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Modeling projects are a pedagogical strategy involving learners working in small groups to propose a problem based on a topic of their interest and solve it using mathematics. As a part of a course focusing on this pedagogical approach, a group of prospective teachers focused on modeling the growth of stray cats and dogs. The aim of this study was to identify the tensions that the group encountered during the development of their project and to identify the learning opportunities that emerged during this process. The results reveal tensions in the modeling process due to difficulties with the mathematics involved in solving the problem. To address these challenges, the group simplified the variables in the original problem and developed an exponential model to reflect the growth of stray dogs and cats. Throughout the process, the group developed STEM skills to address a real problem and solve it mathematically. Moreover, the project generated important reflections on responsible pet ownership.
*Address all correspondence to: jeannette.galleguillos@uv.cl
1. Introduction
The present study addresses the modeling project strategy outlined by Borba and Villarreal [1], in which prospective mathematics teachers, working in small groups, propose a problem based on a topic of their interest and develop it with the help of the teacher and digital technologies. Using this strategy, it is possible to establish solid connections between mathematics and the real world. In the development of projects aimed at promoting knowledge and skill acquisition, learners are asked to propose a question or a problem based on a real-life situation they deem relevant [2], which should contrast with typical textbook problems that can be solved using classic procedures. The development of a project with these characteristics requires advanced knowledge and information that can be obtained through Internet searches and with the support of the teacher.
According to the current STEM goals, individuals are required to develop knowledge, attitudes, and skills to identify real-world problems and to apply interdisciplinary knowledge and skills to shape their own learning experience [3]. To achieve STEM goals in K − 12 education [4], it is essential that teachers are empowered with real-world learning experiences and problems. To this end, a modeling project strategy was adopted in this study, focusing on an actual project implemented as a part of a course taken by prospective mathematics teachers.
Regarding the implementation of modeling in the teaching and learning process, Schukajlow and Blum [5] recognise two broad categories in the pertinent literature, an instruction-oriented approach along behaviourist lines and a constructivist-oriented approach, which may comprise enquiry learning and problem posing. In both cases, teacher preparation is required that goes beyond merely observing how students solve problems. The authors thus highlight the need for instructional principles of self-regulation in problem-solving, including providing students with opportunities to engage in social learning by encouraging peer and small group discussions, as well as teaching students how to use problem-solving strategies. In the same vein, the modeling project strategy offers opportunities for the learners to be actively involved in the modeling process while working collaboratively with their peers and the teacher, and utilising relevant online resources.
One of the difficulties in implementing modeling projects is that some learners may not know how to proceed with their execution [6], so they are overly dependent on the teacher to advance at each step. To address this issue, in this study, a support guide based on [7, 8] was used to assist prospective teachers in advancing the project. The aim of the support guide is to prompt the group working on the project to focus on its topic, question or problem, and then proceed to the mathematisation, explanation and reflections or conclusions. In this work, we see the difficulties of preservice teachers as tensions, and learning opportunities as possibilities for expansion from the perspective of expansive learning [9]. Accordingly, the aim of this study is to identify the tensions within a group of prospective teachers in the development of their project and to study the learning opportunities that the group experienced in this process.
We understand mathematical modeling as any relation between mathematics and the real world (e.g., [10]), being a process that goes from the real world to mathematics. During this process, the concepts of simplifying and mathematising are used as a part of the modeling cycle [11]. According to Blum and Leiß [11], the real situation has to be simplified, structured and made more precise, leading to a real model of the situation, while mathematisation corresponds to the transformation of this initial model into a mathematical model, i.e., “its data, concepts, relations, conditions and assumptions are to be translated into mathematics” [10].
There are different ways of viewing modeling. Particularly, in the modeling project approach of Borba and Villarreal [1], learners work in small groups and choose a topic of their interest to investigate. Once the project topic is determined, they pose a question or a problem that can be solved using mathematics. This strategy allows learners to make connections between mathematics and reality, helping them appreciate the practical utility of mathematics. Moreover, learners may feel encouraged to engage in mathematical modeling because they have the opportunity to work on a topic of personal interest. For this type of strategy to be successful in school mathematics education, (prospective) teachers must develop the skills required to work with projects and to guide their students.
Other studies on modeling and projects involving (prospective) teachers have been published, referring to prospective teachers’ experience of modeling with the use of technologies [12], teachers’ dilemmas and conflicts when they propose a modeling problem [13], prospective teachers using modeling as an evaluation strategy [14] and modeling through the design of task for teachers [15].
