Definitions and examples in elementary calculus: the case of monotonicity of functions

ORIGINAL ARTICLE

Definitions and examples in elementary calculus:the case of monotonicity of functions

Cristina Bardelle • Pier Luigi Ferrari

Accepted: 15 December 2010 / Published online: 21 January 2011

� FIZ Karlsruhe 2011

Abstract This paper is devoted to the investigation of stu-

dents’ understanding and handling of examples in the

framework of an example-based introductory mathematics

undergraduate course. The plan of the course included a wide

use of graphs in standard lectures, tutoring sessions as well as

in examinations. This study deals with the notion of increasing

function, which has been introduced by means of both the

standard definition and a range of examples and non-exam-

ples, most often conveyed through graphs. We have analysed

students’ interpretations of the notion of increasing function

as they applied them in a set of written examination tests. The

data gathered have been completed by a number of interviews

of students whose answers were difficult to interpret. The

outcomes underline the importance of linguistic and semiotic

competence and suggest that the design of innovative teach-

ing paths should take care of the linguistic and semiotic skills

needed to handle the representations involved.

Keywords Mathematical language � Colloquial register �Examples � Pattern imagery

1 Introduction

The relationship between theories and examples is a basic

issue in mathematics education in the transition from high

school to university. At the beginning of undergraduate

mathematics courses, a number of students are urged to turn

from learning patterns based on prototypes and numerical

algorithms to others that require handling general mathe-

matical statements as well. Examples are used in a wide

variety of ways in mathematical education as described for

example in Bills et al. (2006) and references therein. They

play a pervasive role in concept formation and in other steps

of mathematics learning processes (Watson and Mason

2005; Bruner et al. 1956; Mason and Pimm 1984; Peled and

Zaslavsky 1997). Goldenberg and Mason (2008) address

examples as ‘‘major means for ‘making contact’ with

abstract ideas and a major means of mathematical com-

munication whether ‘with oneself’, or with others’’ (p. 184).

What makes examples so important is that they are the link

between the general and the particular.

This paper deals with the teaching and learning of the

first concepts and methods of calculus in mathematics

courses designed for science freshman students.

Traditionally, these concepts were introduced through

theory. By means of a well-organised sequence of defini-

tions and theorems, students were expected to grasp the

basic ideas of calculus. This path was often complemented

by tutoring sessions where more examples were shown and

the ideas learned were applied to solve mathematical

problems. Applications of the theory outside of mathe-

matics were possibly shown later.

In this study and in our instructional setting, we used

examples of concepts to clarify the definitions given: for

instance, examples and non-examples of monotonic func-

tions in order to clarify the boundaries of the concept of

monotonicity. The decision to deal with graphs is linked to

the idea that the coordination of different semiotic systems

is crucial in mathematics learning, as suggested by Duval

(1995) and that, in this account, visual aspects are central to

meaning making (Presmeg 1992, 2006). In particular, we

C. Bardelle (&) � P. L. Ferrari

Dipartimento di Scienze e Tecnologie Avanzate,

Universita del Piemonte Orientale ‘‘Amedeo Avogadro’’,

Alessandria, Novara, Vercelli, Italy

e-mail: [email protected]

P. L. Ferrari

e-mail: [email protected]

123

ZDM Mathematics Education (2011) 43:233–246

DOI 10.1007/s11858-010-0303-4

think that supporting the distinction between critical and

non-critical features of graphs fosters the building of a

correct pattern imagery in the sense of Presmeg (2006),

which in turn leads to grasping of the concept. This prac-

tice is aimed at enriching students’ example space (Watson

and Mason 2005, Goldenberg and Mason 2008). The

example space is in turn aimed at developing students’

concept image and concept definition (Tall and Vinner

1981, Vinner 1983). ‘Concept definition’ is used here as in

the original paper of Tall and Vinner (1981) that is as ‘‘a

form of words used to specify a concept’’ and not to denote

the standard mathematical definition.

In our experience, teaching paths of traditionally

designed courses have hardly proved successful with sci-

ence freshman students. The reduction of the time given to

mathematics in most Italian undergraduate curricula, the

reduced inclination of many students to work steadily

throughout the course, and the diversity of their initial

levels of achievement induced us to search for other means.

So, we have designed teaching methods based on

examples, with care for definitions even from the linguistic

viewpoint. The number of proofs given (and required for

passing the final examination) has been significantly

reduced. This decision is closely related to the needs and the

potential of our students and does not entail any criticism

against the role of proofs in mathematics education.

When a definition was given (e.g., that of increasing

function), a range of representations were provided, from

verbal texts (including both accurate versions and more

colloquial ones) to symbolic expressions, with warnings

about the slippery steps (if any). Moreover, examples and

non-examples of the definition were provided, also through

a wide use of figural representations (mainly graphs). Stu-

dents were asked to take care of the reasons why examples

satisfy the definition, whereas non-examples do not.

