Year 26 / N° 42 / 2024 /
DOI: https://doi.org/10.36995/j.recyt.2024.42.010
Teaching mathematics in pre-service biology teachers: logistic model of
population growth
Enseñar matemática en la formación de Profesores en Biología: modelo
logístico de crecimiento poblacional
Gretel A. Fernández von Metzen1, *; Verónica,
Parra 2, 3; Jonathan M. Schuster 1, 3
1- School of Exact,
Chemical and Natural Sciences (FCEQyN). National
University of Misiones (UNaM). Argentina.
2- School of Exact
Sciences, Mathematics Education Research Centre (NIEM). National University of
the Centre of the Province of Buenos Aires (UNCPBA). Argentina.
3- National Scientific
and Technical Research Council (CONICET).
*E-mail: gretel.fernandez@fceqyn.unam.edu.ar
Received:
06/05/2024; Accepted: 14/08/2024
Abstract
This paper describes and
analyses the construction and reconstruction process of mathematical and non-mathematical
knowledge that emerged while studying the logistic model of population growth.
This experience was carried out in first-year students of the Biology Teacher
Training Course (PUB, for its name in Spanish) of the School of Exact, Chemical
and Natural Sciences (FCEQyN, for its name in
Spanish) of the National University of Misiones (Argentina) in the subject
Mathematics. This study is part of the design and implementation of a didactic
device called study and research courses
(SRC). This device is framed in the Anthropological Theory of Didactics (ATD),
which aims to overcome the knowledge monumentalisation phenomenon and contribute to a paradigm
change in mathematics teaching. This experience seeks to transcend the
traditional approach where the teacher explains, and students repeat mechanised techniques. So far, the results obtained are
encouraging since this method of learning mathematics has given back a sense of
usefulness to mathematics studied by the biology teacher training students.
Keywords:
Logistic model of population growth; university; biology teacher training
course; mathematics; Study and Research Course.
Resumen
Este trabajo describe y analiza el proceso de
construcción y reconstrucción de los saberes matemáticos y no matemáticos
emergidos durante el estudio del modelo logístico de crecimiento poblacional.
Esta experiencia se llevó a cabo por estudiantes de primer año del Profesorado
Universitario en Biología (PUB) de la Facultad de Ciencias Exactas, Químicas y
Naturales (FCEQyN) de la Universidad Nacional de
Misiones (Argentina) en la asignatura Matemática. Este estudio forma parte del
diseño e implementación de un dispositivo didáctico denominado Recorrido de estudio e investigación
(REI). Este dispositivo, se encuadra en la teoría antropológica de lo didáctico
(TAD), con el que se pretende soslayar el fenómeno denominado monumentalización del saber, y contribuir a la promoción
de un cambio de paradigma en la forma de enseñar matemática. Se pretende, a
partir de esta experiencia didáctica, trascender el enfoque tradicional donde el
profesor explica y los estudiantes repiten técnicas mecanizadas. Hasta el
momento los resultados obtenidos son prometedores pues, esta manera de estudiar
Matemática ha permitido a los estudiantes para profesor en Biología encontrar
un sentido y una utilidad a la Matemática estudiada.
Palabras
clave: Modelo logístico de crecimiento poblacional; Universidad; Profesorado
en Biología; Matemática; Recorrido de Estudio e Investigación.
1.
Introduction
There are several pieces
of research in the field of Didactics of Mathematics which refer to the
different contributions that emerge from the analysis of practices involving
the use and interpretation of mathematical models at the university (Bassanezi and Biembengut, 1997
[1]; Bosch et al., 2006 [2]; Barquero et al., 2006
[3], 2010 [4], 2014 [5]; Barquero, 2009 [6]; Biembengut and Hein, 2004 [7]; Ruiz Higueras
and García García, 2011
[8]; Costa, 2013 [9]; Parra, 2013 [10]; Oliveira-Lucas, 2015 [11]; Sureda and Rossi, 2022 [12]). In Barquero
et al. (2014) [5], it is stated that, presently, there is no discussion of the
convergence of thought around the need to incorporate mathematical modelling in
mathematical practices at all levels of schooling. However, these authors state
that, under the influence of the dominant epistemology, this is the concrete
way in which society, the university as a teaching institution and the study
community (teachers and students) understand what mathematics is, how it is
constructed and used; modelling has been restricted to the level of applicationism.
From this perspective, mathematical modelling activity has been identified as
“...a mere ‘application’ of previously constructed mathematical knowledge or,
in its most extreme case, as a simple ‘exemplification’ of mathematical tools
in certain extra-mathematical contexts artificially constructed for this
purpose” (p. 89).