The modeling skills emphasised by the Chilean education system require the teacher to consider using, applying, selecting, comparing and evaluating models, focusing on the objectives to be achieved at grade levels 1–10 [16], along with building models of real situations, required at grade levels 11 and 12 [17]. However, most of the activities proposed in didactical books on modeling correspond to the applications and use of models rather than model development.
Thus, by involving prospective teachers in investigating a topic of interest under a modeling project, they are encouraged to participate in the modeling activity and to adopt a strategy that breaks away from a behavioural approach. In addition, the modeling project strategy can be approached in a way that uses mathematics familiar to students as well as learning new mathematical knowledge, adding multiple skills to pose a problem and solve it using technologies [1]. Considering the challenges for teachers to integrate modeling (and projects) into the teaching and learning process, in a socio-cultural manner [18], this study used an activity theory perspective to understand the experience of a group of prospective teachers in the modeling process.
2.2 Activity theory
The cultural-historical theory of activity, also known as activity theory, was established from the studies conducted by Vygotsky [19], Leontiev [20] and later Engeström [9], among other authors. Vygotsky essentially studied the development of human thought, contributing to the development of the notion of mediation, in which subject−object relations are mediated by artefacts. In a broader sense of the notion of mediation in learning, it is established that learning occurs in a dialectical process, in which the subject is transformed by the world, and, in turn, the world is transformed by the subject.
According to the activity theory, people actively participating in joint activities to solve problems are subjects in an activity system [9]. On the other hand, the object is considered the real purpose of the activity system (e.g., to solve a certain problem). Artefacts correspond to the means that subjects use to achieve their object [9]. In the interaction of the subjects in the activity system, different perspectives on how to solve their problem (reach their object) may arise. These different opinions are expressed through words or gestures. Generally, opposing ways of approaching a situation collide with each other and reflect the existence of internal contradictions in the system. This indicates that internal contradictions are the cause of the tensions. The contradictions are not easily perceived, but can be identified by analysing people’s words or gestures [21]. In this work, we view tensions as manifestations of contradictions [21]. Learning by expanding [9] corresponds to the emergence of tensions and their successful resolution, leading the subjects to the construction of a new system that offers a solution to the initial problem. The resolution of contradictions is an indication of learning or progress in the system.
This study was developed under the parameters of basic qualitative research [22], in which a phenomenon is studied through the experiences reported by the subjects who were involved in that phenomenon. The aim was to identify tensions encountered by prospective teachers who worked as a group on a modeling project as part of an activity included in a mathematical modeling course offered in the context of an initial education course for mathematics teachers at a Chilean university. The course is integrated into the curriculum in the seventh semester (out of a total of ten), being a compulsory course with three hours of classes per week. The project at the focus of this investigation was developed in 2021, and students were given 1 month for its implementation, but due to the restrictions imposed by the Covid-19 pandemic, work was conducted online.
Six prospective teachers took part in the course, and by the time the project was initiated, they had already worked on modeling linear and exponential problems and were also asked to propose modeling problems related to school mathematics at grade levels 7–12. They were then given examples of modeling projects as a preparation for using the modeling project strategy. Thus, prospective teachers were asked to form two-member groups based on their own affinity, according to Borba and Villarreal [1], and in coherence with Schukajlow and Blum [5]. The resulting three groups were asked to choose a relevant topic to investigate and propose a problem. They chose topics related to the environment, bees and animal care. In this work, focus is given to the animal care project designed by a group of two prospective teachers, Anne and Martin (fictitious names). In accordance with the activity theory framework [9] and the modeling project strategy [1], the project was developed by the group in collaboration with the course teacher while using digital technologies.
The instruments used in this study include a project report document, a short video presentation of the project, and a narrative written by the group about the experience of developing a modeling project. The document contains information about the project, such as topic and information found on the Internet, problem or question, mathematising, explanations and reflections or conclusions. The group presented their work in an 8-minute video summarising the project. A narrative was written by Martin but expressed the experience of both members of the group. The narrative was guided by the following prompts: What was your experience developing a modeling project like? How was the idea of the theme born? What modeling stages do you consider you experienced and which ones you did not utilise?
As the group chose animal care as their project topic, they proposed modeling the growth of stray dogs and cats, which is a prominent issue in Chile and generated reflections that allow us to confront this form of animal mistreatment. The analyses presented in the following sections focus on detecting the tensions of the group participants in developing their project through qualitative analysis of the narrative. Another aim was to identify possible developmental (or learning) opportunities in the modeling process. Thus, from Engeström’s perspective on activity theory, the purpose is to evidence the occurrence of expansive learning [9] in the group, i.e., to visualise tensions and contradictions in the development of their project, as well as to observe whether the group was able to resolve the tensions by developing a model that solves their problem [23].