Through the course, we administered tests to evaluate

students’ ability in dealing with definitions. The tests asked

students to recognise examples and non-examples of given

definitions and usually included plenty of figural repre-

sentations (graphs). The focus was on the meaning of the

statements (grasped also through a rewording process)

rather than on the treatment of symbolic expressions.

From the beginning, the tests gave quite puzzling out-

comes. A number of answers at first sight looked incon-

sistent or very difficult to interpret. The relationship

between definitions and specific examples has proved a

critical point. So, we tried to arrange subsequent tests to

gather more evidence on the reasons why some students

chose some specific answers in place of other ones. In some

cases, we interviewed students who had given answers that

were difficult to understand.

The question we have addressed in this paper is: based on

which criteria do students recognise (or fail to) that some

given objects (represented in different ways) are or not

examples of given definitions? In other words, what are the

reasons why students fail to recognise this? The examina-

tion of a large number of protocols in the past few years has

suggested that language-related difficulties might play a

major role as sources of difficulties. So, we paid particular

attention to how the subjects involved in our study used

language and representations in their solution processes.

2 Theoretical framework

Our teaching experience and previous research have

pointed out the importance of language. On the one hand,

communication problems often played a major role; on the

other, the improper use of semiotic systems (verbal texts,

symbolic expressions, graphs, …) proved a serious obstacle

to students when doing mathematics. We underline that our

students’ difficulties involve verbal language and the

overall organisation of verbal texts too, and not just the

meaning of some words. In this regard, a range of studies

(Ferrari 2003, 2004a, b; Morgan 1998; Pimm 1991) have

pointed out the shortcomings of interpretations of the lan-

guage of mathematics taking into account vocabulary and

symbolic expressions only. The claim that the language of

mathematics is rigorous, formal and exact, whereas the

ordinary one (sometimes misleadingly referred to as ‘nat-

ural’, see at the end of this section for more details) is

ambiguous, informal and inaccurate does not properly

explain the complex interplay between the different ways

of using language. The same studies have underlined the

need for investigating these phenomena with the means

provided by pragmatics (see Leech 1983 for a concise

survey), which deals with not only syntactic and semantic

aspects, but also the interplay between text and context. In

particular, we, as well as Morgan (1998), have found very

appropriate the means provided by functional linguistics, a

theoretical perspective that focuses on the functions of

texts as opposed to form, intended as internal organisation.

The main reference for functional linguistics is the book

of Halliday (1985), whereas many ideas useful for the

application to mathematics (although the book does not

refer to mathematics at all) have been drawn by the study

of Leckie-Tarry (1995). The construction that in functional

linguistics links text to context is register. A register is a

linguistic variety related to use. It is formed through the

selection of the linguistic resources available for a subject,

related to her/his goals. A basic distinction is between lit-

erate and colloquial registers (Leckie-Tarry 1995). Ferrari

(2004a, b, 2006) showed that the registers used in mathe-

matics (as defined by Morgan, 1998) are extreme forms of

literate registers and share with them a large amount of

properties. Colloquial registers are mainly used in spoken

234 C. Bardelle, P. L. Ferrari

123

communication, but also in written informal notes, e-mail

messages or SMS. Literate ones are generally used in

books, textbooks and in a wide range of written documents,

but possibly also in some lectures or formal speeches.

The difference between colloquial and literate (and thus

mathematical) registers is not just a matter of vocabulary or

symbols, but involves the general organisation of texts. For

example, colloquial registers are focused on interpersonal

relationships between the participants more than literate

ones. In colloquial registers, texts are generally more

concrete and situation dependent, and meaning is built as a

process. Literate registers (including mathematical ones),

on the contrary, are more abstract and less situation

dependent, and meaning is built as an object with less

opportunities for negotiations. Moreover, the role of syntax

is more important in literate registers than in colloquial

ones. This is especially true for mathematical registers and

symbolic notations too, where syntax sometimes is the only

means to convey meaning.

According to this framework and our previous research

(e.g., Ferrari, 2004a, b), students with a good mastery of

literate registers are much more likely to succeed in dealing

with mathematical registers than others. This should explain

the reasons why we consider the expression ‘natural lan-

guage’ misleading, as it might suggest that there is a gener-

ally shared level of linguistic competence, which is sufficient

to deal with mathematics, at least informally. Our experience

and research suggest that generally this does not occur. The

mastery of literate registers seems not natural at all, but has to

be built through appropriate teaching actions, preferably in

primary school, as suggested by Ferrari (2006).

