O curso “Ideias Matemáticas” como projeto de inovação para cidadãos matemáticos críticos

Resumo

Tradicionalmente, a matemática é acusada de ser um dos fatores preponderantes que explica o abandono de estudantes nas universidades. Parece que essa apreciação é violenta, no entanto, não está tão longe da realidade. Sentimos falta de aspectos relacionados à empatia com as novas gerações de estudantes e um uso mais próximo da matemática na incorporação de novos discursos. Portanto, nesta comunicação, discutimos alguns aspectos relacionados à nossa compreensão da matemática para "não matemáticos" na universidade. Especificamente, reinterpretamos a ideia de treinar cidadãos matematicamente críticos e propomos o design inovador de uma disciplina de matemática universitária ministrada na escola de ciências humanas de uma universidade colombiana.

1. Introduction

In the context of a pedagogical innovation project of the Universidad del Rosario, the specific needs of the School of Human Sciences have been initially addressed to incorporate different spaces where the students of some programs of the School could learn mathematics and understand its use in each of the sub-disciplines. After collecting these needs, the referred course has been designed and developed. Currently, it is at an evaluation and sustainability phase.

Traditionally, mathematics has been one of the main factors for dropping out of Colombian higher education. There is also the notion that mathematics turns out to be ‘useless’ in diverse disciplines. As an institutional effort, we have proposed technology-mediated innovations that bring students closer to the applicability of what they have learned (from the mathematical context) to break down these beliefs that lead to obstacles in the learning of mathematics. The “mathematical ideas” course is aimed at students of programs that do not have a mathematical focus. In this regard, it seeks to enable an approach, an environment of reencountering, and in some cases a “reconciliation”, with mathematics. In class, basic concepts are addressed with important applications in everyday life in order to have a more critical understanding of situations that involve using mathematics; in other words, understanding the use of mathematics as a tool to analyze and make decisions that have a positive impact on their lives and society. In addition, there is a series of virtual support activities that have proved to be very useful as complement to face-to-face classes, accompanying the students’ out-of-class work; that is, use of a blended-based learning methodology.

There is constant discussion about the implications of knowing how to use mathematics and the responsibility that this entails, always focusing on – let us put it this way – fair use. Therefore, we focus on forming mathematically critical citizens, an idea that is closely related to what Paul Ernest (2002) denominates “critical mathematical citizenship”. In addition, students learn through projects facilitated by the teacher, in which they can solve real tensions in society. Hence, we seek to raise the levels of social sensitivity of our students in the mobilization of content and skills for the solution of real problems.

In this communication, we will show the motivations that have allowed the creation of the course. We will describe in detail the scope of the course, the use of different disruptive methodologies and virtual scenarios, and a reflection from a qualitative perspective (through a case study) that gives prominence to the voice of students who have taken the course.

There is evidence of the positive effect of this course in the face of the perceptions that students had about mathematics. Students who have taken this course have been empowered and have managed to mobilize the content and skills learned for the solution of real problems that they have detected and modeled and that they have solved using mathematics. This is based on a project that has been in operation since the beginning of the course and that is being improved as it progresses. The results of this research are made public in the framework of Pi Day, an activity that seeks to give visibility to mathematics within the university and that allows students to connect with their intrinsic motivations.

2. Objectives

2.1. General objectives

To encourage the education of critical and mathematically educated students for non-scientific academic programs.

2.2. Specific objectives

• To create a virtual classroom that supports blended learning processes
• To create a space for reconciliation with mathematics
• To provide students with tools for the mobilization of mathematical content when solving real problems
• To enable students to use mathematical knowledge for solving problems in their immediate environment that they themselves identify and toward which they feel empathy

3. Theoretical framework

3.1. Withdrawal from university

The report on withdrawal determinants of the System for Prevention and Analysis of Withdrawal from Higher Education Institutions, presented in 2014 by the Center for Economic Development Studies (Centro de Estudios sobre Desarollo Económico) of the Universidad de los Andes, states that according to “the statistics released by the System for the Prevention of Withdrawal from Higher Education Institutions (SPADIES), 48.47% of the students who entered higher education in the first semester of 2000 did not reach the tenth semester, while 57.2% of those who entered the first semester of 2008 did not do so.” (Centro de Estudios sobre Desarrollo Económico, 2014, p. 9) These figures are worrisome from an educational and social point of view, as this is saying that (by default) one in two people does not finish his or her university studies.

