical Definitions into the Conceptual Understanding
In mathematics, definition plays an extremely important role for the construction of
a mathematical concept. The interest about the definition of mathematical concepts
arises from the commonly observed difficulties met by students entering advanced
levels of study (Morgan, 2005). The way one considers definitions depends on the
view he/she has about the mathematical experience, and then the view about proof
(Ouvrier-Buffet, 2010). Tirosh (1999) suggests that mathematics rely heavily on
primary notions, axioms and definitions due to its deductive nature. In fact, Tirosh
(1999) indicated that BA definition of a concept captures and synthesizes its essence
and provides tools for discriminating between instances and non-instances of the
concept^ (p. 341). The ways in which definitions appear in school mathematics
vary significantly with the type of mathematics involved and with the age of the
intended student (Morgan, 2013), starting from informal situations to more formal.
In higher education, students often are asked to memorize the definition (even if it is
not understandable for them) in the course and they are given credit in examinations
for being able to repeat it (Ewards & Ward, 2008). Freudenthal (1973) criticized the
practice of presenting definitions of geometric concepts to students, as it Bkilled^
the chance to participate in the entire activity from the starting point of the
development of a concept to the finish.
Vinner and Hershkowitz (1980) distinguished between the terms Bconcept image^
and Bconcept definition^ in order to describe individual’s conceptions of concepts. In
Tall and Vinner’s (1981) work, the construct of concept image is defined as Bthe total
cognitive structure that is associated with the concept^ (p. 152) and consists of mental
pictures, processes as well as properties evoked in our mind when we think of a
concept. On the other hand, the notion of concept definition refers to Ba form of words
Function’s definition, representation and problem solving

4. used to specify that concept^ (Tall & Vinner, 1981, p. 152) and may be grasped by a
learner in a rote or meaningful manner. A concept definition being formulated by an
individual comprises a Bpersonal concept definition.^ This kind of concept definition
may differ from a formal one (Vinner, 1983), which is given to an individual as an
aspect of a mathematical theory and is broadly accepted by the mathematical commu-
nity. For example, in the context of mathematical functions, the concept definition may
be as follows: Ba relation between two sets A and B in which each element of A is
related to precisely one element in B^ (Tall & Vinner, 1981, p. 153). For some learners,
this concept definition may stimulate a concept image of a rule or formula. For others, a
graph or a table of values may serve as the concept image that corresponds to this
concept definition.
Furthermore, many scholars have used the term Bpotential conflict factor^ to declare
that a part of a concept image or concept definition may conflict with another part of
this concept image or concept definition (Tall, 1988). According to Fischbein (1978),
two contradictory conceptions may exist at the same time with the learner being rather
unconscious of this conflict. Vinner and Dreyfus (1989) indicated that students may not
connect a formal definition to their mental images and they may use partially the
concept image with a consequent not satisfactory conceptual understanding. Students
base their solutions at problem solving tasks on the concept definition which may be
correct or incomplete. Rasslan and Vinner (1998) studied Israeli students’ concept
definitions and images of the function. Most of them could state the definition, but only
few of them applied the definition successfully.
Mathematical definitions are relevant for research in mathematics education in
general (Mosvoldn & Fauskanger, 2013) and we believe that it could be used as mirror
of the understanding, as students’ concepts are not Bopen^ to direct study by the
researcher. At the present study, we consider the presentation of a definition by a
student as a statement given in order to know which of the main elements of the
constructed concept image are.
The Role of Different Representations into the Conceptual Understanding
BConceptual understanding refers to understanding the relations between units of
knowledge in a domain and of the principles that govern a domain^ (Larsson, 2013,
p. 52). The abstract nature of mathematics underlines the necessity to examine the
mathematical conceptions which can be assessed only through their representations
(Kilpatrick, Swafford & Findell, 2001). According to the results of several researches
(i.e. Dufour-Janvier, Bednarz, & Belanger, 1987), although the students can produce
and use the mathematical representation on demand, they do not have the attitude of
turning to these as tools to help them solve problems. An important aspect of knowl-
edge about a mathematical concept is the different ways of approaching it due to the
different forms of representing it. Recent research (i.e. Deliyianni, Gagatsis, Elia &
Panaoura, 2015) reports that students’ representational flexibility and problem-solving
abilities are found to be major components of students’ representational thinking.