This framework provided us the means to effectively

analyse the texts produced by students and helped us to

take notice that many students interpret and produce texts

in conformity to colloquial use, without realising that

mathematical registers work in a quite different way. The

interference between colloquial and mathematical use of

language has been recognised by some researchers, such as

Tall (1977), Cornu (1981), Mason & Pimm (1984), Ferrari

(2004a, b), Kim et al. (2005) and Bardelle (2010) as a

factor affecting students’ performances.

This interference might depend on the lack of linguistic

competence (such as poor grasp of literate registers) or their

beliefs on the (usually undervalued) functions of language

in mathematics and the ensuing negative attitudes. The

possibility that students may lack basic linguistic or semi-

otic skills is largely undervalued by current research in

mathematics education. The same belief seemingly holds

for graphs too, as remarked by a number of researchers (see

for example Eisemberg 1991). Through this paper, we are

not assuming that for any student, the interpretation and

production of texts, symbolic expressions and graphs is an

easy task, with no troubles and problems.

Since our tasks widely involve graphs, visual thinking is

unavoidably involved. In this regard, the construct of pat-

tern imagery introduced by Presmeg (1992) has proved

useful. The ability of depicting a pure relationship in a

visual–spatial scheme, i.e., the ability of grasping the

essence of its structure from an image, stripping it of

concrete details, is crucial to recognise monotonicity

through graphs. In particular, Presmeg (1992) describes

how prototypical images adopted by students are a hin-

drance in building pattern imagery (metonymic usage of

diagrams). The reference to factors related to language and

imagery is also a way to give a more precise account of the

reasons why some subjects switch from interpretations

based on the definition to others usually classified as

‘intuitive’, in the sense of Fischbein (1987).

We also adopt the theoretical construct of pivotal-

bridging example as introduced in Zazkis and Chernoff

(2008): ‘‘An example is pivotal for a learner if it creates a

turning point in the learner’s cognitive perception or in his

or her problem solving approaches; such examples may

introduce a conflict or may resolve it. When a pivotal

example assists in conflict resolution we refer to it as a

pivotal-bridging example, or simply bridging example, that

is, an example that serves as a bridge from learner’s initial

(naıve, incorrect or incomplete conceptions) towards

appropriate mathematical conception’’. Clearly, each sub-

ject will exploit pivotal-bridging examples more or less

effectively, according to her or his own potential example

space.

3 Methodology

The experiment was performed with a group of about 150

freshman science (biology, chemistry, environment)

students.

We adopted a mixed method strategy, in the frame of the

sequential mixed method as described by Cresswell (2009).

We gathered the protocols of four ongoing examination

tests administered between October 2009 and February

2010, to the same cohort of students. We took into con-

sideration only the subjects who attended all the tests. Tests

1–3 were standard examination papers, designed according

to the customary pattern. Test 4, which was still an

examination test, was slightly modified by the inclusion of

questions such as Q3 (see below).

After each test, we interviewed about 10 students,

selected among the subjects whose answers appeared very

difficult to interpret.

We considered a wide range of patterns of answers given

by our students. Some of them were adopted by a large

numbers of subjects (e.g., more than 30% of them), while

others were adopted by small groups or individuals only.

The case of monotonicity of functions 235

123

We believe that all the data are important, but also that

an investigation willing to be relevant from the educational

viewpoint and useful for the improvement of the design of

the course has to deal not only with single episodes, but

also with a large number of students as well, even though

we do not use the methods of quantitative research.

When examining each student’s answer to each task, we

took into account:

1. the answer in itself (right or wrong);

2. the reasons explicitly given by the subject to justify the

answer (if any);

3. the ideas implicit in the answer given (e.g., the

meaning implicitly adopted for ‘increasing function’);

4. the overall quality of the text produced from the

linguistic viewpoint.

The latter characteristic has been evaluated according to

the criteria described above (see Sect. 2) and related to the

distinction between colloquial and literate registers.

We decided to try to evaluate the linguistic quality of the

answers, because we thought it could deeply affect students’

performances, not only when they make some mistake, but

also when they do not answer or do not give any explana-

tions because of their lack of confidence about language.

3.1 Tasks and instructional setting

We report some patterns of the questions included in the

tests. Questions like Q1 and Q2 have been customarily

included in all the tests. Question Q3 has been included in

the last test only (end of February 2010).

Q1

Tell which among the graphs shown below do not repre-

sent an increasing function on the given interval. Explain

your answer.

Q2

Is the function defined by equation y = -3x2 ? x ? 1

decreasing? Justify your answer.

Yes No

Q3

Mark any of the sentences below that correspond or are

equivalent to the definition of increasing function in the

interval [a, b]. Justify your answer.

(a) For any x1, x2 [ [a, b], f(x1) \ (x2) holds.