The project was described in the report document and in the video following the steps based on the supporting guidance [7]: project topic, problem or question, mathematisation and reflections. Some images of the video presentation of Anne and Martin are shown in Figure 1.
Figure 1.
Images of video presentation.
4.1.1 Project topic
The group was interested in caring for animals and improving the conditions of those who are abandoned and suffer from diseases, hunger and thirst. As explained by Anne, their topic was “Rescue and sterilisation of stray animals”—particularly dogs and cats—and combining the data provided by the organisation “Stray animals without voice” with the information available on the Internet to analyse and model the problem.
4.1.2 Problem or question
The group formulated the problem as follows:
If stray animals are not rescued or sterilised, their population will increase. In 2020, 343,000 abandoned animals were registered in Chile, 88,000 of which were cats and 255,000 were dogs.
Accordingly, they posed the following questions to aid them in their modeling task:
If in 2021 121,440 cats and 323,850 dogs were registered, what is the percentage of growth in cats? What is the percentage of growth in dogs? What will the number of stray animals be in 2025?
The organisation “Stray animals without voice” on average rescues 32 animals per month to reduce overpopulation and improve their quality of life. Thus, they must pay for vaccines and food, with the following costs (see Tables 1 and 2).
Cats
Expenses per cat (in Chilean currency)
Sterilisation
$ 15.000
Vaccines
$ 30.000
Food
$ 4.530 (per month)
Table 1.
Expenses per cat.
Dogs
Expenses per dog (in Chilean currency)
Sterilisation
$ 22.000
Vaccines
$ 30.000
Food
$ 5.400 (per month)
Table 2.
Expenses per dog.
With the data provided above, what would be the annual expense assuming that 12 cats and 20 dogs are brought to the facility each month?
4.1.3 Mathematisation
Using the data for 2020 and 2021, the group found 38% growth in stray cats and a 27% growth in stray dogs, thus answering the first part of Question 1. Then, with some interactions with the teacher of the course, they projected the growth in these populations by 2025, assuming that animals are not sterilised (Tables 3 and 4), finding the exponential function for both cats and dogs.
Year
Growth in the number of stray cats: f(x)
0
88,000
1
88,000·(1 + 0.38)
2
88,000·(1 + 0.38)2
3
88,000·(1 + 0.38)3
…
…
n
88,000·(1 + 0.38)n
Table 3.
Growth in stray cat population.
Year
Growth in the number of stray dogs: g(x)
0
255,000
1
255,000·(1 + 0.27)
2
255,000·(1 + 0.27)2
3
255,000·(1 + 0.27)3
…
…
n
255,000·(1 + 0.27)n
Table 4.
Growth in stray dog population.
From this, the group established the exponential growth functions—Eqs. (1) and (2)—which were explained by Martin in the video:
f(x)=88,000·(1.38)xE1
where f represents the annual growth in the cat population, and
g(x)=255,000·(1.27)xE2
where g represents the annual growth in the dog population.
It was projected that, under the conditions of the proposed problem, in 2025 there will be 440,431 abandoned cats and 842,478 abandoned dogs, totalling 1,282,909 stray animals. Then, they represented both graphs, observing how these populations would increase in that period (Figure 2). They also calculated the costs of sterilisation, vaccines and food to support the stray animals, which are not reported in this work.
Figure 2.
Exponential growth in the population of abandoned pets.
4.1.4 Reflection
In the video Anne shared the group’s reflections, commenting on the rampant growth of stray animals, stressing the importance of caring for stray animals and emphasising responsible ownership and care of pets.
4.2 Tensions and learning opportunities
The following excerpts are taken from the group’s narrative written by Martin:
(…) Mathematising the project involved a lot of work, since, when the problem was posed, we were not clear about the mathematics that should be used to model the growth of the population of these stray animals, since at the beginning we wanted to have different factors to make the model, such as birth rate, mortality, abandonment and annual rescue, as well as the medical price of sterilisation. For this reason, it was not possible to find the mathematics that would fit all the factors that we wanted to take into account, so we had to focus on one variable, which we believed to be the most important, which was the annual birth rate.
