The Concept of Derivatives Through Eye-Tracker Analysis

Christian Casalvieri

, Alessandro Gambini

, Camilla Spagnolo

and Giada Viola

Department of Mathematics, Sapienza University of Rome, Piazzale Aldo Moro 5, Rome, Italy

Faculty of Education, Free University of Bozen, Viale Ratisbona 16, Bozen, Italy

Department of Mathematics, University of Ferrara, Italy

Keywords: Eye-Tracker, Derivative, Tangent Line, Semiotic Registers, Maths Education.

Abstract: In this paper we present a qualitative study on data collected by an eye-tracker tool regarding a Calculus task.

One purpose of this research is to highlight the differences and similarities between visual observation of

expert and non-expert groups. Analysis of the way of reading a text can provide a lot of information about

cognitive processes carried out to solve the task. Moreover, the aim of this study is to analyse, through the

eye-tracker tool, the difficulties of students concerning the concept of derivatives and to understand what may

trigger a wrong answer to the task.

1 INTRODUCTION

In recent years, there has been an increase in the use

of eye-tracking rools for research in the field of

Mathematics education. This technique, used also in

other fields such as Psychology, Neuroscience or

Linguistics (Ferrari, 2004) as well as about cognitive

process creativity (Schindler & Lilienthal, 2020),

provides information about the way a person looks at

a visual stimulus. Thanks to the eye-tracker tool, it is

possible to study eye movements while an individual

is observing a stimulus. In particular, it is interesting

to analyse the eye movements of a person while

performing a mathematical task. The way to read a

text offers a lot of information about the problem-

solving process. Some studies are done in

mathematics in high school (Spagnolo et al., 2021)

with the eye-tracker tool, while little research is

carried out at university level.

In this research study, we focused attention on the

problem-solving process of a calculus task involving

the concept of derivatives. The task chosen was part

of the international survey TIMSS Advanced of 2008.

The choice of this task was based on the results of

standardised assessment tests (Gambini et al, 2020).

In fact, the results show that students have difficulties

with the concept of derivatives and the concept of

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slope of a function. They have difficulties

understanding these concepts and explaining the

meanings of their cognitive process in a mathematical

task (Ferrari, 2017). Therefore, one of the purposes of

this research is to understand, using the eye-tracker

instrument, students’ difficulties regarding these

mathematical objects (Almfjord & Hallberg, 2020).

In addition, the aim of this analysis is to highlight the

difference between the gaze of experts in

mathematics (high school teachers, PhD students,

academics) and that of non-experts (Andrà et al.,

2009; Inglis & Alcock, 2012). The non-expert group

is composed of students of scientific faculties who

attended a calculus course in the first academic year.

Moreover, we wish to observe what has changed

since taking the calculus course. Therefore, in this

study we wish to make a comparison between

standardised assessment results and the responses of

the candidates of our sample. Thanks to data collected

by the eye-tracker tool and answers to interviews, it

is possible to analyse the process carried out by

individuals when solving the task. In this way, we can

study which elements led candidates to a solution and

understand how they did so.

378

Casalvieri, C., Gambini, A., Spagnolo, C. and Viola, G.

The Concept of Derivatives Through Eye-Tracker Analysis.

DOI: 10.5220/0011941100003470

In Proceedings of the 15th International Conference on Computer Supported Education (CSEDU 2023) - Volume 2, pages 378-385

ISBN: 978-989-758-641-5; ISSN: 2184-5026

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

2 THEORETICAL FRAMEWORK

Eye-tracking allows us to track what a person

observes while performing a task. Recently,

researchers have used an eye-tracker tool to analyse

cognitive processes (Schindler & Lilienthal, 2019). In

particular, this tool has been used in research into

areas of Mathematics such as Geometry (Schindler &

Lilienthal, 2017, Simon et al., 2021), Algebra

(Obersteiner & Tumpek, 2016) and interpretation of

motion graphs (Ferrara F. & Nemirovsky R., 2005).