Therefore, teachers can evaluate whether students understand mathematical ideas or
not by examining the representational models students choose to use (Lamon, 2001).
The concept of function admits a variety of representations, while several represen-
tations of the concept offers information about particular aspects of the concept without
A. Panaoura et al.

5. being able to describe it completely (Gagatsis & Shiakalli, 2004). Lack of competence
in coordinating multiple representations of the same concept can be seen as an
indication for the existence of compartmentalization, which may result in inconsis-
tencies and delay in mathematics learning at school. The particular phenomenon reveals
a cognitive difficulty that arises from the need to accomplish flexible and competent
translation back and forth between different kinds of mathematical representations
(Duval, 2002). Translating (or converting) a semiotic representation into another one
cannot be considered either as an encoding or a treatment. A translation (conversion)
involves two modes of representation. From the modes equation and graph (used also
in this study) we have the translations Bgraph to equation^ and Bequation to graph^
(Janvier, 1987). Janvier (1987) used a 4 × 4 table in order to explain the translation
processes between different representations of a situation. He included the translations
among tables, graphs, formulas, and verbal descriptions. The translation processes are
developed effectively if the students are asked to make translations from the source to
the target and vice versa in a symmetrical way.
Yerushalmy (1997) claims that most teaching approaches do not take into
consideration the passage from one type of representation to another which is a
complex process and relates to the generalization of the concept at hand. For
example, to deal with functions point wise means to plot and deal with discrete
points of a function either because we are interested in some specific points or
because the function is defined on a discrete set. There are times that we need to
consider the function in a global way, for example, when we want to sketch the
graph of a function given in symbolic form, or when we want to find a function
which is defined on the real numbers.
Sierpinska (1992) indicated that students have difficulties in interpreting graphs and
manipulating symbols related to functions. Sfard (1992) found that students are unable
to bridge the algebraic and graphical representations of functions. Makonye (2014)
discusses how the notion of a function can be developed through the use of different
representations and models. Some students’ difficulties in the construction of concepts
are linked to the restriction of representations when teaching (Elia & Spyrou, 2006).
The Concept of Function in the Mathematics Textbooks
The textbook is a source of potential learning and it expresses what has been called the
intended curriculum. Mesa (2004) investigated the conceptions of function enacted by
problems and exercises in 35 mathematics textbooks from 18 countries participating in
the third International Mathematics and Science Study. Cyprus was not included in this
analysis because of researcher’s difficulties with the language. In general, five concep-
tions of function were identified as promoted in the textbooks: symbolic rule, ordered
pair, social data, physical phenomena, and controlling image.
We present a brief overview of the school mathematics textbooks in Cyprus,
regarding their contents for teaching the concept of function at gymnasium and lyceum,
without trying to use any content analysis similar to the work of Mesa (2004). In the
first and second grade of gymnasium, the concept of variable is included in algebra
teaching. It is used for defining and solving equations with one variable (most often
referred to as the Bunknown^). The definition of function is presented at the second
grade by the following up way:
Function’s definition, representation and problem solving

6. Function f is a process (rule) from a set A to a set B, in which each element from
set A matches in one exactly element of set B. Set A is the said field of definition
or domain of the function f.
The relation is usually represented by the small Latin letters: f,g,h, etc. If a function f
from the set A to the set B then x∈A (x belongs in the set A) matches to y∈B, then we
write: y=f(x) and read «y equal to f of x». The set, that has all the elements of f(x) for all
x∈A, is the said range set of the values of f and symbolizes with f(A).
Then, the concept of the function appears in the third grade of gymnasium in the
eighth section of the mathematics textbook, entitled BGraphs.^ Teachers are instructed
to complete this section within 10 teaching periods (a teaching period endures 40 min).
This section’s contents are the following: (a) rectangular system of axes, (b) graph of
function with formula y=ax+b, (c) graphical solution of the equation ax+b=0, (d)
graph of function y=ax, (e) special cases, (f) gradient coefficient (slope of a line), and
(g) parallel lines. The next section (nine teaching periods are suggested) entitled
BEquations^ involves solving second-degree equations with two variables by various
methods starting with the graphical method.
