Artefacts Teach-Math. The Meaning Construction Of Trigonometric Functions

DOI: 10.1478/AAPP.99S1A15
AAPP | Atti della Accademia Peloritana dei Pericolanti
Classe di Scienze Fisiche, Matematiche e Naturali
ISSN 1825-1242
Vol. 99, No. S1, A15 (2021)
ARTEFACTS TEACH-MATH. THE MEANING CONSTRUCTION
OF TRIGONOMETRIC FUNCTIONS
ANNAROSA SERPE a AND MARIA GIOVANNA FRASSIA b ∗
ABSTRACT. Trigonometry is an area of the high school mathematics curriculum strictly
related to algebraic, geometric, and graphical reasoning. In spite of its importance to both
high school and advanced mathematics, research has shown that trigonometry remains a
difficult topic for both students and teachers. A new approach the teaching of trigonometry
– calibrated for the 21st century – opens the question of the place and nature of trigonometry
in contemporary high school mathematics. In this prospective, the authors have carried
out a study aimed at explaining connections between research and teaching practice of
trigonometric functions emphasizing conceptual understanding, multiple representation
and connections, mathematical modelling, and mathematical problem-solving. This paper
shows a teaching approach for meaning construction of trigonometric functions with the aid
of technological artefacts.
1. Introduction
Trigonometry is a standard component of high school mathematics curriculum in all
countries, usually in the latter half of secondary years. It is the precondition for understand-
ing more advanced concepts it is the basis of advanced learning in mathematics and it is
necessary for the formation of a mathematical language.
Trigonometry is also of paramount importance in daily life, it has attracted research atten-
tion due to its historical development and its current importance in mathematics education.
Unfortunately, many high school students do not understand its benefits, historical usage,
or application to daily life (Tuna 2011). From a more optimistic viewpoint, students may
consider trigonometry as a component of mathematics, rather than as an independent subject
(Adamek et al. 2005; Dündar 2015). Yet despite its importance to both high school and
advanced mathematics and science, research has shown that trigonometry remains a difficult
topic for both students and teachers (Weber 2005; Brown 2006; Thompson et al. 2007).
According to some research studies, these difficulties find a reason for a traditional teaching
of trigonometry based on two separate and unrelated approaches: the trigonometry of
triangles and the trigonometry of periodic functions (Orhun 2005; Brown 2006). According
to “Standards and Principles for School Mathematics” by the National Council of Teachers
of Mathematics (NCTM 2000) in a new approach to teaching of trigonometry –calibrated

A15-2 A. SERPE ET AL.
for the 21st century– is it fundamental that emphasizes a conceptual understanding, multiple
representation and connections, mathematical modelling, and mathematical problem solving.
This would be made possible a technology-rich environment that encourages reasoning,
communication, problem-solving, and confidence building (Hohenwarter and Lavicza 2007;
Güyer 2008).
Mathematical modelling applications provide to develop students’ analytical thinking
abilities, problem-solving abilities, and the qualifications needed in this technology-based
information (Serpe 2018).
In this prospective, the authors have carried out a study aimed at explaining the connec-
tions between research and teaching practice related to trigonometric functions with the aid
of technological artefacts. The research question set by the authors is as follows: how to
build the meaning of a trigonometric function?
Just as taking a square root or cubing, a number can be thought of as operations applied
to numbers, the terms sine, cosine, and tangent can be thought of as mathematical operation
applied to angles. The trigonometric functions, in most cases, are presented as ratios that can
be applied to labeled right-angled triangles. In fact, an approach used by many teachers to
introduce trigonometric ratios is to have students draw many right-angled triangles with the
same angles and to explore the ratios of sides. Such an activity continues to be worthwhile,
although it is tedious and, of course, somewhat error-prone when student measurements are
used.
An excellent means of using this idea with the aid of technology involves constructing a
dynamic triangle using a Dynamic Geometry Software (DGS) and hence automating the
exploration to some extent. Through manipulating such a triangle, students can experience
the key relationship that the ratio is independent of the size of the triangle, of fundamental
importance when defining trigonometric ratios. For example, the use of DGS such as
GeoGebra greatly helps didactics, as it lends itself both to visualization and exemplification
and/or exploration (Hohenwarter et al. 2009; Robutti 2013; Serpe and Frassia 2020).
Below, after reporting the basic lines of theoretical framework followed for the research
study, the authors show a teaching approach for the construction of meaning of trigonometric
functions with the aid of artefacts. The teaching practice design involves three phases
in an active learning environment, and by building on students’ prior and developing
understandings and ways of thinking, the sequence of activities is designed to support the
development of coherent meanings for several trigonometric concepts and ideas, including
trigonometric functions.
The paper is structured as follows. Section 2 presents the theoretical framework used.
Section 3 covers all steps of the teaching practice; and finally, in section 4 conclusions and
recommendations are suggested.
2. Theoretical framework
The theoretical framework is grounded on the instrumental approach and the semiotic
mediation approach. The first theoretical element considered is the instrumental approach,
based on Rabardel’s work (Rabardel 1995), and dedicated to a cognitive approach to
contemporary instruments, mainly those connected to new technologies, focusing on the
tool-user relationship. Rabardel’s work is based on the distinction between artefact and
Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat., Vol. 99, No. S1, A15 (2021) [13 pages]