Didactic research linked
to mathematical modelling within the framework of the Anthropological Theory of
the Didactic (ATD) has been oriented to investigate, on the one hand, how
mathematical modelling processes could improve the teaching and understanding
of mathematical concepts. Moreover, on the other hand, how to get students to
develop modelling skills associated with a specific professional field, if
mathematical modelling is conceived as a means and an end to teaching
mathematics, establishing a relationship with a particular discipline of the
professional field in question (Bosch et al., 2006) [2]. In this way, mathematical
modelling is thought of from a co-disciplinary perspective, in which intra- and
extra-mathematical modelling converge.
Furthermore, Barquero et al. (2006) [3] emphasise
that although the faculties linked to experimental sciences constitute an ideal
space for teaching mathematical modelling, the knowledge taught generally
represents finished, crystallised mathematical organisations. These make almost no reference to the
context and origin in which they were created. In the process of didactic
transposition, what is linked to the modelling activity is lost. Barquero (2009) [6] points out that research on
mathematical modelling, in general, is divided into two categories: those that
consider mathematical modelling as content to be taught and those that consider
modelling as a means to teach mathematics. In this regard, Bassanezi
and Biembengut (1997) [1] propose the concept of
"simplified mathematical modelling" to distinguish the modelling
process carried out in the scientific community from that carried out at the
school level. To this end, the authors
suggest adapting the mathematical modelling process according to the contents
established in a regular course syllabus.
Following the points described above, the concern arises to investigate
which mathematical, biological, chemical and physical organisations
can be reconstructed through the study of a solid problematic question and, in
turn, in what ways these would allow the development of mathematical modelling
practices in the subject Mathematics of the Biology Teacher Training Course.
This paper aims to describe and analyse the
process of construction and reconstruction of mathematical and non-mathematical
knowledge that emerged during the study of the logistic model of population
growth within the framework of the implementation of a didactic device called
Study and Research Course (SRC) with first-year students of the Biology Teacher
Training Course.
2.
Theoretical framework: The Anthropological Theory of the Didactic (ATD)
The theoretical
underpinnings that guided this research work are circumscribed in the ATD. This
theory conceives mathematical activity as a human activity that can be modelled
through praxeologies or mathematical organisations. This term includes two blocks of organisation of knowledge: that of praxis and that of
logos. The first corresponds to 'know-how', specifically to the problems and
techniques constructed and used for their treatment. Meanwhile, the second
comprises the discourse that describes, explains, justifies and even produces
the techniques used (Chevallard, 1999 [13], 2019
[14]).
ATD questions the
approach to topics in school institutions since, in general, they do not
provide a problematisation of the origin and
functionality of their existence in mathematics curricula. Therefore,
mathematics is conceived as a set of organisations or
mathematical works already constructed, disconnected among themselves and from
other disciplines, lacking meaning and functionality where it is only possible
to visit them but not to question and construct them. Chevallard
(2007 [15], 2017 [16], 2019 [14], 2022 [17]) explains this situation by using
the analogy of a visit to a museum,
where mathematics plays the role of a work of art, the students, the role of
museum visitors, and the teacher, the tour guide. Under this conception,
mathematics teaching is described as a visit to a work of art that the visitors
cannot manipulate and is there to be admired and revered. This phenomenon is
called knowledge monumentalisation.
In this sense, ATD promotes a paradigmatic change oriented towards the paradigm
of questioning the world, where mathematical modelling acquires a primordial
place in the teaching of mathematics (Gascón, 2022)
[18]. Here, mathematical modelling is thought of from a co-disciplinary
perspective, in which intra- and extra-mathematical modelling converge.
The notion of study and research courses (SRC) is
then presented as a didactic device whose primary function is to bring
mathematical modelling to life in teaching systems and, in addition, to promote
a paradigm shift in the way mathematics is taught. The SRC are characterised by the construction (and reconstruction) of
mathematical (and non-mathematical) knowledge as an answer to a question in a
strong sense called the generative question.
Questions that come from the discipline in question (biology in this case) and
whose answer is not the simple search for information on Internet sites,
textbooks or by going to the teacher, but requires the generation of other
sub-questions (called derived questions)
and the construction of models that allow to bring into play, collectively,
mathematical and non-mathematical knowledge.
3.