Many variables have been studied to account for what influences students not to stay in college until the end of their studies. Withdrawal has been and “is, par excellence, a problem of the educational system, intimately related to its surroundings, contours and content, such as the educational environments, family situations, and environmental and cultural demands that directly affect the dropout.” (Páramo & Maya, 1999, p. 71, our translation).

There is evidence that mathematics is precisely one of the areas in which students find greater difficulties and weaknesses: “[T]his is a problem that occurs in many higher education systems in the world. In the case of the United States, Herzog (2005) shows that one of the factors that influence the probability of enrolling in the second year of studies is having passed a first-year math course. Herzog indicates that mathematics is an indicator of withdrawal risk – students who come with greater academic disadvantages in the area of mathematics have a higher risk of dropping out.” (Portales et al., 2015, p. 2)

In general, achieving the continued presence of students has been an active concern at the Universidad del Rosario, as has been providing them with tools to make them feel comfortable with their work in mathematics. This has been a constant task of the university’s department of mathematics and computer science. On more than one occasion, we have asked ourselves about the factors that affect students to the point of being willing to give up on mathematics. Consequently, we have worked on different experiences and projects, seeking to contribute to a good relationship between students and mathematics.

3.2. Predisposition toward mathematics

Since we have lived inside the classroom, we know that some of our students arrive at their first math classes carrying a ‘heavy baggage’ of prejudices and fears, such as “mathematics is only for the smartest”, “it is a boring subject”, “it does not have much relation to reality”. These imaginary thoughts prevent an attitude toward a… why not call it “a happy learning of mathematics”?

This is not a novelty. Since the 1980s, the study of students’ beliefs about mathematics and their impact on the learning of this science began to intensify. At the same time, greater interest was aroused by the study of concepts and recently by the study of perceptions and imaginaries. Such research recognizes that “beliefs are as necessary as the psychological functions of cognition and metacognition to achieve effective learning” (Andrews et al., 2008, p. 326), and that they play a role as regulatory system, indicator, inert force and prognostic character. In addition, it is mentioned that beliefs can greatly influence student learning and the way they use mathematics, so they can also be an impediment in their learning process (Törner & Pehkonen, 1996). In the same line of thought, Campos (2008) deals with affections – beliefs, attitudes and emotions – in mathematics education, and Da Ponte (1999) works on the topic of beliefs and conceptions as key studies in mathematical education as well.

A more recent field is the study of imaginaries in education: collective representations strongly related to experience, with intuition and feeling, which influence our ways of perceiving and acting (Suavita-Ramírez, 2017) in relation to, for example, math class.

These investigations have resulted in several general observations regarding how to get closer to achieving a better disposition for students in relation to mathematics. It is expected that with better disposition students are able to enjoy classes and improve their attitude and results, which at the same time would contribute to their remaining in the university.

3.3. Critical citizens

One way to “hook” students with mathematics is by showing them that it is not devoid of social meaning, in other words, it has powerful meaning and implications for social and political life, for their lives and society. In this regard, critical mathematics education “argues that the values of openness, dialogicality, criticality towards received opinion, empowerment of the learner and social/political engagement and citizenship are necessary dimensions of the teaching and learning of mathematics if it is to contribute towards democracy and social justice.” (Ernest, 2001, p. 1)

Paul Ernest (2002) introduces the idea of critical mathematical citizenship, which aims at the formation of mathematically critical citizens, that is to say, citizens able to discern through critical judgment, and to make important political and social decisions. A mathematically critical citizen has the necessary mathematical tools to analyze a situation and understand it, before taking a definitive position with respect to it; for example, it is possible to make a critical reading of mathematical instruments used in a newscast or in an election campaign and the way in which they are presented. A mathematically critical citizen can also make everyday decisions using mathematics, for example, in relation to personal economy – to compare discounts, understand interest rates, make a comprehensive reading of public utility bills. Therefore, a mathematically critical citizen is expected to contribute to the construction of a fairer and better society.