In the textbook for the fourth grade of lyceum, the first sections revise the concepts
about graphs that were taught in the third grade of gymnasium. In the BResults^ section
entitled BFunctions,^ the meaning of function is discussed. This section (16 teaching
hours) included the definition of function, the concept of graph, the concepts of
domain, range, and set of values and then methodology for the graphical representation
of functions of the form f(x)=ax+b, f(x)=ax2
+bx+c and f x
ð Þ ¼ a
x is developed. At
the end of this section, a large number of exercises using a variety of representations are
found. In the fifth grade of lyceum of the common core, the contents of the fourth grade
are repeated in a more simplified form, such as linear function graphic representations
and graphic solution of systems and then the verbal and symbolic definition of function
is given and concepts like equality functions, functions, operations, 1-1 functions,
inverse function, and composition of functions, limits, and continuity of functions are
developed. For this section, 16 teaching periods are provided.
Methodology
Participants
The participants of the study were 756 students of gymnasium and lyceum from
different schools in Cyprus. Specifically, there were 315 students attending the third
grade of gymnasium (15 years old), 258 students attending the fourth grade of lyceum
(16 years old), and 183 students attending the fifth grade of lyceum (17 years old). At
the educational system of Cyprus, the first, second, and third grades of secondary
education constitute the gymnasium level, while the lyceum level is consisted of the
fourth, fifth, and sixth grades. We assumed that the samples were from comparable
cognitive level of students because all the schools were urban and there were not any
comparative national examinations at Cyprus which could enable us to discriminate the
schools’ level. There was not any randomized procedure to select the sample; on the
contrary, mathematics teachers accepted voluntarily to spend the necessary time for
A. Panaoura et al.

7. their students’ participation at the study. A limitation of the study was our inability to
control the teaching method which was used for the specific concept. However, in
Cyprus, there is a common curriculum, one common textbook for students, and
teachers are receiving the same instructions and in-service training for the teaching
methods they have to follow.
Research Instruments
A test was developed and administered to students which consisted of different types of
items. To eliminate the factor of students’ tiredness, the test consisted of two parts, in
order to be completed within two different teaching periods. There were not any
qualitative differences between the two parts of the test. We had asked mathematics
teachers to evaluate the construct validity of the instrument in order to be sure the
mathematical content of the tasks would be suitable for all the groups of students.
Initially, the first part of the final test was administered and in a span of a week, the
second part was given to the students. Each part of the test was administered in a 45-
min session, by their teachers. The correct answers to the tasks were validated by
mathematics professors. A list of guidelines about the correct answers and the scoring
of the test were provided to the researchers involved in the correction of the test.
Correct and wrong answers were scored by the researchers as 1 and 0, respectively. No
answers were presented as missing cases.
The tasks focused on defining and explaining the concept of function, recognizing,
manipulating, and translating functions from one representation to another (algebraic,
verbal, and graphical) and solving problems. The tasks were divided into four groups:
(1) definition, (2) recognition, (3) interpretation, and (4) problem solving. In fact, the
first test asked students to (i) present a definition of function and an example, (ii)
explain their procedure for recognizing that a graph does not represent a function and
present a non-example of function, (iii) write the symbolic representation of six verbal
expressions (e.g. Bthe area E of a square in relation to its side^), (iv) draw a graph to
solve a problem, (v) recognize graphs of functions, (vi) explain a graph in terms of the
context, and (vii) examine whether symbolic expressions and graphs represent func-
tions. In the second test, students had to (i) present their procedure for examining
whether a graph represents a function, (ii) draw graphs of given functions, (iii)
recognize graphs of given verbal or symbolic expressions, and (iv) write the symbolic
equations of given graphs. The reliability for the total of the items on both tests was
high (Cronbach’s alpha = 0.868).