Methodology
This is a qualitative,
exploratory and descriptive research. This type of study allows us to gain in-depth
knowledge of specific characteristics, properties or profiles of particular
objects, groups of people or phenomena that we wish to investigate (Hernández Sampieri et al., 2014) [19]. In this case, the aim is to characterise the process of construction and reconstruction
of mathematical and non-mathematical knowledge that emerged during the study of
the logistic model of population growth in a regular course with first-year
students of the Biology Teacher Training Course (PUB, for its name in Spanish)
of the School of Exact, Chemical and Natural Sciences (FCEQyN,
for its name in Spanish) of the National University of Misiones (Argentina) in
the subject Mathematics. The subject is a year-long, and before the
implementation of the SRC, the first units of the course syllabus had already
been studied. The mathematics syllabus begins with an introduction to
propositional logic and set theory, then continues with the study of real
numbers, and ends the first four-month period with the development of real functions
of real variables.
The
information presented and analysed in this paper is
part of a research project belonging to the FCEQyN of
the National University of Misiones. It is also part of the development of a
doctoral thesis project being carried out by one of the authors.
Additionally,
the data analysed in the discussion section of the
results come from the written or audio records of the observations made in 6
(six) classes out of 28 (twenty-eight) classes during the second four-month
period of 2023. These six classes were selected rather than others because they
were the ones where the construction of the logistic model of population growth
was carried out. The study and investigation of this model are part of the
design and implementation of the SRC, the genesis of which lies in the
generative question:
Q0: How can it be determined whether
certain bodies of water are suitable for aquatic life?
Based
on the vast array of derived questions that emerged, one of the decisions taken
by the study community was about which path to take to study the generative
question. A collective decision was to study the dynamics of a population since
the problem question refers to bodies of water that are suitable for aquatic
life, so they considered it necessary to investigate the population behaviour of the different living organisms present in a
body of water.
The
implementation of the SRC began at the beginning of the second four-month
period of the year 2023 in the subject of Mathematics in the first year of the
PUB. To this end, we had the presence and collaboration of a graduate in
Genetics, who contributed to the study of biological knowledge during the
development of the lessons.
Twenty-three students
participated in the implementation and were divided into seven study groups:
four groups of four students, one group of three students and two groups of two
students each. Data recording was carried out using different instruments. The
classes were recorded in general videos of the course, field notes were taken,
and each group had to prepare a field log of what was being investigated and
studied regarding the problematic question and the derived questions.
Throughout the four-month period, they made three digital deliveries per group,
and, specifically in the case of the logistical model, they made multimedia
presentations about what they had researched on this model, which were
presented orally.
Finally, the extracts of
the students' productions presented here correspond to the final presentation,
in digital format, of a group field log made throughout the implementation of
the SRC.
4.
Analysis and discussion of the results
The classroom
implementation of the generative question generated a vast array of derived
questions that, for the sake of length, are not presented in this paper. This
section only describes the students' responses to the derived questions
associated with the logistic model of population growth. It should be clarified
that prior to working with this model, the students asked several derived
questions linked to the study and investigation of the exponential growth
model, but due to the length of the manuscript, it is not possible to present
their analysis in this paper.
After working with the
exponential growth model, the teacher suggested the question in Figure 1. This
derived question arose because the students stated that in reality, a
population does not grow indefinitely over time since, if this were the case,
the availability of resources in their environment would be unlimited.
Figure 1: Question derived from the Q0 question, which emerged after studying the
exponential growth model.
To provide an answer to
this question, the students investigated how to represent, through a
mathematical model, these environmental and resource constraints. Maximum
carrying capacity was one of the key concepts explored for this response.
Figure 2 contains part of one of the answers outlined by Group 4.
Figure 2: Excerpt from the answer to the derived question
given by group 4.
Group 2, on the other
hand, in addition to questioning about carrying capacity, asked other derived
questions that allowed them to delve deeper into the problem under study. These
questions refer, for example, to how carrying capacity is defined, what happens
if the population exceeds the carrying capacity, what overpopulation means, and
how this would affect the model, as shown in Figure 3.
Figure 3: New derived questions asked by group 2 based on
the question in Figure 1.
During the sharing of
each group's research, there was a discussion about what carrying capacity
means for a population and how it is expressed in the mathematical model they
presented. However, in the answers provided up to that point, there was no
suggestion as to how this model was related to the one previously worked on for
the exponential growth model. For this reason, the teacher in charge of
implementing the SRC recommended working hypotheses linked to the birth rate
and death rate of the size of a population, together with their respective
variability. The teacher presented both rates and analysed
how they behave in the exponential model (they remain constant) as opposed to
the logistic model. Figure 4 presents this teaching intervention.
Figure 4: Hypotheses suggested by the teacher to study
the logistic model of population growth.
By incorporating this
hypothesis and these new variables, the logistic model of population growth
could be worked on based on the modifications made to the exponential growth
model.