(b) For any x1, x2 [ [a, b], such that x1 \ x2, f(x1) \ f(x2)

holds.

(c) There exist x1, x2 [ [a, b], such that if x1 \ x2, then

f(x1) \ f(x2) holds.

(d) f(a) \ f(b)

(e) There exist x1, x2 [ [a, b], such that f(x1) \ f(x2)

holds.

3.2 A priori analysis

The questions were chosen to investigate the effects of

some factors that appeared crucial from previous obser-

vations and in particular:

1. the potential conflict between the definition of increas-

ing function and the overall behaviour of the graph

(such as graph A in Q1);

2. the handling of discontinuous graphs, or graphs

defined only in a portion of the domain;

3. the effects of switching from functions presented

through graphs to functions defined by an equation;

4. the knowledge of the definition.

(B)(A)

(D)(C)


123

All of the questions require knowledge (possibly partial)

of the definition of increasing function, which means some

grasp of the role of the quantifiers (or the corresponding

verbal expressions) occurring in the definition of increasing

function or in the distractors of question 3. Question 1

requires skills in the reading of graphs, such as properly

associating the points in the diagram with their coordinates.

Q1 and partially Q2 as well require a careful reading of the

text, since students are requested to recognise functions

that are not increasing, whereas from the context they

might expect the reverse question. Justifying answers

requires awareness of the methods adopted and linguistic

skills. It involves students’ beliefs and attitudes as well, as

they might regard the task of justifying as improper or less.

4 Outcomes

The results presented in this paper do not cover all the

behaviours we found in our study, but are aimed at high-

lighting the interplay between definitions and the use of

examples. For more on this topic in a similar setting (see

Bardelle 2010).

As far as we have used standard evaluation tests, we

should expect that the outcomes are affected by a variety of

factors, such as the (welcome) progress of students, who,

through the course and the tutoring activities involved,

might have improved their grasp of the subject matter.

However, there is also the possibility for students to adopt

alternative methods to pass the test, such as cheating or

adopting pseudo-analytic strategies, i.e., strategies not

aimed at understanding the questions involved, as descri-

bed by Vinner (1997).

Here, the graphical system plays a central role and so the

results depend on this particular characteristic we gave to

our research. This is a consequence of the design of the

course, as outlined in the introduction.

Let us start with the description of four main different

ideas of increasing function seemingly adopted by students

to provide an answer to tasks such as Q1, Q2 and Q3:

(1) the proper definition;

(2) a function f is increasing in [a, b] if f(a) \ f(b);

students just compare the values of the function at the

end of the interval;

(3) a function is increasing when the increasing pieces of

a graph are predominant compared to the decreasing

ones;

(4) as in (1), but applied to connected portions only of

graphs of discontinuous functions.

Pattern (3) is not a clear-cut one, and it seems mostly

based on visual aspects. In the sequel, we try to explain

how definitions were adopted by our students.

According to the results, we categorised some graphs of

functions that were considered to be increasing by students

into three different grades of exemplarity.

Information about the reasons for students’ behaviours

was gathered from their written explanations and interviews.

We found three main reasons leading to ideas like (2)–

(4). Firstly, almost all students showed poor understanding

of the standard definition or difficulty in its application.

Secondly, some mathematical terms borrowed from

ordinary language such as ‘increasing’, ‘decreasing’ and

‘constant’ were interpreted according to their colloquial

meaning and not to the mathematical one. More than one-

third of the sample showed this kind of problem. Thirdly,

obstacles strictly related to the handling of graphs were

shown by not \30% of the sample. In particular, the

influence of (visual) prototypes along with poor quality of

the examples space is an obstacle to the building of a

correct pattern imagery related to monotonicity.

4.1 Understanding the definition

Since recognising a monotonic function (as in task Q1 and

Q2) has proved to be a very difficult task, we introduced

the question Q3 in the last examination test of the course.

The aim was to understand possible connections with the

knowledge of the standard definition of monotonicity.

Notice that the symbols V and A were not used.

Less than 25% of the sample chose the proper answer

(b), even if all students were allowed to look up books or

hand notes in the examination. The low number of correct

answers shows poor or unstable knowledge of the defini-

tion. Actually, most of the students marked other items in

addition to the correct choice (b). In some cases, students

chose (c) or (d) in addition to (b), because these statements

contained something true about the increasing functions.

For example, a student explicitly told us in his interview:

‘‘I marked also (d) [in addition to (b) in Q3] because

for an increasing function d) is also true.’’