The tasks of the first group (definition) are about defining and explaining the
concept of function. These tasks were divided in four subgroups according to the
type of the definition or the type of example of function that was asked. The two
first groups concern the type of definition, whereas the last two groups are about the
presentation of examples or non-examples of functions. For the definitions, we set a
group named as Bdefinition in action^ based on the term of Vergnaud (1994). He
defines theorems-in-action as Ba proposition that is held to be true by the individual
subject for a certain range of the situation variables. It follows from this definition
that the scope of validity of a theorem-in-action can be different from the real
theorem, as science would see it. It also follows that a theorem-in-action can be
false. But at least it can be true or false, which is not the case for concepts-in-
Function’s definition, representation and problem solving

8. action^ (p. 225). In our tasks, Bdefinition in action^ refers to theorems and/or
properties used by students, which are adjusted to the way the students recognize
or not whether a graph represents function or not. In this case, students apply in
action the theorems and/or properties in a procedural way and recognize success-
fully whether a graph shows a function or not, without necessarily relating the
formal definition of function.
There were five tasks in this first group (A1, A2, A3, B1, B2). In all these tasks,
besides task A1 which is given in a verbal form, different types of representation are
involved related to the concept of function. Task A2 asks students to express how they
can recognize that a graph defines a function. It is anticipated that students will try to
draw some graphs to test whether they represent functions or not. On the contrary, they
may express the rule of the vertical line, having a mental image of a graph in which a
vertical line intersects the graph in one point (and then decide it is function), or two or
more points (and decide that it is not function). Question A3 is mostly expressed in a
symbolic form, accompanied by few verbal expressions. Thus, it is also related to
different modes of representation. On the other hand, questions B1 and B2 are
complementary to task A. In fact, in B1, students are asked to express a definition
about how we can decide that a graph represents function. Consequently, students are
expected to give an answer either in a symbolic form or possibly by drawing a graph.
Task B2 asks for an example of a relation that is function and an example of a relation
that is not function. In this task, we are sure enough that students’ answers will be
provided in a symbolic form, without excluding the possibility of the students having a
mental image of a graph in their minds.
In the second group (recognition) of tasks, students’ ability in recognizing functions
in different types of representations was examined. Actually, students were asked to
recognize functions either given in a symbolic (task A10) or a graphic form (tasks A7,
A12, and A13). The third group (interpretation) of tasks was about manipulating and
interpreting functions through different types of representations. In these tasks, students
are mainly given a verbal description of a graph. Students have to interpret these graphs
and find which one is in line to the given verbal description. Thus, what characterizes
this category of task is the articulation between graphical and verbal representations
(A4, A8, B4, B5, and B7). These tasks were grouped according to the data they contain
(arithmetic or not). The following example falls in the category of not arithmetic data,
as in this task, students have to choose the graph which is in line with the given verbal
description that contains no arithmetic values: BThere was a fire in a forest. At the
beginning the fire was slowly burning the forest. Then, the fire flared up and burned the
forest much faster. Then the fire was extinguished. Which of the three presented graphs
describe this situation?^
The tasks in the fourth group (problem solving) asked students to solve three
verbal function problems (A6, B6, and B8), which provided some arithmetic
elements according to which students have to construct the algebraic model of
function and then the corresponding graph. An example is the following: Costas has
20 € and spends 1 € per day. His sister has 15 € and spends 0.5 € per day. (i) Find the
function expressing the amount of money (y) each person will have in relation to the
number of days (x). (ii) Design the graph showing this function for each person. In
how many days will the two brothers have the same amount of money? What will
this amount of money be?
A. Panaoura et al.

9. Statistical Analysis
Primarily, in order to confirm the structure of students’ cognitive abilities in the concept
of functions at secondary education, a confirmatory factor analysis (CFA) model was
constructed using the Bentler’s (1995) EQS program. The model included all the tasks.
The tenability of a model can be determined by using the following measures of
goodness of fit: x2
/ df < 1.95, CFI > 0.9, and root mean square error of
approximation (RMSEA) < 0.06. At first, the emphasis was on confirming a second-
order factor model which would present students’ understanding on the concept of
function (research objective 1) and then a second model indicating the interrelations
among the factors at different educational levels (research objective 2). Secondly, for
the descriptive analysis of the collected data based on students’ performance in the
tests, the success percentages were accounted for each item by using the statistical
software SPSS. Additionally analysis of variance (ANOVA) was conducted in order to
examine the differences (research objective 3) at the performance between the three
different groups of students (third, fourth, and fifth grade).