Moreover, some research studies have been carried

out on use of the eye-tracker in high school (Spagnolo

et al., 2021). The hypothesis, known as the Eye-Mind

Hypothesis, claims that eye movements are linked to

cognitive and learning processes. What is observed

by the subject offers important information about

what is processed by him or her (Just & Carpenter,

1980). Thanks to data collected by eye-tracker

instruments, it is possible to analyse the cognitive

processes carried out by individuals.

In a mathematical text there are elements

belonging to different registers of representation

(Duval, 2006). Therefore, when people read a

mathematical text, they have to be able to switch from

one register of semiotic representation to another

(

Giberti et al., 2023). Some studies show that the

ability to solve a task is related to the ability to read

different semiotic representations. Thanks to eye

movement analysis, it is possible to study the ability

to switch between different representations of

mathematical objects (Andrà et al., 2009; Andrà et al.,

2015). Moreover, thanks to the eye-tracker tool, it is

possible to identify which part of the text attracts

more fixations. It enables scholars to study which

objects catch students’ attention and to obtain more

information about their learning process. In recent

research by Andrà et al. (2009), a comparative

analysis was carried out concerning the approach of

experts and non-experts to mathematical

representations. Through the eye-tracker tool, they set

out to study the pattern of eye movements of the two

groups. Following on from this study of experts and

non-experts, we wish to carry out a qualitative survey

on an analytical concept that seems to present

significant difficulties for students. They, in fact, had

many problems with the concept of derivatives and

slope of a function in a 2008 TIMSS Advanced

survey and in an INVALSI task. INVALSI is the

institution that provides periodic and systematic

testing of Italian students' knowledge and skills; in

particular, it manages the National Assessment

System (SNV).

The annual tests involve all Italian students of

grades 2, 5, 8, 10 and 13. In recent research, the

authors have analysed why students encounter

difficulties in some tasks. They argue that in cases

where students have to apply only one procedure,

they are able to give the correct answer more easily.

When they have to interpret the meaning of the

concept, they are in great difficulty (Gambini et al.

2020). To improve understanding of a concept, Tall

(Tall, 2003) suggests working on its meaning in the

graph. In particular, Tall talks about three

mathematical worlds in which the mathematical

concept takes shape: the embodied world, symbolic

world and axiomatic world. In the embodied world,

in fact, an individual learns through perception. In

this case, it is useful to work on the graph to show the

meaning of the derivative, and then to delve into

symbolic and axiomatic meaning. The INVALSI’

results, in fact, show that in tasks where the concept

of the derivative was linked to the concept of velocity,

students were able to answer more correctly than in

the task presented in this paper (Gambini et al., 2020).

The aim of this research study is to investigate the

problem-solving process as performed by experts and

non-experts, and to analyse their eye movements.

Thanks to the responses to the interviews, it was

possible to understand candidate awareness about

what they looked at while performing the task. In

addition, it is possible to examine which elements

caught the attention of the candidates and what

changed after attending the Analysis I course.

The research questions are:

● Is there a difference between which elements

caught the attention of experts and non-

experts?

● How do these elements influence the problem-

solving process of the candidates?

● Is there a difference between the results of

standardised assessment and the solution

proposed by the non-expert sample?

We predict that the observational approach of the

experts and non-experts is different. We think the

experts' viewpoint focused most on the angular

coefficient of the line equation. Instead, we believe

that non-experts looked mostly at the area of the

graph where lines meet the curve and the ordinate of

the tangency point.

3 METHODOLOGY

This paper presents a qualitative analysis based on

data obtained using the eye-tracker instrument. The

The Concept of Derivatives Through Eye-Tracker Analysis

379

subjects of this research were students of scientific

courses, for example Chemistry, Engineering,

Physics or first year of Mathematics, and

Mathematics graduates, PhD students or high school

teachers of Mathematics. The group composed of

scientific faculty students who took Calculus in the

first year is called the “non-experts” group, while the

term “experts” is used to refer to the group composed

of PhD students, high school teachers or Mathematics

graduates. The two types of candidates were

compared in order to investigate tracked cognitive

processes and highlight similarities and differences

between the two groups.

