2. Let’s Go Shopping

Fanny Aseneth Gutiérrez Rodríguez and Yael Carolina Rodríguez Moreno

“Let’s Go Shopping” is a landscape of investigation about economic and financial education framed by the socio-critical perspective of mathematical modelling. The landscape developed as a result of discussions among students from a school in Bogotá about high-interest charges and loans. Our aim was to identify possibilities and challenges in the use of a socio-critical perspective on mathematical modelling when promoting citizenship education. To do this, we addressed the discussions by the students and considered students’ intentions, backgrounds, and foregrounds. The landscape included investigations of families’ economic conditions, as the students related the mathematics classroom activities to their families’ financial issues.

The school day is underway in the Bogotá capital city of Colombia in a school located in Neighbourhood 18 called Rafael Uribe Uribe, and the ninth-grade students are preparing to receive their mathematics lesson. However, some of them have had an argument about a loan that one had taken out to purchase an item. The interest on the loan is due, but the interest rate is very high and amounts to 100% of the loan itself. This discussion energises the participation of several students as they compare their own families’ economic situations. After these discussions, some questions emerge, such as why it is that in some cases people end up paying up to twice the cost of the purchased item.

Such questions were inevitably related to some of the political, social, and economic situations of the students and their families, as is supported by accounts of the interactions that are typical between teachers and members of the community, and the institutional documents (ID)¹ that the school has (Gutiérrez and Rodríguez, 2015, p. 15). We found that the families of the school community belong to the socioeconomic classification of Stratum 1 and 2,² wherein the parents have a non-stable job. As a consequence, several parents are in the informal sector and there is a medium level of unemployment.

Regarding family monthly income, this was estimated to be around the equivalent of the current legal minimum wage (616,000 Colombian Pesos for the year 2014)³ or in some cases even less. Furthermore, in the community, there is evidence of problematic situations related to the sale and consumption of illegal substances, safety issues, domestic violence, forced displacement, poor waste management, gang violence, teen pregnancy, school dropouts, and prostitution (Gutiérrez and Rodríguez, 2015, p. 17). However, at the same time it is also important to highlight that in this community there are young people with a strong commitment to studying, a spirit of solidarity with their families, and hope and aspirations for their future.

The school had established agreements with public and private institutions to promote formative processes with the students in different social spaces such as cultural houses, football schools, health institutions, among others. Regarding the academic field, the mathematics curriculum is divided into four terms for the school year. In this curriculum, we find Ministry of Education (MEN) guidelines about competence with respect to the different components of mathematics. These guidelines are framed around specific topic areas. The common strategy is to evaluate through achievements and indicators, using workshops, guides, tests, and other activities that are undertaken in the mathematics class (Gutiérrez and Rodríguez, 2015, p. 16).

The above situation, where students expressed their concerns about the credit system, caught our attention as teachers and researchers, as it created the possibility of negotiating exciting and unusual possibilities for subsequent lessons. In addition, one of us was taking a Ministry of Education course about the economic formalisation of education.

Without hesitation, this became the starting point for the construction of our landscape of investigation “Let’s Go Shopping”, where consumer lending was the aspect to be modelled mathematically.⁴ We focussed on the perspective of critical mathematics education by considering some political and social aspects, and we followed the approach of such authors as Skovsmose (1990, 2012), Valero (2006), Blomhøj (2009), among others.

On the research side, our first step was to consider the backgrounds, intentions, and foregrounds of our students, taking into account what happened in the classroom as a result of the discussion of the money loan. Due to this, we moved on to study consumer loans that their families had, especially the one that is linked to the public utility service of electricity supply. We projected a landscape of investigation that allowed us to identify the challenges and possibilities of mathematical modelling in the form of citizenship education through the analysis of student discussions.

Our intention was to establish a learning milieu of mathematical modelling, which creates the possibility of producing discussions about the students’ life-worlds, as stated by Barbosa (2006, 2007) and Lerman (2001). A process of negotiation with the students led us to create nine scenes that constitute the landscape of investigation entitled “Let’s Go Shopping”.

Scene 1: Tell Me What You Need, and I Will Tell You How to Get It!

The mathematics class started to move towards a different approach from that which was customary. The students were invited to leave their traditional classroom and move to an auditorium to watch a video and to talk about it. The smiles on their faces reflected that they enjoyed what was happening—something that was not common in their previous mathematics lessons. With this in mind, we started the proposal.

The dialogue between the students and us as teachers encouraged the possibility of establishing participation agreements inside and outside of the classroom. We hoped that the blog “Modelling Your Finances”⁵ could promote a place for interactions different from that which was common in the regular classroom, and that it would also foster written discussions.

Our students inhabit a world that is highly visual, so for this activity they watched a short video called “Let’s Go Shopping” that had been collated by us. In this video, a young 17-year-old girl explores the possibility of opening a savings account. At the end of the video, there is a series of short fragments of advertisements that show people’s reactions on acquiring a new product, and several invitations from the financial market relating to consumer lending.