ARTEFACTS TEACH-MATH. THE MEANING CONSTRUCTION OF TRIGONOMETRIC FUNCTIONS A15-3
instrument: an object created by a person remains an artefact until the user devises a scheme
for its use. In other terms, an artefact is a material or abstract object aimed at sustaining
new human activity in the performance of a type of task, while an instrument is what the
subject builds from the artefact (Maschietto and Trouche 2010).
In his book, Rabardel starts from the observation that no artefact is given directly to
the user; rather, the user personally elaborates it through what the author calls activities.
Rabardel defines instrumental genesis as the process which enables the user of an artefact
to build the instrument. Instrumental genesis consists of two complementary processes:
instrumentation and instrumentalisation. The two processes differ in the direction of the
activity: in the instrumentation, the activity is directed from the artefact towards the user
involved in the learning of socially shared usage schemes; in instrumentalisation, the activity
is directed from the subject towards the artefact component of the instrument. The user can
create new functions while preserving characteristics and properties of the artefacts. Another
essential theoretical element to consider for the planning and analysis of experimentation
is the use of cultural artefacts as instruments of semiotic mediation for the construction
of mathematical meanings, developed by Bartolini Bussi and Mariotti (Bartolini Bussi
and Mariotti 2008) in a Vygostkian perspective. Bartolini Bussi and Mariotti take into
consideration the complex system of semiotic relations which can be established between
four essential elements: an artefact, a task, a part of mathematical knowledge and processes
of teaching-learning in class. In a similar system, on the one hand, the artefact is connected
to a task accessible for the learner, while, on the other, it connects to the mathematical
knowledge that the teacher wants to elicit. The main issue is how to establish communication
between the physical interaction with the artefact by the learners who are executing a task,
and subject knowledge.
In this context we propose a distinction between three categories of signs (Maschietto and
Trouche 2010):
• artefact signs: often signs of a subjective nature that refer to the artefact’s context
of use;
• mathematical signs: signs connected to the subject meanings, part of the cultural
tradition;
• students’ mastery of them represents one of the goals for the teacher;
• pivot signs: intermediate representations, between artefact signs and mathematical
signs.
Taking a semiotic perspective means to interpret the teaching-learning process recogniz-
ing the central role of signs (Arzarello et al. 2009) in the construction of knowledge, and
focusing on the link that can be established between artefacts and mathematical knowledge.
In a mathematics class context, students can be guided to produce personal signs, which
can be put in relation with mathematical signs. Then, the role of the teacher is very important,
because he or she should act as cultural mediator able to enhance the possible relationship
between the three different types of signs which concur in equal measure to the acquisition
of subject knowledge (Maracci et al. 2013).
The present work fits into this theoretical framework and puts forward an example of
teaching practice design for meaning construction of trigonometric functions with the aid

of technological artefacts. In this way it is, also, highlighted how the Mathematic interacts
with Physics and provides the conceptual tools needed to solve the problem.
3. Teaching practice design
This uses a constructivist approach for developing students’ creative, logical, and analyti-
cal thinking skills. Intended to enhance the strong link between mathematics and reality, the
approach is centred on problem-solving activities with artefacts so students can experiment
how to apply the acquired skills in order to find out or verify proprieties, using inductive
and deductive reasoning. According to Dündar (Dündar 2015), a significant factor in the
efficient performance of problem-solving activities is the type of problem, its structure, and
its representation form (verbal, visual, symbolic). Representations play an important role in
developing mathematical understanding. Use of representations in teaching trigonometry
reinforces students’ understanding when constructing concepts and solving problems.
The teaching practice design –based on a simplified modelling cycle process– consists
of three phases with several important goals: to help students connect concepts in geometry
and algebra, to help them achieve an understanding of how mathematics ideas develop, and
help these students see that mathematical principles and properties have foundations in logic
and are not just arbitrary rules –contrary to what many students believe.
In Figure 1, we can see the framework for teachers’ roles in modelling (Carlson et al. 2003).
FIGURE 1. Framework for teachers’ role in modelling.
On the basis outlined above, the three phases of the experimented teaching practice
are shown below. Each phase is always correlated with the others and is explored in its
interrelationship. Starting from a problem, we approach the goniometric functions of the
angle starting from an acute angle with the use of GeoGebra; then, in a gradual manner, we
finally arrive at the resolution of the problem.
GeoGebra allows the simulation of the model and related affordances. Classic heritage is
transformed into information technology activities by this software, and the advantages of
their use in Mathematics teaching and learning have been clearly highlighted (Hohenwarter
et al. 2009; Frassia and Serpe 2017).
3.1. Pose Question. Mario’s dad has to cut down a dead poplar tree in his garden before
winter arrives. The tree is located at a point near two perpendicular fences that must not be
damaged. Moreover, in the garden there are plants, flowers and two benches, so that the