In this study a screen-based eye-tracker

instrument was used, which can collect gaze data at

60 Hz. This tool is designed for fixation-based

analysis, and it consisted of a binocular camera with

a precision of 0.10° RMS and an accuracy of 0.3°

under optimal conditions. These values of precision

and accuracy were necessary to obtain the heat map

as in the following figures (for example, see Figure

3).The method is based on a collection of images of

both eyes by camera; in this way, it is possible to have

a better position of the gaze in the space and the

diameter of the pupil. The eye-tracker tool was linked

to a computer to analyse data collected with software.

This software provides tools of analysis like creation

of heatmaps, gaze plots and video recording of eye

movements. In this way it was possible to perform a

comparative analysis between the data of candidates.

The heatmap is a graph in which the most interesting

areas are represented with a warm colour. These areas

were observed for many times or for a long time;

therefore, these areas captured most attention from

the candidates. The gaze plot provides information

about the trajectory of the eye movements on the

screen. Fixation durations are used to represent time

spent watching the visual stimulus. The eye-tracker

instrument was calibrated for each subject. In fact,

before being shown the stimulus, the candidate had to

follow with his/her eyes the cursor to calibrate the

tool; the test started after this phase.

Candidates were given a Calculus task on the

concept of derivatives with no time limit to solve it.

Candidates read the text of the task on a monitor

where an eye-tracker camera was placed. The eyes of

the candidates in this research, while performing the

task, were monitored by the eye-tracker instrument.

Therefore, candidates knew that they had to keep their

eyes on the screen throughout the test. The eye-

tracker detected and recorded eye movements while

the subjects were performing the task. After

candidates had solved the task, a blank screen was

shown to them, so that the recording of eye movement

data was stopped, while keeping the candidates’ eyes

on the screen at all times. Afterward, candidates were

subjected to an interview to understand the problem-

solving process chosen. In addition, during the

interview of the subjects, the task was shown to them

again to detect and record their eye movements

during this phase. Data collected by an eye-tracker is

useful to understand the cognitive processes of

candidates and motivation of the problem-solving

strategy chosen. During the interview, the following

questions were asked:

● What did you look at the most - the graph or the

text of the task?

● Which elements in the text most caught your

attention?

● Which elements in the graph most caught your

attention?

● Which element did you start from when looking

for the solution?

● What element enabled you to find the solution?

● Did you first read the text of the task and then

look at the graph, or vice versa? Why did you do

this?

Candidates were vocally recorded for later analysis of

their answers to the interview questions. In this way,

the data collected by the eye-tracker were

reconnected to subjects’ answers, making it possible

to analyse the cognitive process triggered by

candidates.

4 ANALYSIS OF RESPONSES TO

THE TASK

This task was included in the TIMSS Advanced

survey of 2008. Moreover, a similar version of it was

used in the pre-test of grade 13 in the INVALSI

survey.

In this experimentation, thanks to data collected

by the eye-tracker tool, it is possible to carry out a

qualitative analysis organised in levels.

The text of the task is as follows:

“The line of equation 𝑦=





𝑥−2 is tangent at

point P with abscissa equal to 2 to the graph f in the

image. What is the value of f ‘ (2)?”

In the first macro-level of analysis, it is possible

to divide candidates into three categories:

● in the first category, there are candidates

who prefer to focus their attention on the

text of the question. They almost

completely ignored the graph of the task.

We call this category “type T”;

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380

Figure 1: Graph of the task.

● in the second category, there are

candidates who read the text quickly and

after spending more time on the graph,

try to solve the task through the graph

information. We call this category “type

G”;

● in the third category, there are candidates

who favour neither the graph or the text.

Eye movements of individuals move

between text and graph with quick

saccades. We call this category “type

TG”.

In the micro-level of analysis of these categories,

we tried to highlight the distribution of expert and

non-expert students. It is possible to observe that the

expert candidates belong to the first category (we call

these candidates T-E). In contrast, non-expert

candidates are subdivided into category type T (T-

NE), category type TG (TG-NE) and there is one

individual who belongs to the category type G (G-

NE).

Apart from the clear division in these categories

between expert and non-expert candidates, it is

important to point out that the task is an open-ended

question, and, moreover, the answer is in the

stimulus. Analysis of the graph is not crucial to find

the correct solution to the problem. In fact, to solve

the task correctly, it is necessary to connect the

concept of derivative in a point, expressed by f ‘(2) in

the text, with the angular coefficient of the tangent

line, expressed by the equation 𝑦=





𝑥−2.