It seems there is little correlation between the answers to

Q3 and to the problems with graphs as Q1. Actually, most

of the students did not use the definition to solve problems

like Q1. This fact was explored in the interviews, where

some students conceded it. For example, one of them said:

‘‘…in order to answer [Q1] I looked directly at the

graphs and I did not think at the[standard] definition

[of increasing function]’’,

and another one:

‘‘…I never used the [standard] definition to answer

[problem Q1] because it cannot be immediately

applied.’’


123

We would like to point out that only a good coordination

between the verbal/symbolic system and the graphical one

allowed some subjects to state that some piecewise con-

tinuous function (like the ones shown in Fig. 1 below) were

not increasing functions, providing a suitable example to

show that the definition of increasing function was not

satisfied. Actually, they found a couple of real numbers x1,

x2, such that x1 \ x2, but f(x1) C f(x2). Students who

recognised that piecewise functions as below were not

increasing, in their interviews, showed the ability of

switching from the definition (verbal or symbolic) to the

visual properties of increasing functions and vice versa.

This fact agrees with Duval’s (1995) characterisation of

‘coordination of symbolic systems’.

We have pointed out previously that most of our stu-

dents did not use the standard definition to recognise if a

given graph represented a monotonic function. The same

holds for functions given by formulae as in Q2. In this case,

the main common strategy is based on a sketch of the graph

of the function (for other strategies adopted see Bardelle


123

2010). It seems that many students prefer to work with the

visual system. In this regard, it seemed that the transition

from formulas to graphs was highly facilitated since the

functions given, such as parabolas and trigonometric

functions, had been largely dealt with through the course.

The visual system probably allows subjects to get a quick

insight into the values assumed by a function and to use

this to say something about its monotonicity.

We can also highlight another issue we encountered

when examining students’ behaviours. In written respon-

ses, about 33% of the students mentioned the definition of

monotonicity, although they did not use it to give an

appropriate answer.

We have found seemingly correct responses such as:

‘‘The function is not increasing because it does not

match the definition, given two points x1 \ x2 it must

be f(x1) \ f(x2).’’

‘‘A function is defined increasing in [a, b] if for any x,

t [ [a, b] with x \ t one has f(x) \ f(t)’’

We have also found clearly wrong descriptions such as:

‘‘A function is increasing if considered two points in

the interior of an interval the first is always smaller

than the second.’’

‘‘A function is said decreasing in a given interval if

for any x, t belonging to the given interval if x [ t,

then f(x) [ f(t).’’

‘‘The only increasing function in the given interval is

(c) [Q1] because it is the only one for which it holds

for any couple of points (x1, y1) and (x2, y2) one has

f(x1) \ f(x2)’’

The previous excerpts are just some of the examples of

incorrect writings about the definition of monotonic func-

tions. Such excerpts of incorrect answers show a difficulty

in the handling of the definition and its understanding. In

this regard, we present the case of Marta. She correctly

answered Q2 showing good mastery of the definition of

increasing function and, on the contrary, in an unsatisfac-

tory way to the Q1-problem. To the question:

‘‘Is the function y = 3x2 ? x ? 1 increasing?

Explain your answer.’’

she answered ‘‘No’’ and commented:

‘‘…because f(0) = 1 \ f(-3) = 25. f is increasing if

for all x1, x2

x1 \ x2 ) f(x1) \ f(x2).’’

To the question (Fig. 2)::

‘‘Tell which among the following graphs do not

represent an increasing function on the given interval.


she answered:

−6 −4 −2 2 4 6

−4−3−2−1

1234

x

y

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

x

yFig. 1 Piecewise continuous

functions

−6 −4 −2 2 4 6

−4−3

−2−1

1

234

x

y

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

x

y

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

x

y

−4 −2 2 4

−4

−2

2

4

x

y

(A)

(C) (D)

(B)Fig. 2 Increasing and non-

increasing functions


123

‘‘The function represented by graph A is always

decreasing, for example f(-2) = 2 \ f(4) = -2.

f is increasing if for all x1, x2

x1\x2 ) f ðx1Þ\f ðx2Þ’’.

Her interview confirmed that she did not use the stan-

dard definition to state if a graph represented an increasing

function.

I: How did you decide which graphs represent or not

represent increasing functions?

M: I compared the values assumed in the extremes of the

interval shown…if the value of the right extreme is

higher than the left one the function is increasing

I: How did you answer to Q2? Did you draw the graph of

the function?

M: I used directly the increasing function formula…I did

not think to sketch the graph.

I: Why did you not use the same formula in the problem

Q1?

M: …I don’t know…perhaps it dealt with graphs…but I

don’t know…

Marta’s behaviour is typical. From the interviews, we

understand that the subjects mentioned the definition of

monotonicity in their written answers just to give an

explanation suitable for the instructor, and their responses

did not actually describe how they obtained their solution.