poplar tree can be knocked down only in directions perpendicular to the two fences (Figure
2).
Mario’s father does not know the height of the tree and has only two instruments: a
metric wheel (50 meters) and a clinometer. How can Mario help his father determine the
height of the poplar tree? Having determined the height of the tree, can Mario’s dad manage
to cut down the tree without causing damage to the two fences?
FIGURE 2. Plan of the garden.
After reading the text, the teacher starts a discussion with the students to examine the
problem. Students are intrigued by the clinometer because it is an instrument they do not
know. The teacher shows the students a clinometer which is better known as the Abney1
level after its inventor (Figure 3).
FIGURE 3. Clinometer.
Students are therefore encouraged to carry out an appropriate research on the Internet in
order to acquire a more complete cognitive picture of the clinometer.
The research is also aimed at proposing a more complete approach to the historical-
scientific heritage, looking at it as a whole, which is to say in the three types of material
that constitute it, not only the collections of instruments, but also the book collections and
archival records.
This research leads the students to the Hoboken Historical Museum 2
where in the online
Collections Database they have the possibility to consult the Manual: Abney Levels. Uses &
Applications with Appropriate Tables (K&E Manual 1950) and download a PDF file of it.
An example from this is in Figure 4 which shows page 1. 3
.
1
Sir William de Wiveleslie Abney (24 July 1843 – 3 December 1920) was an English astronomer, chemist,
and photographer.
2
https://www.hobokenmuseum.org/
3
credit https://hoboken.pastperfectonline.com/archive/FAE698C2-6811-4FBC-82EA-186223979366

FIGURE 4. The pages 1 of book Manual: Abney Levels. Uses & Applications
with Appropriate Tables (K&E Manual 1950).
From reading the manual the students learn that the clinometer is an instrument used
for measuring angles of slope (or tilt), elevation, or depression of an object with respect
to the direction of gravity. Thanks to its characteristics, the Abney level can be used to
determinate the height of an object and as a levelling instrument also.
For example, in the Figure 5, the instrument in its entirety is in a truly horizontal position.
FIGURE 5. The Abney Level in a horizontal position (Usill 1901).
In contrast, the Figure 6 shows the instrument being used for the angle of acclivity (which
in this case in 34 deg. 15 min.), and that of declivity, or 19 deg. 30 min. with the horizon
(Usill 1901).

FIGURE 6. The different positions of the Abney level with the horizon.
Gradually, thanks to the type of research carried out (Physics and History) the students
grasp the versatility and extraordinary nature of this tool and the role it has played in the
various fields of science.
This historical excursion, connected to a careful observation of the characteristics of the
instrument, constitutes an opportunity for students to acquire further knowledge concerning
the connection between the development of physical knowledge and the historical and
philosophical context in which it developed.
At this point, the teacher will want to give further emphasis to the fact that the tool is
based on geometric techniques and principles that are transformed into practical protocols.
For this purpose, to become familiar with the tool, the teacher can organize an activity that
involves measuring the height of a tree in the schoolyard.
3.2. Return to the initial question. Students’ attention is redirected to the initial problem
because it is necessary to identify possible strategies for finding solutions (Build solutions).
The teacher starts a discussion with the students.
In Table 1 an extract of the discussion protocol is shown.
TABLE 1. Extract of discussion protocol between students (S) and teacher (T).
Urged on by the teacher, students perceive the importance of making a sketch of the
problematic situation. In fact, being able to associate the corresponding geometric object