One of the issues analysed by the class group arose from a new question
derived question posed by the teacher:
What does it mean that in the equation
?
Students in group 1
stated that this meant that the birth and death rates were in equilibrium, as
outlined in Figure 5 (students do not use subscripts in this case). Following
the response given by the students, they were asked to expand on what it meant
to them that the birth and death rates were in equilibrium. At that point, they
could only state that the population growth rate was zero when the birth and death rates were equal,
but for that part of the SRC, they were still unable to relate that information
to the carrying capacity.
Figure 5: Excerpt from the answer given by students in
group 1 to the new derived question.
This question and the
answer given by the students were taken up in later classes when studying the
growth and decay of functions, critical points, and the concavity of a
function.
In this part, it is
essential to point out that the mathematical model presented by the students
involves the study of mathematical organisations
referring to initial value problems, non-linear first-order differential
equations, average growth rate and instantaneous growth rate, derivatives of
real functions of a real variable and indefinite integrals, among others. The
students had already worked on some of these mathematical organisations
during the development of the exponential growth model; however, from the
investigation and study of the logistic model, this mathematical knowledge
could acquire a new meaning for them. At the same time, this model enabled the
construction of new knowledge linked to increasing and decreasing functions,
critical points, extreme values of a function, concavity, inflection point and
horizontal asymptote in real functions of a real variable.
The discussion around
the information provided by the mathematical model of logistic growth continued from
what emerged from the presentation of group 2, who early on investigated what
happens to population growth by analysing the sign of
the first derivative. This is shown in Figure 6 and Figure 7.
The information
presented by the group made it possible, on the one hand, to undertake a
qualitative analysis of the logistic model. This means obtaining information
from the mathematical model about the population's behaviour
without the need to solve the associated differential equation. On the other
hand, it also made possible the construction of mathematical knowledge linked
to the growth of a real function of a real variable, which the class group had
not studied until now. The latter is considered an extremely valuable
contribution to the study community since it highlights the value of the
qualitative treatment of a mathematical model in relation to analytical
resolution techniques, which are generally prioritised
in the teaching of differential equations, as pointed out by Moreno Moreno and Azcárate Giménez (2003) [21] and Artigue
(1995) [22].
Figure 6: Excerpt from the presentation of group 2 on
population growth in the logistic model.
A collective discussion
about the meaning of the statements outlined both on the slide and on the
blackboard was triggered following the presentation of group 2. Also, the
question of where these statements are visualised in
the graphical representation of the logistic model emerged. It should be noted
that during the study and investigation of the exponential growth model, the
graphical representation of the logistic model appeared in some groups.
Figure 7: Excerpt from the study conducted by group 2 on
the behaviour of a population under the logistic model.
Based on the discussion
carried out by the study community, it was decided to introduce the derived
question: What do and
mean? And to advance the study of the growth and decay of a function
according to the sign of the first derivative.
For this question, the
students stated that the information found on Internet sites was not complete,
so it was suggested that they research a specific bibliography on differential
calculus of real functions of a real variable, some of which are available in
the university library and others in digital format.
Following the search for
information on the sign of the first derivative, questions arose about whether
the population growth rate behaves in the same way along the entire logistic
growth curve or whether it undergoes changes. This led the students to
investigate the derivative of the population growth rate and what information
the sign of the second derivative in the logistic model provides. Mathematical
knowledge also emerged concerning the inflection point and concavity of a real
function of a real variable. Figures 9 and 10 show part of the study carried
out by the students.
Figure 8 shows that the
students state that ‘the rate of change reaches its maximum’ at the inflection
point and indicate that a change of concavity occurs at that point; however,
during the oral presentation of their research, they do not relate this
statement to the study of the second derivative, nor what information the sign
of this derivative provides concerning how the growth of a population occurs
under the logistic model.
Figure 8: Excerpt from the presentation of group 4 on the
study of concavity and the inflection point.
Group 2 states that the
inflection point occurs in the middle of , but they do not justify this statement. In addition,
they indicate that ‘the change in concavity is a change in the growth rate’ and
associate it with the second derivative. However, there is confusion with the
information provided by the sign of this derivative. This is shown in Figure 9.
Figure 9: Excerpt from group 2 on the study of concavity
and the inflection point.
Since the growth of a
function had previously been studied by analysing the
sign of the first derivative, group 2 relates the sign of the second derivative
to the study of the growth of a function. Still, they do not indicate which
function they are referring to: the population function N or the function
associated with the derivative of this function.
Given these questions, the class group discussed together what it means
that and
, and which function is increasing or decreasing,
depending on the sign of this derivative. To do this, the students in group 1
made a graphical representation of the logistic curve and pointed out where the
second derivative of the function N shows such behaviours.