More generally, some seemingly inconsistent answers

(such as the claim that any function is either increasing or

decreasing) might be explained by the subjects’ need for

providing an answer anyway rather than as manifestations

of their actual conceptions.

4.2 Pivotal examples

From the interviews, it was clear that some examples and

non-examples were, as they played a significant role in the

discovery of incorrect concept definitions about

monotonicity.

We illustrate two cases of examples playing a pivotal

role.

4.2.1 The parabola pivotal example

We describe the case of Valeria. You can find some

excerpts from her answer to a version of a Q1-question

The English translation of her answer is:1

‘‘The functions not increasing are:

(B) because in the interval between � 12

and þ 12

x1 is

smaller than x2 but f(x1) [ f(x2)

(C) because at point (1;2) the function stops, and

anyway because x1 \ x2 but f(x1) [ f(x2).

The graph (A) increases in the given interval but

anyway there are points where x1 \ x2 but

f(x1) [ f(x2).’’

Such a response seems to show an acceptable use of the

standard definition of monotonicity, even if she did not

provide numerical counterexamples and gave some strange

explanations. Regarding the expression ‘‘the function

stops’’, we refer to Sect. 4.4. The expression ‘‘The graph

(A) increases’’ is in conflict, mathematically speaking, with

‘‘there are points where x1 \ x2 but f(x1) [ f(x2)’’. It seems

that her answer has not been influenced by couples of

points x1, x2 acting as counterexamples, but rather by

something else. The interview was aimed in particular at

understanding if the perceptive nature of the graph was the

cause of her inconsistent answers. In fact ‘‘The graph

(A) increases’’ seems to be based on a notion of ‘increasing

function’, which merges the standard definition and the

everyday life meaning of the word.

In the interview a Q2-question proved pivotal to Valeria.

She answered ‘yes’ to the Q2-question

‘‘Is the function y = -3x2 ? x ? 1 decreasing?


with the explanation:

‘‘because x1 \ x2 gives f(x1) [ f(x2).’’

She justifies her answer as follows:

V: From the formulae [y = -3x2 ? x ? 1] I recognised

it deals with a parabola. Such function increases in the

first piece and decrease in the second one…I don’t

know whether to answer yes or no…I chose yes

randomly and I wrote x1 \ x2 gives f(x1) [ f(x2) in

order to give an explanation

I: Can you give an example of a non-increasing function?

Valeria started to draw a graph like (A) in Fig. 3 and

said:

V: This is a function increasing on the whole but it is not

increasing because the law x1 \ x2 gives f(x1) [ f(x2)

is not respected.

I: Is the law respected in the parabola?

V: No because…but I cannot decide which part wins, the

increasing or the decreasing one.

From this piece of interview and from other graphs she

gave as examples of non-increasing function, it was evi-

dent that for her a non-increasing function was a function

with decreasing parts that were very small such as (A) and

1 Here, we give English translations of students’ texts. We would like

to point out that there are properties of texts that get lost through the

process of translation, no matter how accurate it is.


123

(B) in Fig. 3. Her concept image about an increasing

function was a function in whose graph the increasing

pieces visually predominate over the decreasing ones (3).

Probably this image was built through an everyday life

interpretation of the terms ‘increasing’ and ‘decreasing’

and an inadequate understanding of the standard definition.

In particular, such a problem seems to be related to the

understanding of the role of the quantifiers in the standard

definition. It is important to highlight that Valeria referred

to graphs representing non-increasing functions, according

to her idea (3), sometimes as ‘‘non-increasing because

there are points that do not respect the law [definition of

increasing function]’’ and sometimes as ‘‘increasing even

if there are points that do not respect the law’’ or

‘‘increasing on the whole’’. She used conflicting words,

from the mathematical point of view, to express the same

concept. The example of the parabola was pivotal for

Valeria: the perfect symmetry between the increasing and

decreasing parts of the parabola generated a conflict in her

concept image. Her idea of increasing/decreasing function

(3) was not sufficient or useful anymore to achieve the

answer. In the interview, she became aware of this and of

her need for finding another way to decide whether the

parabola was increasing or not. The parabola was also a

bridging example for her, since she understood that it was

‘‘not increasing just because of two points not satisfying

the law’’. We think that for the resolution of her conflict,

mastery in handling the symbolic definition and coordi-

nating it with the graphs played a significant role. The

achievement of the proper pattern imagery of monotonicity

was also confirmed by the following excerpt from her

response to a further Q1-task. In this task, she claimed that

all the graphs provided were not associated with increasing

functions, with the one exception of (C), showing a more

appropriate understanding of the role of the definition

(Fig. 4).

The translation of her answer is:

‘‘I exclude all the graphs, except C, because in all of

them there exists at least two points in which

x1 \ x2 ? f(x1) [ f(x2) is not respected.