with the data of the problem constitutes the first step towards the mathematical modelling
process.
FIGURE 7. Students’ sketch.
At this point, the students using the sketch made (Figure 7) can work out a possible
solution.
TABLE 2. Extract of the next discussion protocol between students (S)
and teacher (T).
The abstract of the aforementioned protocol shows a first geometric modelling of the
problem.
3.3. Build Solution. In this phase the student must model the real situation with the use of
a representation component. The introduction of a practical and representative dimension
by means of the GeoGebra software leads the students to structure the actions performed
and their effects. Students now have to implement the algorithm inherent in the geometric

construction of the problem with GeoGebra.
The steps in solving the algorithm are as follows:
1. Draw three points non-aligned A, B and C in the Euclidean plane;
2. From the A point trace the rays as follows: f passing through B point, and g passing
through C point;
3. Choose a P point on ray g;
4. Draw the perpendicular line t to the ray f and passing through P point;
5. Draw the H point of intersection of the straight line t and the ray f.
Find the values of following ratios:
PH
AP
;
AH
AP
;
PH
AH
; (1)
FIGURE 8. Output of geometric modelling.
The proposed modelling involves the elementary geometric entities, so at first it may
seem detached from the initial problem posed. In fact, the students often reveal concerns
about the usefulness of this construction for the resolution of the problem.
The teacher, for his/her part, pretends to ignore the students’ worries and asks them the
following question: how do the ratios, previously calculated, vary with the variation of
point P on the ray g?
The question is aimed at making students aware of the fact that the aforementioned
ratios remain constant. With the formalization of the algorithm, the students will see how
variations of point P along the ray g do not vary the values of the calculated ratios, i.e., they
remain constant. This awareness becomes a further object of discussion: many students
agree that these ratios must have ’some link’. The teacher encourages students to observe
more criticality. The action seen in an algorithmic way (dynamic displacement of point P

along the ray g) leads the students to affirm that: “As P changes on g, all the points move,
but the angle and the calculated ratios remain the same”.
3.4. Validated conclusions. The introduction of a practical and representative dimension
by means of a DSG like GeoGebra leads the students to structure the actions performed
and their effects into a specific descriptive framework that makes the simulation a useful
learning process. The teacher now provides the missing numerical data for the problem:
α = 50° , β = 40° , AB = 9,24 m , AC = 13,09 m (2)
so that students can solve it autonomously.
In Table 3 an extract of the validated conclusions is shown.
TABLE 3. Extract of the validated conclusions between students (S).
4. Conclusions
The use of artefacts adds value to mathematical learning, because their use as teaching
aids facilitates the ability to make conjectures and construct even complex demonstrations
on their properties and functioning.

The use of technological artefacts is helpful in presenting trigonometric concepts or
events as cognitive tools in constructing understanding.
In particular, the introduction of a practical and representative dimension by means of the
software GeoGebra leads the students to structure the actions performed and their effects
into a descriptive framework, a process through which they gradually acquire familiarity
with arguments and demonstrations of increasing complexity.
In this research, the students exhibited an increased level of comfort in articulating their
meanings, thinking, and reasoning, and critiquing the thinking and reasoning of others -
including their teacher and the researchers. Furthermore, the sequence of the phases allowed
students to think multidirectionally and transfer information; this strategy also ensured the
formation of conceptual knowledge and an accurate image of the trigonometric functions
not just from a technical point of view, but as fundamental tools for research and planning.
Furthermore, the Mathematics and Physics become immediately perceptible, so this phase
becomes a special opportunity for reflection on the inextricable connections between the two
disciplines and on the process with which a mathematical model is it constructed starting
from a real physical problem (Serpe and Frassia 2018).
The chosen methodology and the synergic interaction between the disciplines increases
students’ motivation facilitating true learning; in particular, they effectively improve the
ability to formulate and solve problems, as well as demonstrate the unitariness of technical
and scientific knowledge.
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a Università della Calabria
Dipartimento di Matematica e Informatica
Via Pietro Bucci, Cubo 30/A, 87036 Rende (CS), Italy
b IIS IPSIA-ITI “E. Aletti”
Via Spalato, 87075 Trebisacce (CS), Italy
∗ To whom correspondence should be addressed | email: annarosa.serpe@unical.it
Paper contributed to the international workshop entitled “New Horizons in Teaching Science”, which was held in Messina, Italy
(18–19 november 2018), under the patronage of the Accademia Peloritana dei Pericolanti
Manuscript received 03 December 2020; published online 30 September 2021
© 2021 by the author(s); licensee Accademia Peloritana dei Pericolanti (Messina, Italy). This article is an
open access article distributed under the terms and conditions of the Creative Commons Attribution 4.0
International License (https://creativecommons.org/licenses/by/4.0/).

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