3.4. Blended learning

Electronic learning, or e-learning, accelerates the educational model and contributes to its effectiveness by promoting self-directed learning in a guided way (Rosenberg, 2001). The use of e-learning is in continuous growth, an expansion that is due to the fact that e-learning solves geographical, temporal and demand problems. However, this type of learning has drawbacks, such as that there is little or no interaction between teachers and students, the feedback on processes that are not automated can be slow and therefore untimely, it can be more difficult to rectify errors in both the teacher’s and student’s materials, it could entail a decrease in the quality of training, and students may develop feelings of loneliness, impersonality or isolation.

The Mathematical Ideas project is part of the lines of innovation for learning to learn, inclusion and equity, social impact, internationalization, use of ICT and integral training. The use of ICT is introduced under a blended methodology in which learning processes are supported by technology through work in virtual classroom activities.

Partial face-to-face learning or b-learning (blended learning) consists of the integration of face-to-face learning experiences with online learning experiences. These methodologies generate several benefits compared to e-learning:

• Students are encouraged to appropriate the critical discourse obtained by listening to a teacher and discussing content in class
• Learning becomes independent, self-reflective and self-taught
• Students can learn at the pace that best suits their knowledge
• Students receive training in digital competencies (search, selection, collection of resources, elaboration, extraction of ideas, sharing of information, etc.).
• The content can be arranged in a more interactive way with the use of information technologies.
• Students are encouraged to use mobile devices as a guided learning tool.

3.5. Modes of inquiry

The Mathematical Ideas class arose as a project of technological mediation in the teaching of university mathematics as a result of a call for research project proposals at the Universidad del Rosario. This proposal has been an important challenge that obliged us to redesign and rethink strategies and content in such a way that allowed us to identify and delimit the extent to which face-to-face teaching was taking place, in order to establish a learning process that would also integrate the technological resources available – one that would be functional.

A consolidated pentagonal scheme was used in the phases of (1) sensitization, (2) ideation, (3) construction, (4) implementation; and (5) evaluation, as shown in Figure 1.

Figure 1. Pentagonal creation scheme.

Phase 1: Sensitization

• Dialogue with directors and students: throughout a semester, meetings were held with chairs of the School of Human Sciences to recognize opportunities for improvement in the mathematical academic offerings of this School. Given that a course on fundamentals of mathematics was offered and its impact on student learning presented major obstacles, we decided to approach the students – the target population for action – to explore the problems they were experiencing with this course. Having held some focus groups with students, it was concluded that the course on fundamentals of mathematics was designed in a way that the students could not motivate themselves or connect positively with the content and competencies linked to the course. It was further concluded that much of the content offered is not relatively relevant for this particular type of student. This formative dialogue held with both chairs and students allowed the emergence of a “seed of research” that would ultimately develop into an interesting course, mediated by technology, where students really connect with the content and competencies, using them to solve real problems of interest to them.

Phase 2: Ideation

• Why e-learning and projects? The courses were not designed properly, in other words, they did not take the students into consideration, aiming solely to encompass all of the course contents as the teachers thought was best. The course design was not inclusive. It was necessary to think about offering students an innovative alternative for their reconciliation with mathematics. Taking into account that this generation of students is one of digital natives, we decided that technological mediation would be an efficient tool to guarantee their academic success, achieving, among other things, their commitment to the course, knowing that they would have to perform relevant academic work at home via virtual classrooms. Moreover, we decided to make (tangential) use of project-based learning, to ensure the mobilization of content and competencies for the solution of social problems of interest to students.
• Levels of social sensitivity: this course also sought to raise the social sensitivity levels of our students. This means that during the exercise of solving problems students should be the protagonists in the collection of data and design of models that represent real problems they find distressful.