4.2.2 The piecewise continuous function pivotal example

Non-increasing piecewise continuous functions such as that

in Fig. 5 were introduced to verify if students were able to

build a correct pattern imagery related to the monotonicity

of functions.

Fig. 3 Valeria’s task - 1


123

As remarked above, 80–90% of the subjects regard such

functions as increasing. Many of them were skilled stu-

dents and in particular they succeeded in all the other kinds

of graphical Q1 and also the Q2 and Q3-questions. How-

ever, they failed in tasks involving this kind of graphs.

Such examples proved to be crucial for a complete

understanding of monotonicity. We stress that their power

to create a cognitive conflict and possibly resolve it

depends on the individual characteristics of the subject. In

particular, we observed their crucial role in proficient stu-

dents. We report here the case of a bright one: Giulia

(Fig. 6).

The protocol clarifies her process of thinking as very

appropriate. She used directional arrows to mark increasing

or decreasing pieces of a function. At first, she justified her

choice as follows:

‘‘the graphs A, B, C, E do not represent strictly

increasing functions because they have intervals

where the function decreases’’

Afterward, she added:

‘‘for graph A f(-4) [ f(-2); for B f(-1) [ f(4); for

C f(-3) [ f(-2.5) and for the graph E f(-2) [ f(-

1).’’

Giulia in her interview confirmed that she used the

counterexamples not to decide about monotonicity, but to

give a response that was acceptable to the instructor.

Fig. 4 Valeria’s task - 2

−6 −4 −2 2 4 6

−4

−3

−2

−1

1

2

3

4

x

y

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

x

y

Fig. 5 Non increasing

piecewise continuous functions


123

Moreover in the interview, she was invited to think about

possible counterexamples to D) and she suddenly found

some proper couples of numbers. Finally, she added:

‘‘So the function [D)] is not increasing. I did not think

[in the written examination] to find counterexamples

for D) but I thought of the two pieces separately’’.

She looked for ‘descending parts’ of a graph to decide

that a function was not increasing; indeed, her prototypical

image of a decreasing portion of a function is given by an

explicit, continuous line that monotonically decreases and

which corresponds to (4). This could arise from the fact

that in human communication when there are no signs there

is nothing to communicate. The lack of signs might be a

sign itself, but it is much more difficult to detect and read,

as it needs more complex kinds of inference (see Ferrari

2004a, b for an outline of the role of inference in text

interpretation). Notice that the lack of some ‘descending

part’ of the graph, although perfectly correct from a

mathematical viewpoint, is not of much help and makes the

representation unfit from the viewpoint of pragmatics, at

least for subjects adopting colloquial interpretations of

mathematical texts or representations. The example

(D) was pivotal-bridging for Giulia: she realised that her

initial idea of increasing function (4) was insufficient and

she had to think of the values of the function in order to

give correct answers also for discontinuous function.

4.3 Language-related difficulties

In many cases, subjects answered using a colloquial

interpretation of the terms ‘‘increasing’’ and ‘‘decreasing’’

without realising that they had a precise mathematical

meaning and without grasping the functions of mathemat-

ical terms in doing mathematics. For example, a number of

subjects claimed that functions like those represented in

Fig. 7 were increasing ‘‘because they tend to assume higher

and higher values’’.

Functions such as sine and cosine are considered to be

not increasing, because they take values between -1 and

?1. Some excerpts of this kind have been given by Bar-

delle (2010). The interpretation of the terms ‘increasing’

and ‘decreasing’ according to the everyday usage leads to

definitions as in (2) and (3). For example, in the

Fig. 6 Giulia’s answers


123

explanations of Valeria we see a continual intertwining

between the use of the standard definition and a colloquial

interpretation of ‘‘increasing function’’.

Moreover, mathematical terms such as ‘‘stationary’’ and

‘‘constant’’ are widely misused to claim that a function as

sine or cosine takes values between -1 and ?1 only.

Furthermore, the terms ‘‘positive’’ and ‘‘negative’’ are used

in place of ‘increasing’ and ‘decreasing’, respectively, as

the following interview shows (Bardelle 2010; Fig. 8):

I: What is an increasing function?

A: A function that little by little assumes more positive

values

I: Does the graph on the right represent an increasing

function in the given interval?

A: yes

I: …but it takes only negative values…A: …when I say more positive values I mean higher values

Behaviours like these are very common in the responses

of students and they are due to a use of colloquial registers

instead of mathematical ones (Ferrari 2004a, b, 2006). For

other examples on this topic (see Bardelle 2010).

This is not just a matter of meaning of single words: the

use of words according to meanings that are explicitly

stated is typical of literate registers, whereas colloquial

ones generally adopt vague meanings that, if necessary,

may be made more precise through negotiations between

the participants of the exchange.