Results
The Structural Organization of the Conceptual Understanding of Function
In order to confirm the structure of students’ cognitive abilities in the concept of
function at secondary education, a confirmatory factor analysis (CFA) model was
constructed. We assume that this model is adequate for representing the conceptual
understanding of functions taking into the account the important role of definition in the
development of the concept and the role of representational flexibility when using and
coordinating the multiple representations of the concept of function.
The second-order model (Fig. 1) that is considered appropriate for interpreting
students’ abilities involves four first-order factors. The first factor (F1) expresses
students’ ability to define functions either in a formal or an informal way and their
ability to present examples in order to explain the concept by using verbal, diagram-
matic or symbolic representations. The second factor (F2) consists of their ability to
recognize function into different forms of representations (graphical, algebraic), while
the third factor (F3) consists of their ability to interpret (and translate) the concept from
one type of representation to another. The fourth factor (F4) stands for the students’
problem solving abilities.
Those four first-order factors are regressed on a second-order factor expressing the
students’ understanding of the concept. The same structure was confirmed both for
gymnasium and lyceum (gymnasium: x2
/ df = 137.506/86 = 1.59, CFI = 0.975, and
RMSEA = 0.031 and lyceum: x2
/ df = 149.038/92 = 1.619, CFI = 0.973, and
RMSEA = 0.031), but with different factor loadings (in the figure, the first number
in each case is for the gymnasium and the second for lyceum). When the loadings are
higher, the relations between the variables that constitute a factor are stronger. It is
important to note that the model consists of four first-order factors whose loadings are
above 0.400. The loadings of the first-order factors on the second-order factor are
extremely high (0.830 – 0.997), indicating that all the factors are important dimensions
Function’s definition, representation and problem solving

10. of the development of the concept. Interpretation of the concept and problem solving
are more strongly regressed on the second-order factor than the other two dimensions.
Moving a step forward, we aimed to explore the way these four dimensions of
the model were interrelated. Thus, a second confirmatory factor analysis was
performed for tracing the statistically significant interrelations among these dimen-
sions. All the interrelations among the four second-order factors were statistically
significant. However, there were important differences in respect to the loadings of
these relations (Fig. 2).
Function
Conceptual
Understanding
F1: Definition
.830 , .846
F3: Interpretation
F2: Recognition
F4: Problem Solving
A1
Α2
Β1
Β2
Α3
Α4
Α8
Β4
Β5
Β7
Α7
Α13
Α10
Α12
Α6
Β6
Β8
.538 , .536
.732 , .728
.930 , .986
.626 , .629
.454 , .541
.669 , .657
.371 , .429
.514 , .469
.550 , .637
.508 , .648
.579 , .576
.429 , .544
.544 , .540
.486 , .486
.580 , .571
.480 , .580
.470 , .600
.912 , .926
.997 , .995
.381 , .470
Fig. 1 Structural organization of the conceptual understanding of function
A. Panaoura et al.

11. F1: Definition
F3: Interpretation
F2: Recognition
F4: Problem Solving
A1
A2
B1
B2
A3
A4
A8
B4
B5
B7
A7
A13
A10
A12
A6
B6
B8
.513 , .548
.708 , .738
.626 , .627
.459 , .558
.676 , .665
.429 , .528
.471 , .475
.544 , .634
.493 , .446
.562 , .577
.421 , .650
.563 , .615
.495 , .654
.587 , .554
.456 , .480
.577 , .560
.381 , .477
.841 , .821
.926 , .848
.779 , .779
.945 , .987
.907 , .868
.847 , .731
Fig. 2 Interrelations between the dimensions of the conceptual understanding of function
Function’s definition, representation and problem solving

12. The second model was confirmed for gymnasium and lyceum as well (gymnasium:
x2
/ df = 307.677/168 = 1.83, CFI = 0.957, and RMSEA = 0.036 and lyceum: x2
/
df = 333.814/169 = 1.975, CFI = 0.954, and RMSEA = 0.039). The loadings of the
tasks on the first-order factors still remain higher than 0.400 in almost all cases. The
interrelations are statistically significant in both levels of education. Those interrela-
tions are higher in the cases of gymnasium (with only one exception). The interrelations
between the factors show that students who are able to define the concept of function
are also able to interpret the concept (gym = 0.841, lyc= 0.821), to recognize the
concept (gym= 0.779, lyc= 0.779) and to solve problems (gym= 0.847, lyc= 0.731).