Phase 3: Construction

• Instructional design: the Mathematical Ideas course was structured in four fundamental thematic blocks. A first block explicitly addresses the art of posing and solving problems.  In this regard, factors to be taken into account when addressing a situation are explored – for instance, reading the problem carefully, understanding its meaning, knowing what information is available and what information is sought. Problems are solved by inductive reasoning and solution strategies are developed. Variational thinking constitutes the second thematic block. In this part of the course, we mainly work toward understanding the concept of proportionality (ratio, proportion and variation, and percentages) and its implications in daily life in order to raise the students’ awareness to the uses of this important idea. The third block provides an introduction to counting and probability, specifically, counting techniques, permutations and combinations; counting problems that include ‘no’ and ‘o’ are also addressed. Finally, the course closes – in a fourth block – with an introduction to statistics. In both blocks, we have used problems in which such concepts are applied.
• The final project: a fifth block was also proposed, entitled ‘projects’. It refers to an assignment that students carry out from the beginning, throughout the course, until its final delivery and defense to conclude the program. The aim of this project is that the students address one or more contents of the four major themes by studying a real situation in which they show what they have learned in the course and presenting a solution to a problematic situation. These student projects are presented in poster format, within the framework of the Universidad del Rosario’s Pi Day (see Figure 2). This activity aims to give visibility to mathematics and empower students, allowing them to publicly display their mathematical findings in the different mathematics courses.

Figure 2. Pi Day at the Universidad del Rosario.

• Virtual objects: in this phase of ideation, we decided to create simple, but empathetic, virtual learning objects that would be embedded into virtual institutional classrooms (Moodle). Some views of these objects are shown in Figure 3, Figure 4 and Figure 5.

Figure 3. Main page of the virtual classroom of Mathematical Ideas.

Figure 4. Panel for access to the activities according to the thematic blocks.

Figure 5. Thematic block 1: The art of solving problems.

Phase 4: Implementation

Once the virtual classroom was created, the course was offered for students of the School of Human Sciences in the second semester of 2017, including a group of 15 students from Philosophy, Anthropology and Sociology programs. During the first and second semesters of 2018, the groups comprised 11 and 15 students, respectively. Currently, there is a group of 16 students.

The classes were carried out according to plan, and the students, in general, responded positively by responsibly attending the classes and working in the virtual classroom activities. While this was happening, a report was prepared during each semester on aspects that could be improved, especially in relation to the virtual objects. At the end of the semester, the virtual classroom was reformed considering these observations and making the appropriate adjustments to the activities.

In relation to the teacher in charge of the course, it is important to highlight that he/she has the constant support of the university’s e-learning center and its guidance for any questions regarding both the blended learning methodology and the operation of the virtual classroom. In this way, we have attempted to ensure proper implementation of the Mathematical Ideas class independently of switching teachers.

Phase 5: Self-evaluation

To evaluate the Mathematical Ideas project, two focus groups were established with students who had taken the course in two different terms (the first and second semesters of 2018). In this way, we were able to get closer to what the students had experienced and thought about the Ideas course and mathematics.

An instrument was created as a guideline for the coordination of the focus groups. For this guideline, we took into consideration aspects such as the image of the Mathematical Ideas course constructed by the students, the methodology they associated with how the class was conducted, their idea of mathematics and whether this idea had changed after class, and the use given or that should be given to mathematics according to their constructions in class.

4. Letting different voices speak – Results

Some student comments from the focus groups held in the self-evaluation phase of the course are presented below.