This is shown in Figure 10.
Figure 10: Graphical representation made by group 1 for
the study of the concavity and the inflection point in the logistic model,
which reads: “The inflection point takes
place when reaches its maximum value, at half of K, N =
The change in concavity is a change in the
growth rate (second derivative).”
From the graphical
representation, the students in group 1 pointed out that at the
inflection point, and that
is verified in
the part of the curve that is concave upwards.
Analogously, they said that in the part where the curve is concave
downwards, it is realised that
. This analysis allowed further exploration of the
meaning of the statement ‘the change in concavity is a change in the growth
rate’. Collectively, it was concluded that the change in the growth rate is
expressed by the derivative with respect to the time of the derivative of the function
. The teacher wrote the symbolic representation
on
the blackboard, following the conclusion developed by the study community.
Next, to analyse the statement that for the function is
increasing and that for
the function is
decreasing, the teacher proposes that the students go back to the right-hand
side of the equation
and analyse the function
.
It is worth mentioning
that when studying the exponential population growth model, the students had
studied initial value problems, so it was familiar to them to recognise in the expression on the right-hand side of the
logistic equation a function that depends on . However, it was the teacher who introduced the
discussion about the behaviour of this function. To
do so, she wrote
on
the board. This is illustrated in Figure 11a.
Figure 11: a) Graphical representation made by the teacher
to analyse where the maximum population growth
occurs. b) Analysis of the sign of the second derivative from the graphical
representation of the function of the first derivative.
This equivalent symbolic
expression of the function was recognised by the students as a quadratic function whose
graphical representation is given by a downward concave parabola since the
quadratic coefficient is negative. However, for the analysis of the roots and
the determination of the vertex of the parabola, it was necessary to remember
how to find such points of the curve for a quadratic function in general
. Thus, the students and the teacher determined that
the roots were at
and
, and that the coordinate corresponding to the
horizontal axis of the vertex was at the midpoint of the roots, given by the
expression
.
Based on this, the
statement written on the blackboard was brought up again: ‘The inflection point occurs when
reaches its maximum
value, at half of
,
’. This was analysed in the
graphical representation of the function
as the value that the vertex of the parabola reaches
for the coordinate corresponding to
. Then, a student from group 1 suggested drawing the
tangent line at the vertex of the parabola and affirmed that the derivative at
that point is zero. In response to this statement, the teacher asked which
derivative is zero. A member of group 2 said that it is the derivative of the
derivative that cancels out. As shown in Figure 12b), the teacher wrote the
following on the blackboard:
. Then, similarly, what happens with the slope of the
tangent line for those points of the parabola that are at
and for
, as shown in Figure 12b) was analysed.
Based on this analysis, it was concluded that for values of
within
the derivative function
is
increasing and that for
the derivative function
is decreasing.
To conclude the class,
this analysis was retrieved in the logistic curve graph shown in Figure 11.
This allowed us to justify the statement that, although the function is
increasing throughout its domain, its maximum growth occurs for
, and that in the part where
, although the population presents an increasing behaviour, it does so at a lower rate than for those values
located to the left of the inflection point. It was also concluded that, for
these values, where the function is concave upwards, the curve follows an
exponential growth, as had been studied in the classes where population
dynamics under the exponential growth model was discussed.
5.
Conclusions
According to the results
obtained with the implementation of the generative question in general and what
emerged from the study and research of the logistic model in particular, we
conclude that this type of didactic experiences are aimed at transcending the
traditional approach to the teaching of mathematics. From this point of view,
it is understood that this way of studying has allowed the students of the
Biology Teacher Training Course not only to find meaning and utility in the
mathematical knowledge studied, particularly in this case, in the logistic model
but also to construct the model themselves throughout the study process. A
certain autonomy has even been achieved on the part of the students and an
interest in learning mathematics. On the other hand, this change of paradigm
promotes substantial modifications in the teaching practices of mathematics in
the university environment, questioning not only the way in which it is
conceived as a science but also the epistemology that underlies the usual
practices of teaching mathematics.
The study and research
of the logistic model of population growth, in this implementation, has
contributed to overcoming the inherent problem of the loss of meaning that
frames the study of Mathematics in degree courses where the study of
Mathematics is not an end in itself, but a modelling tool that allows answers
to problems specific to the area of training, as in the case of PUB. It is
understood that future implementations of the generative question will enrich
the construction and reconstruction of new mathematical models in addition to
those mentioned in this work, which will make it possible to deepen the
research being carried out.
6.
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