Thus, students’ ability to define a concept acts as a predominant factor which controls
or influences positively other dimensions of the understanding of the concept. The
inter-individual differences among students in respect to their ability to define the
concept remain the same concerning their ability to recognize and interpret the concept
and to their ability to solve problems.
Similarly, the ability to interpret the concept is related with students’ ability to
recognize the concept (gym= 0.926, lyc= 0.848) and their ability to solve problems
(gym= 0.945, lyc= 0.987). Finally, the ability to recognize the concept is related
significantly with their ability to solve problems (gym = 0.847, lyc = 0.731).
Obviously, problem solving is the most difficult activity for students, as the rest of
the dimensions of the model (define, interpret, and recognize) act as presuppositions for
the successful solution of problems on functions, in both levels of education.
Descriptive Analysis of Students’ Performance in the Definition Tasks
Results of the descriptive analysis on students’ performance at the task indicated that from
the total sample, only 159 students were able to present a correct definition of the concept,
242 presented a wrong definition, while most of them (358 students) did not give any
answer. Most of the students who presented a correct definition (Table 1) and present an
example in order to explain the definition of a function were at the fourth and fifth grade.
Only 401 of the students presented an example, and of these, 323 were correct.
Students were also asked to explain their procedure for identifying a graph that does not
represent a function and provide a relation that does not represent function and addition-
ally, they were asked to explain their procedure for identifying a graph that represents a
function and to provide an example of a function. Table 2 indicates the percentages of
correct answers for the specific tasks. In all cases, the results were higher in the fifth grade,
as it was expected, but there were especially negative results for students in the third grade,
as they did not give an answer (missing cases for the four tasks: 197, 212, 268, and 279
from 315) and many of those who answered presented a wrong procedure.
Table 1 Students’ correct definitions and examples
Grade Correct definition (n = 159) (%) Correct example (n = 401) (%)
3rd 8.2 23.0
4th 58.5 57.0
5th 32.7 55.7
A. Panaoura et al.

13. The comparison of the percentages at the four tasks indicates that it was easier for
students to present an example of a function rather than an example of a relation which
was not a function and they were more able, in all cases, to present an example rather
than to explain verbally the cognitive procedure they had to follow. However, even in
lyceum, more than half of the students were not able, in all cases, to describe
mathematically the procedure with the stages they had to follow in order to decide
whether a graph presented a function.
Students’ Performance for Each Dimension of the Concept Understanding
One of the objectives of the study was to identify statistically significant differences
(p < 0.05) concerning the concept of function in respect to students’ grade level in order
to investigate the developmental aspect of the notion’s understanding. ANOVA analysis
was used for comparing students’ means in presenting the definition of the concept and
giving an example of a function in respect to the categorical variable of their grade. In
the first case, Scheffé analysis indicated that there was a statistically significant
difference (F2,397 = 33.396, p < 0.01) between the students at the third grade with
the students at the fourth and fifth grade (x3 = 0.11, x4 = 0.53, x5 = 0.48). Although the
highest mean was at the fourth grade, there was not any statistically significant
difference between the two grades at the lyceum. Concerning the presentation of an
example in order to explain the definition of the concept, the difference was statistically
significant (F2,398 = 5.896, p < 0.01) only between the third and the fourth grade
(x3 = 0.72, x4 = 0.88, x5 = 0.78). The same analysis was conducted concerning
students’ ability to describe a procedure for recognizing an equation which did not
represent a function, in respect to grade level. There were statistically significant
differences between the third grade and the two other grades (F2,292 = 33.510,
p < 0.01) with a large difference between the means (x3 = 0.12, x4 = 0.47, x5 = 0.55).