The first concept, belonging to a purely analytical

representation, is linked to the second one (belonging

to a purely geometric representation) through the

concept of angular coefficient (algebraic/analytical

representation) of the tangent line (geometric

representation), thus expressed through a "mixed"

representation, according to the following scheme

(Figure 2):

Figure 2: Scheme of representations involved.

4.1 T-E Candidates

The heatmap in Figure 3 (below) shows that the

fixations of the expert candidates focused on those

parts of the text connected with the angular

coefficient. In fact, these parts are sufficient to solve

the task.

Figure 3: Heatmap of the T-E candidates.

These fixations are linked to the cognitive process

of the expert candidate, who does not perform a

mental calculation. This result is confirmed by the

subsequent interviews. More than one expert

candidate states: “the elements I looked at the most

(in the text, NdA) are: angular coefficient and the

question of the task”. The occasional gazes at the

graph are connected to the ease of the task. In fact,

this represents a disorienting element for experts, so

they check more closely the request of the task and

make control evaluations. These could be due to the

subject’s anxiety, which is inversely proportional to

the perceived difficulty of the task. In fact, an expert

candidate says: “I saw that the task was very easy and

that is why I thought that there was a trap […] I

checked that the abscissa of P was 2”.

The candidate's heatmap, recorded during the

interview (Figure 4), is the one drawn while the

subject is solving the task. However, there is a

fixation (the only noteworthy example) of the

candidate on the point of tangency. The rest of the

graph was almost ignored.

The Concept of Derivatives Through Eye-Tracker Analysis

381

Figure 4: Heatmap of the T-E candidates during the

interview.

This action can be justified because the expert

candidate gives the local value of the derivative of the

function at a point. It is possible to say that expert

candidates see the task’s graphic register as a

confirmational element. In fact, they are able to

highlight essential elements of algebraic or analytical

nature useful in providing an answer based on simple

definitions, decreasing the phase of calculation or

graphic/geometrical analysis. This process of

reduction of useful information is clear by the textual

part “at point P with abscissa equal to 2” was

observed less by the candidates, because this

information belonged to “f ‘ (2)”.

4.2 T-NE Candidates

Non-expert candidates who looked mostly at the

textual area of the task have a heatmap which varies

little from that of T-E candidates. However, their

conclusions are different; this means that the use in

the cognitive process of the visual elements, obtained

by eye exploration phase, is different, interpreting

incorrectly acquired information.

Figure 5: Heatmap of the T-NE candidates.

Although the candidate’s attention is focused on

the same textual elements, it is possible to observe

that it is more uniformly distributed across the text

(Figure 5). This marks a lower ability to select the

elements useful in solving the task. Confirming this

weakness, non-expert saccades are shorter than those

of expert candidates and their fixations have a shorter

duration (as can be seen from the gaze plot, which we

have not reported here due to limited space).

Although the abscissa of the tangency point was

ignored, it is the only part of the graph which may be

considered essential. In contrast with the expert

candidates, observation of the graph belongs to the

exploratory phase, and it was soon abandoned,

because no useful elements were identified to solve

the task. This is evidenced by the saccades between

the textual part and graphical part, which were almost

absent. This indicates the absence of any cognitive

process of providing links between textual and

graphical data.

By analysing the textual register, it is possible to

observe many saccades between the line equation and

the demand of the task. This can be justified by the

way the T-NE candidates perform the problem: they

compute the line’s derivative and determine the

solution by equating the function’s derivative at line’s

derivative in the same point. Therefore, the candidate

knows the link between derivative and tangent line,

but not between derivative and angular coefficient.

Therefore, he needs to carry out explicit calculations

to solve the task, indicating a need for formal

justification to determine the solution of a

mathematical problem. This is also confirmed by a

discomfort, expressed in the following interviews,

about inexplicit knowledge of the function f(x). One

candidate states: “I could not explicitly compute the

derivative of the function in 2, I calculated the

derivative of the tangent line in 2 and I thought the

two values were equal”. Therefore, candidates T-NE

prefer an analytical/algebraic method of solving the

problem and this leads them to do analytical

calculations to determine solution. The graph is an

irrelevant element for them.