(…) Throughout this process, we were able to identify the problem we wanted to work on, in a real-life context, and it was also a problem of social relevance, which generated even greater interest. After researching all this data, we were able to mathematise the problem and this led to the creation of the relevant model, which allowed us to make a synthesis of the problem, returning to real life, raising awareness of the problem of pet abandonment and what the disproportionate growth would be like if the animals were not sterilised.
In the process of constructing a modeling project, tensions as well as learning opportunities were experienced by the participants. First, the narrative shows that when the initial problem was posed, the group was not clear about the mathematics to be used to model the growth of stray animals due to the fact that too many variables were considered in the problem. Thus, tension was generated by the contradiction of posing a problem based on reality but not knowing how to solve it (part 1 of the problem). Moreover, the prospective teachers were expected to be able to propose problems that could be solved using the mathematics at grade levels 7–12.
The group pointed out that, in order to resolve the contradictory situation, it was necessary to make a decision regarding the variables to be addressed in the problem, which from the notions of modeling is understood as simplifying the problem. Thus, they considered only the variable “annual birth rate of the animals,” as evident from the following comment: “For this reason, it was not possible to find a mathematics that would fit all the factors that we wanted to take into account, so we had to focus on one variable, which we believed to be the most important, which was the annual birth rate.” These assertions reveal both the contradiction and the decision to choose the most influential variable that allowed the group to refine the problem and answer the question. In this way, the prospective teachers resolved the contradiction by varying the problem statement and simplifying it, so that it was based on a single variable; thus, an expansive learning process took place [9].
The results also reflect the development of STEM skills in prospective teachers, since during the process Anne and Martin were able to identify a socially relevant problem taken from the context of their country, with real data taken from the Internet, and construct a problem related to a relevant topic. This is reflected in the following statement: “Throughout this process, we were able to identify the problem we wanted to work on, in a real-life context, where it was also a problem of social relevance, which generated even greater interest. After researching all this data, we were able to mathematise the problem and this led to the creation of the relevant model, which allowed us to make a synthesis of the problem, returning to real life.” They further indicated that developing a problem pertaining to their chosen topic generated greater interest in the modeling process.
The project addresses the problem of pet abandonment in Chilean society and can be presented or developed in schools to generate deep reflections on this issue, which can lead to social transformation. The group pointed out that they chose this particular project as a way to raise awareness of the problem of pet abandonment and lack of sterilisation in Chile. In addition, the project presents an opportunity to enhance mathematical knowledge by using percentage calculations and finding patterns.
As the project objectives were achieved, the contradiction situation was resolved, which implies that expansive learning took place [23], as reflected in the group’s approach to a real problem that could be solved with school-level mathematics while bringing focus to the problem of animal abandonment in the streets. The learning opportunities resulting from this project include skills to identify a problem of social relevance, constructing a problem and solving it using mathematics and the Internet (all of which are vital STEM skills), as well as reflections that prompt the school community to face social problems and respect animals.
The study results show that a contradiction emerged in posing a real-life problem that has too many variables and is thus outside the scope of school knowledge. The contradiction was resolved by taking the decision to choose the most important variable for the problem, and simplifying it, which corresponds to expansion. The implementation of modeling projects, in which the learners propose the problem, showed most strongly the need for simplifying the situation.
Second, the results show a greater interest in engaging in modeling situations when the topic has social and cultural relevance, concurring with the findings of other authors [1, 8].
Third, the Internet was a crucial resource for obtaining information on the topic as well as accurate data to propose the problem, acting as an active component in an activity system [24, 25]. The large amount of information led the group to consider all pertinent variables, which resulted in a contradiction (formulating a problem which they cannot solve with school-level mathematics). The resolution of the contradiction goes hand in hand with the decision to choose the most useful and necessary variable to consider in the problem statement, with a focus on knowledge that can be addressed in the school environment and in their future teaching work.
The activity of posing a real-world problem and solving it using the Internet and mathematics led to the enhancement of crucial STEM skills, showing that expansive learning took place [9]. In addition, very important reflections were made on a social problem in Chile, where pets are regularly found abandoned in squares and streets without scruples and despite the municipal efforts and the organisation of care for stray animals for feeding, sterilisation and adoption. It is therefore, a social problem that, based on this situation and the proposed project, can provoke reflections in children and young people in the school environment, addressing socio-critical aspects.
The author thanks the Project ‘Bienes y equipamiento Red 21.995 (IESED-CHILE) - RED21995 B-1123’, for support in the financing of equipment for this research.
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Written By
Jeannette Galleguillos
Submitted: 19 June 2024Reviewed: 02 July 2024Published: 04 October 2024
Open access peer-reviewed chapter - ONLINE FIRST