All the items were grouped in order to be able to further analyze students’ perfor-
mance concerning the specific aspects which were the main interest of this study: (i)
propose definition, (ii) present examples to explain a concept or a procedure, (iii)
recognition of the concept, (iv) interpretation and translation of the concept from one
representation to another, and (v) problem solving tasks. Table 3 presents the means
and standard deviations of students’ performance on these specific dimensions.
Then, ANOVA analysis was conducted for examining the statistically significant
differences for each of the above aspects of a conceptual understanding of function in
respect to the three grade levels. Statistically significant differences (p < 0.05) were
found only concerning three of the dimensions. There was a statistically significant
Table 2 Percentages of students’ correct answers at specific tasks
Tasks 3rd 4th 5th Total
Procedure indicating that a verbal expression or an equation is not function 2.2 38.4 42.0 24.3
Procedure indicating that an expression is function 2.5 32.6 47.5 23.7
An example of a function 19.4 50.4 55.7 38.6
A non-example of a function 3.8 38.4 38.3 23.9
Function’s definition, representation and problem solving

14. difference (F2,182 = 7.678, p < 0.01) concerning the proposed definition of the
concept. The Scheffé analysis indicated that the difference was between the third grade
with the fourth and fifth grade (x3 = 0.33, x4 = 0.75, x5 = 0.71). The second statistically
significant difference was for the students’ ability to recognize functions presented in
different forms of representations (F2,570 = 18.926, p < 0.01). The difference was
between the students at the third grade in comparison to the students at the fourth and
fifth grade (x3 = 0.48, x4 = 0.41, x5 = 0.40). It was unexpected that in this case the
performance at the third grade was higher than the two other grades. The third
statistically significant difference was for the students’ ability to interpret the functions
(F2,165 = 11.077, p < 0.01). The students at the third grade had a lower performance
than the older students (x3 = 0.343, x4 = 0.547, x5 = 0.702).
Discussion
The results of the present study confirm previous findings that students face many
difficulties in understanding function at different ages of secondary education (Sajka,
2003; Tall, 1991). Findings revealed serious students’ difficulties in proposing a proper
definition for function or a tendency to avoid proposing a definition due to their
possible belief that the intuitive and informal presentations of their conceptions cannot
be part of the mathematical learning. This students’ difficulty is in line with the results
of Ko, Lu and Tso (2013), who found that very few students are able to exactly grasp
definition of function.
Although formal definitions of mathematical concepts are included in the mathe-
matics textbooks for secondary education, mathematics teachers probably do not focus
on definitions. Instead they promote the use of algorithmic procedures for solving tasks,
as actions and processes (Cottrill, Dubinsky, Nichols, Schwingedorf, Thomas
&Vidakovic, 1996), and underestimate the word and meaning (Morgan, 2013) as
interrelated dimensions of the concept. We believe that the teaching practice concerning
the use of definition has to be divided into stages in respect to different ages. In the first
grades of secondary education, the definition has to be taught and examined in a
constructive perspective of presenting verbally by the accurate expressions the mental
image of the concept. The role of teachers has to be concentrated on insisting on the use
of the accurate mathematical language, form, and symbols. In the upper grades of
secondary education, the definition has to be a formal part of the mental image which
has to be used as a tool for understanding a representation, for problem solving
procedure, and in proving.
Table 3 Means and standard de-
viations of students’ performance
Task X SD
Definition 0.718 0.261
Examples 0.837 0.231
Recognition 0.449 0.152
Interpretation 0.619 0.212
Problem solving 0.393 0.169
A. Panaoura et al.

15. Secondly, it seems easier for students to present an example for explaining a
mathematical concept, rather than using a non-example in order to explain the respec-
tive negative statement. At the same time, it is easier for them to use an example for
explaining the concept of function rather than explaining the procedure they follow for
determining whether a graph represents a function. This is probably a consequence of
the teachers’ method of using examples in order to explain an abstract mathematical
concept. Thirdly, students had a higher performance on manipulating the concept in
graphical form and translating from one type of representation to another, than on
recognizing functions in algebraic and graphical forms. Symbolic equation solving
follows the graphical work (Llinares, 2000), and probably for this reason, the perfor-
mance on graphical functions was higher. Finally, the students’ results at the third grade
of secondary education were especially negative, thus we have to further examine
whether their processing efficiency and cognitive maturity prevent us from teaching the
specific concept at that age. Despite the tendency to use the spiral development of
concept in the teaching process, we have to rethink the teaching methods we use at the
different ages and the cognitive demands of tasks at each age. A more informal or
experimental procedure with an acceptance of Binformal definitions^ is suitable for the
lower grades of secondary education, which facilities the use of inquiry-based learning,
while the necessity of using formal definitions and procedures is more suitable for the
higher grades of secondary education as a simulation of the work of mathematicians.