4.3 G-NE Candidates

Non-expert candidates, who focus their attention

mainly on the graphical part of the task, have an

opposite approach from T-E candidates. The analysed

data show some saccades between the line equation

in the textual part (with particular attention to angular

coefficient) and tangency point in the graph part.

Candidates almost completely ignored the rest of the

text, including the request of the task. The candidates

know the importance of the angular coefficient of the

tangent as a solving element, but they are not able to

CSEDU 2023 - 15th International Conference on Computer Supported Education

382

determine a direct link between the coefficient and

the derivative of the function f. Therefore, they are

unable to transfer the information obtained about the

tangent line from an analytical point of view to the

graph of the function represented. They try to use the

knowledge of the angular coefficient obtained in the

textual part to find a connection with the graphical

element in order to enact a cognitive process to solve

the problem.

Figure 6: Heatmap of the G-NE candidates.

The heatmap (Figure 6) shows that fixations of G-

NE candidates are focused on graphical properties of

the point of tangency, which is observed through eye

movements along the tangent line and the behaviour

of the function. This is the only category in which

candidates try to determine graphically the analytical

behaviour of the function, looking for distinctive

visual elements, such as intersection with abscissas

axis or transition at points near the tangency point

(helped also by the presence of the numbers in the

graph). The heatmap highlights many fixations and

saccades in a large (global) area that follow the

behaviour of the green curve. One candidate states: “I

tried to understand what the parabola equation was

…”. From this perspective, we can point out that,

sometimes, a non-expert candidate associates

increasing nonlinear behaviour with a parabola graph.

This probably occurs because a parabola is the most

familiar nonlinear behaviour for high school students.

Therefore, the cognitive process of a candidate G-NE

follows an opposite process to that of T candidates:

they behave as if the graphical register were essential

to obtain all the information needed to solve the task,

and afterwards to translate it into the analytical

4.4 TG-NE Candidates

Candidates of this category display many saccades

between the text of the task and the graph. Their

approach to the execution of the problem is based on

a continuous comparison between the textual part,

with fixations focused on the angular coefficient of

tangency line and question of the task f ‘(2) (similarly

to candidates of category T), and the graphical part,

with fixations focused in particular on the point of

tangency, but with considerable saccades and short

fixations following the behaviour of the function up

to the axis origin (Figure 7).

Figure 7: Heatmap of the TG-NE candidates.

A comparative approach of this type requires a

continuous change of the semiotic register, from

algebraic to graphical. This transition occurs through

a mental process that requires the transformation of

two registers using analytical knowledge that one

should acquire after a Calculus course, which makes

this approach the most complicated. In fact, from this

method it is possible to posit a typology of analytical

mistakes presented by TG-NE candidates, connected

to some misconceptions:

1. confusion between the value of the function at the

point and the value of the correspondent

derivative in the same point: the TG-NE

candidates check the ordinate of the tangency

point and often answer the task question with that

value. One of them states: “I checked that

tangency point was at 2 and as tangent line and

function have the same value in that point, that

value is the solution…”;

2. a wrong attribution of globality to the local

problem: visual attention (and cognitive) to the

behaviour of the function even at the points far

from the tangency point, are a feature. A TG-NE

candidate states: “…I tried to observe only the

tangency point, but not knowing the behaviour of

the function, I was not able to understand the

derivative”;

3. (linked to previous point) the lack of distinction

between the value of the derivative of the function

in one point

The Concept of Derivatives Through Eye-Tracker Analysis

383

f ‘(x

) and derivative of function f ‘(x). This is

highlighted also in the language used: one

candidate states that “the first derivative of the

function at that point is a tangent line to the

function at that point, identifying a number

(derivative in one point) with a curve (tangent

line). From this perspective the behaviour of the

supposed parabolic (mentioned earlier) and the

supposed analytic quadratic behaviour for the

function, can explain why the derivative is a line

(tangent line). This excludes the fact that if a

behaviour was exponential or logarithmic, the

tangent curve could not be a straight line.”