1. Instructional design: the Mathematical Ideas course was structured in four fundamental thematic blocks. A first block explicitly addresses the art of posing and solving problems. In this regard, factors to be taken into account when addressing a situation are explored – for instance, reading the problem carefully, understanding its meaning, knowing what information is available and what information is sought. Problems are solved by inductive reasoning and solution strategies are developed. Variational thinking constitutes the second thematic block. In this part of the course, we mainly work toward understanding the concept of proportionality (ratio, proportion and variation, and percentages) and its implications in daily life in order to raise the students’ awareness to the uses of this important idea. The third block provides an introduction to counting and probability, specifically, counting techniques, permutations and combinations; counting problems that include ‘no’ and ‘o’ are also addressed. Finally, the course closes – in a fourth block – with an introduction to statistics. In both blocks, we have used problems in which such concepts are applied.
2. The final project: a fifth block was also proposed, entitled ‘projects’. It refers to an assignment that students carry out from the beginning, throughout the course, until its final delivery and defense to conclude the program. The aim of this project is that the students address one or more contents of the four major themes by studying a real situation in which they show what they have learned in the course and presenting a solution to a problematic situation. These student projects are presented in poster format, within the framework of the Universidad del Rosario’s Pi Day (see Figure 2). This activity aims to give visibility to mathematics and empower students, allowing them to publicly display their mathematical findings in the different mathematics courses.
3. Virtual objects: in this phase of ideation, we decided to create simple, but empathetic, virtual learning objects that would be embedded into virtual institutional classrooms (Moodle). Some views of these objects are shown in Figure 3, Figure 4 and Figure 5.
4. When we asked our students about how they define the Mathematical Ideas class, they mentioned words such as ‘tool’, ‘usefulness’ and ‘novelty’, and phrases such as “foundations for the inexperienced” or “a different approach”. This made it possible to show that they were noticing what we wanted to communicate, and it was even more interesting when compared to other opinions in which they related the class to words as ‘analysis’, ‘demonstration’ and ‘rigor’, among others.
5. With regard to the perception of what the Mathematical Ideas class is like, we have allowed our students to speak:

GD1_P1: It was interesting for the simple tools that any professional should know, even for everyday life. A novel proposal in the curriculum.

GD1_P3: The class covers the objectives set; it does not represent an impenetrable area of knowledge, although it has some rigor.

GD1_P6: It is interesting and dynamic.

GD2_P4: It cultivates practice and the love for mathematics.

GD2_P2: I loved the experience I had with mathematics. Despite studying for a career in human sciences, mathematics has always caught my attention. This class, besides reminding me of some basic and fundamental things I will need for my life, allowed me to approach again this incredible world of mathematics.

• Some students agree that what they had in mind about mathematics has changed after attending the course. One of them even said, “I usually thought mathematics was only theorems and demonstrations. Of course it has that, but it was interesting to apply mathematical ideas to my areas of knowledge.” [G2D_P3]
• Regarding the course project, during the discussion one of the participants mentioned: “As I said before, mathematics from the beginning has always attracted my attention, and I enjoyed it with this course, especially the final project in which I had to relate mathematical elements to social aspects. It is a very interesting work of analysis and one cannot only learn but also be surprised, since in fact mathematics is everywhere. On the other hand, I think it is important to eliminate the division that has existed for centuries between Exact Sciences and Human Sciences, because they are closely related. In addition, both coexist within the same field. It could be said that sometimes one depends on the other, and vice versa.” [GD_P5]

The figure below (Figure 6) shows a couple of examples of posters for the course final project.

Figure 6. Posters presented by two of the students of the course to defend their final project.

• In general, participants agree that mathematics should be used to solve everyday problems, “from understanding a banking transaction to understanding politics or economics”. They also agree that mathematics serves to interpret reality in a critical way. However, some of the participants are still not very convinced that mathematics can help make better decisions.

5. Discussion and conclusions

According to the opinions collected, the Mathematical Ideas course was considered innovative, with a different approach, while also rigorous – a contrast of opinions which proved revealing. We had aimed for a kind of reconnection with mathematics, but besides that we found that the class naturally contributes to the destigmatization of topics such as that an innovative methodology – of mathematical creation – lacks rigor. In the same way, the students were given the opportunity to experience a more “human mathematics”, and, based on their projects, for which they themselves identified real problems to solve, the students discovered that mathematics can have an impact on their environment, giving social meaning to those problems.

Why is “Mathematical Ideas” so significant for our students? Based on the evidence obtained, we found that the strategic design of this subject, responding to the specific needs of this student population and including non-traditional learning methodologies, has allowed, for example, the average approval rating of this subject to reach 98%, greatly surpassing the approval rates of the fundamental mathematics courses offered by this School in the past.

In addition, we have found evidence of change in the way the students who experienced the course think about mathematics, especially in terms of the tools this science provides for understanding the world and the way it contributes to their development as critical citizens.

Lastly, a group of students empowered with knowledge and satisfied with their learning outcomes is being formed. The inclusion of the projects in the course curriculum has added both intrinsic and extrinsic motivations to the teaching and learning processes of our students.