The structural models which were confirmed indicated that the conceptual develop-
ment of function is related to the definition of the concept, to its recognition, manip-
ulation, and translation through a variety of representations and problem solving
abilities. Previous studies indicated that students understand the different representa-
tions as separate entities while the present study indicates that the definition, the
recognition, and the interpretation of different representations and the problem solving
ability are different dimensions of the conceptual understanding of the concept which
are interrelated. The strong interrelations underline the coherence of the structure and
the possible influence of a dimension on other dimensions of the same concept.
Students who are able to properly define function are able at the same time to recognize
functions, manipulate, and translate them from one type of representation to another.
The interrelations of the ability to define the concept with the three other abilities
indicated the important role which the specific dimension has on understanding the
concept and it is an indication that it belongs to the highest order level of the
understanding of the concept. These interrelations are higher in gymnasium, because
the few students who are able to solve function problems at the specific age are able to
define the concept, interpret it, and recognize, as they seem to be the exceptions that
have a coherent understanding of the concept.
The results of the present study, mainly the descriptive analysis, permit us to assume
that the understanding of the concept of function can be divided into three main levels.
The first level, which is the lower one, consisted of the students’ ability to solve tasks
asking them to use flexibly different forms of representations for the concept of
function, by following a procedure without necessarily understanding it. The second
level consisted of their ability to define the concept, present verbally the procedure they
have to follow in order to prove whether a statement is function or not, and propose
examples in order to explain further their respective conceptions. The third level which
cannot be included at a hierarchy consisted of students’ ability to solve mathematical
Function’s definition, representation and problem solving

16. problems for which they have to use the conceptual and procedural knowledge of the
two previous levels. Coherent understanding of the concept may be indicated primarily
by successful problem solving which is built on the basis of the correct definition and
flexibility in dealing with multiple representation tasks.
Conclusion
Based on the above, the new findings of our study and thus the contribution of this
study for research and mathematics education can be summarized in the following
points: (1) The verification of a structural model indicating the crucial dimension of the
conceptual understanding of the concept of functions; (2) the involvement of represen-
tational flexibility (recognition, interpretation, translation) in a structural model about
the conceptual understanding of the concept of functions; and (3) The involvement of
tasks and examples in relation to the definition of functions in different registers of
representations. In this way, we are able to justify the strong relationship of definitions
with the rest of the three factors.
The assumption of cognitive hierarchy, which is suggested at the present study, has
direct implications for future research as regards the teaching practice of function. The
educational system has to know and understand how students conceptualize the notion
of function and realize the obstacles and misunderstandings of students. Undoubtedly,
function can be treated as a unifying concept among different mathematical domains
and this perspective at the first stages of secondary education will improve students’
understanding.
A serious limitation of the present study is the inability to explain qualitatively the
high percentage of missing cases in many tasks. A possible explanation could be the
tendency of teaching at secondary education in overestimating the formalistic perspec-
tive of mathematics and in focusing on the correct result of a task, rather than paying
also attention to the informal presentation of the students’ conceptions. Therefore, a
future qualitative study could concentrate further on the missing cases using interviews
and observations in order to investigate those students’ thoughts, conceptions, difficul-
ties, and misconceptions. In conclusion, we believe that it is not adequate just to
describe the students’ knowledge of a concept, but it is interesting to design and
implement didactic activities and examine their effectiveness. Brown (2009) suggests
that the construction of concept maps will enable teachers to have in mind all the
necessary dimensions of the understanding of the concept. Thus, the use of multiple
representations, the connection, coordination, and comparison with each other and the
relation with the definition of the concept should not be left to chance, but should be
taught and learned systematically.
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