Definitely, in the approach used by TG-NE

candidates, it is possible to highlight a marked

distance between information acquired from

observation of the text and information obtained by

visual analysis about behaviour of the function.

Therefore, we can say that the graphical register is a

distracting element for these candidates.

5 CONCLUSIONS

This work is part of a more general project, which sets

out to analyse (by means of the eye-tracker tool) the

data obtained from the administration of questions

based on concepts learned in a standard Calculus

course. Two kinds of candidates were involved: the

experts, including university professors, high school

professors, doctoral students and master's students,

and the non-experts, i.e., students enrolled in the first

years of an undergraduate degree course of a

scientific faculty. The basic idea is that, by comparing

the data obtained, it is possible to "reconstruct" the

different approach and cognitive path used to tackle a

mathematical problem. The purpose is twofold: on

the one hand, it is possible to take a "snapshot" of the

delicate transition that a student faces in moving from

secondary school to university; on the other hand, it

is possible to try to derive useful indications to

improve the teaching of mathematics in a first-year

university course. In this paper, a qualitative analysis

was presented of a question based on a quantitative

analysis about an INVALSI task. It concerns the link

between the concept of the derivative, the angular

coefficient of the tangent line and the slope of the

graph of a "smooth" function at a point. The results of

the Italian INVALSI assessment referring to the same

task were as follows: Correct 13%, Incorrect 57%,

Missing 30%. As we mentioned above, the purpose

of the eye-tracking analysis is to figure out the student

behaviour.

The use and interpretation of the different

theoretical concepts used in the test allowed us to

hypothesise the cognitive processes implemented by

the different types of participants. The nature of the

representations involved in the scheme in Figure 2 are

represented by the two different and distinct areas of

interest distributed over the question: the textual part

expressed in analytical/algebraic register and the

figurative part expressed in geometric register. What

was possible to observe is a very clear

characterisation of the four expert candidates (T-E

candidates), who preferred a purely analytical

approach. For them, therefore, the main visual (and

cognitive) area of interest was textual, with particular

attention paid to the question request (f’(2)) and the

angular coefficient of the tangent line, while the

figure assumed only the role of confirmation or

control. For such candidates, the previous scheme is

strongly shifted to the left and the answer to the

question was unanimous and correct. The division of

the nine non-expert candidates was more complex.

Three of these candidates (T-NE candidates) also

followed a purely analytical approach, but the link

between the first derivative and the angular

coefficient of the tangent line was less decisive: they

preferred to calculate the derivative of the equation of

the tangent line and to identify the concept of the

derivative of the function with that of the tangent line.

These candidates answered the question correctly and

the figural register was essentially irrelevant. Four of

them chose an "intermediate" approach (TG-NE

candidates), with areas of interest evenly distributed

between text and graphics. For these candidates, the

diagram was the main focus: they tried to relate the

equation of the tangent line to its graph, losing sight

of the (local) concept of the first derivative at a point

and the (global) graph of the tangent line. Two of

them answered incorrectly, confusing the slope of the

graph of a function with the value of the function at

that point. Finally, one non-expert candidate (G-NE

candidate) approached the question from the opposite

side to that of the experts. His area of interest

containing the figure was clearly predominant: he

tried to determine the slope of the graph from the

slope of the tangent line, calculated using the grid

(although his answer was not correct). The graph

takes priority in his approach and his diagram is

strongly shifted to the right. It was not possible to

deduce from this purely qualitative analysis that there

are statistically significant correlations between the

different methods of approaching the question, the

candidates' prior knowledge and the outcome of the

question itself. To this end, we intend to acquire a

large amount of data in the coming months so that we

CSEDU 2023 - 15th International Conference on Computer Supported Education

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will be able to carry out a more quantitative analysis.

However, the information acquired about the

cognitive processes were important to underline the

observation of a mathematical problem articulated in

different registers, such as the one we experimented,

and the theoretical information that should be

acquired as the primary objective of a basic course in

Calculus, in order to gain useful information on the

best teaching methods that can be used and possible

technologies suited to support such methods.

We think this can also be helpful from a teacher

professional development perspective (Spagnolo et

al., 2022).

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