4.4 Reading graphs

Interpreting graphs on a Cartesian plane can by itself be a

source of errors. In many cases, the criteria adopted in the

responses stem from prototypical images that are visual

examples, or non-examples of increasing or decreasing

functions, and from improper generalisations. We report the

case of Nicolo. He defined a function as increasing when its

graph was in the portion of the Cartesian plane with positive y:

‘‘I saw in a book an example of an increasing func-

tion and its graph was all above the x-axis so I

thought that the increasing functions were all those

with the graph above the x-axis.’’

The example space of Nicolo proved to be very poor,

which induced him to mistake positive function for

increasing ones. This phenomenon was already observed

by Rasslan and Vinner (1998) in the context of an Israeli

Arab high school.

A relevant number of students considered non-continu-

ous functions such as more graphs of functions drawn in

the same Cartesian plane. This may suggest that their

prototypical image of a function is a continuous one.

The case of Alberto is, in this sense, even more sur-

prising. He considered some continuous piecewise function

as more functions. He chose only graph (A) as strictly

increasing and wrote (Fig. 9):

‘‘In D the functions are both decreasing. In C one can

find one increasing and the other one decreasing. In B it

increases but only towards 6 and in E increasing. But in A

we find increasing functions in the interval [-6, 6]’’.

In the interview he explained

‘‘In D, C and A there are two functions starting at the

origin.’’

Alberto was confused about functions and their monoto-

nicity. He considered some functions as in D as ‘‘two func-

tions starting at the origin with one going towards right and

one going towards left’’. These problems in reading graphs

add up to more general language-related problems, such as

colloquial interpretations of mathematical terms and lack of

control of the goals of mathematical expressions.

Finally, we recall the response of Valeria as above.

Valeria stated that C) was not increasing ‘‘because at point

(1;2) the function stops’’. She clarified in the interview that

V: it seems to me it [C)] is not a function because

functions are [defined] on the whole reals, they go on

−6 −4 −2 2 4 6

−4

−3

−2

−1

1

2

3

4

x

y

−6 −4 −2 2 4 6

−4

−3

−2

−1

1

2

3

4

x

yFig. 7 ’Globally increasing’

functions

−6 −4 −2 2 4 6

−4−3−2−1

1234

x

y

Fig. 8 Increasing functions as positive ones


123

I: So why do you wrote ‘‘it stops’’?

V: it stops means that it is not a function and so it does not

make sense to talk about increasing or decreasing…

It seems that functions that are not defined on the whole

real line do not belong to Valeria’s example space of

functions.

The cases reported show a metonymic usage (Presmeg

2006) of graphs, which proved to be a hindrance to

learning, leading students to illegitimate generalisations.

Such behaviour is in most of the cases strictly related to

language-related problems and are both influenced, on the

one hand, by difficulty to deal with the standard definition

of monotonicity and, on the other, by lack of understanding

of the functions of mathematical language.

5 Final remarks

It has been proved that the main obstacles to the correct

handling of examples are the careless use of mathematical

terms (according to colloquial registers), the use of defi-

nitions other than the standard ones, some difficulties in the

handling of graphs and the misuse of prototypes. Attitudes

towards language and mathematics have proved to deeply

affect students’ behaviours. All of these behaviours might

be related to the lack of understanding of the characteristics

and goals of mathematics and its language, which might

affect students’ perception of their control on the tasks they

are dealing with and result in behaviours hardly explain-

able otherwise.

It seems to us that the outcome of this study underlines

the importance of linguistic and semiotic competence.

The design of innovative teaching paths should not

neglect the linguistic and semiotic skills needed to handle

the representations involved, as concrete as they can be. In

other words, the adoption of good teaching methods does

not provide the certainty that more students will learn. In

the same way, some grasp of the main concepts involved

might be helpful in handling the examples and in devel-

oping students’ self-confidence and more effective atti-

tudes towards mathematics.

Fig. 9 Alberto’s task


123

Our results confirm the relevance of the role of pivotal-

bridging examples as already observed by Zazkis and

Chernoff (2008). Such examples should be constructed

carefully, analysing their different linguistic and semiotic

aspects with particular attention, to evoke and possibly

resolve cognitive conflicts in the concept image of the

students. In our opinion, the power of these examples is

also testified by the fact that they tend to impress in the

memory of students, because of their peculiar features, thus

further helping their learning process. Teaching patterns

should then be conceived, including and taking full

advantage of such pivotal-bridging examples. Anyway, it

cannot be concealed that pivotal-bridging examples seem

to work more effectively for students with reasonable lin-

guistic and mathematical skills.

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Definitions and examples in elementary calculus: the case of monotonicity of functions

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