The video caught the attention of all the students, eliciting reactions and comments from them. For example, Jeisson⁶ mentioned that “the video invites us to the culture of consumption and the benefits that they offer us for it” while another student, Marlon, expresses that “in December there is joy and in February we are doomed”⁷ (Gutiérrez and Rodríguez, 2015, p. 54). The previous statements indicate that some students and their families constitute part of a community that actively makes considerable use of consumer lending.

After watching the video, hoping that we could adapt that situation to a classroom activity, the students were organised into groups of four, and they distributed between them the roles of an imagined family from their neighbourhood.

However, during this activity, some students decided to assume the roles of families in socio-economic conditions higher than the ones they came from. As a result, some of the roles selected did not correspond to those in their own neighbourhood. Only three out of the ten groups undertook a reflection based on the economic capacity of their own parents and how they managed to cover their basic needs This information is relevant to what will come in the next scene. Some students who used their own family contexts decided to use the internet on their mobile phones as a way of working with real data. This brought about strong observations to share with the whole classroom, as expressed by Dory, who worked with Jeisson:

There is not going to be much money left from our parents’ salary because they will only be working to pay for our clothes, food, and all that. Our parents should earn a little bit more than the minimum wage, in my opinion (Rodríguez and Moreno, 2015, p. 62, our translation).

Internet access became a tool for the students; it made other sources of information that could help them during the activities accessible. In this case, they discovered the salary that their parents earned, which was previously unknown to them. Thus, some mediation between the contributions of the teachers and the considerations of the students with respect to the knowledge involved in the investigations has taken place. This brings about an insight that goes way beyond mathematics and connects their schooling to their own socio-cultural environment.

The students became involved in an exercise of trust and autonomy by choosing the situations that allowed them to broaden their ideas about their socio-economic contexts. Nonetheless, at this point we realised that neither we nor our students had sufficient experience in creating and executing an educational approach like this one.

Scene 2: Going Shopping Is Amazing!

Once again, the class started with a lot of chatter, which is normal when students are asked to move to another room. This time, the students were asked to go to the computer room because of the opportunities that this space offered for accessing information on the internet.

Our intention for this stage was to provide the students with the opportunity to question themselves about: (a) the items that they wish to buy (branded clothes, the latest model car, a mansion, a private jet, and other luxury things); (b) the reason why all these items are only dreams (high cost of acquisition); and (c) the payment method that they would use to get them (bank loans or credit cards). With the purpose of having a record of their discussions and reflections, we asked them to access the blog and answer some questions in groups. To answer the questions, the students needed to use their own experiences with their families, like in Scene 1. Analysing the situation, they concluded that it was very unlikely that they would ever be able to afford these luxurious items using cash, so they would need to use consumer lending in order to buy them.

When we talked with the students, we recognised the lack of knowledge about consumer lending that they had, and we allowed them to spend some time finding out about the topic. To ensure that all students had a similar experience, we asked them to do a PowerPoint presentation of approximately five minutes. Here, they could present all the relevant information about the selected payment method, and also explain the reason why that method was chosen.

During this time, both students and staff noticed an evident change in the mathematics class. The scene generated an inflection point that challenged the usual routine of consistency on the teacher’s explanations and then solving some exercises. This is one consequence not only of the change of spaces, but also of having access to the internet as a pedagogical tool.

Scene 3: Selecting!

For this scene, we were located in the school audio-visual resources room. This is the place where the students were asked to prepare their presentations of the information they researched in Scene 2. Students found four types of commonly used consumer lending. Among them was one linked to the payment of the electricity bill.⁸ This is the payment on which the whole landscape of investigation was focussed, and we sensed an uncomfortable atmosphere in the classroom that day. The positive and joyful environment that we had seen in the first two scenes seemed to be fading away. The students seemed distracted and not focussed on the objective that we were proposing.

Time kept running out, and the class quickly ended, but it left us researchers thinking about what we were doing. We imagined that we had presented a routine-like exercise of gathering information from the internet. If we look closely at what did happen, the silence that was experienced when the students were asked to come up with questions could have been because each student was only interested in what they were doing in their own group. We can see some evidence of that in the following discussion:

Julian: Well, the documents required for this loan can be identified here [pointing to the image projected in the video beam]. If the person receives a salary, a pension, or is an independent worker, they need to have: an enlarged copy of the ID, a certificate of incomes and deductions, a work certificate, a copy of the last salary payment slip, a copy of the income tax certificate from the previous year, a certificate from the chamber of commerce dated not less than 30 days before, a copy of bank statements for their savings or checking account over the last few months, financial statements—those are some of the requirements to get a loan.

Teacher: OK, do you have any questions about any of the terminology that is being used there, any of the conditions they have talked about… [Silence]. Nothing? Is everything clear? Is everything understood? [Silence]

Steven: Questions, whoever…. [Students laugh]

(Gutiérrez and Rodríguez, 2015, p. 67, our translation)

Based on what was said before, we needed to reflect on what was taking place. At this point, we felt the pressure of not knowing whether we were actually achieving the aim of creating a different type of mathematics education. Some questions emerged, such as: what was the purpose of the presentations done by the students? Why did we ask the majority of the students to develop a similar activity? Maybe we did in order to ease the monitoring of the activity? Or maybe our experience as teachers called us to go back to our comfort zones? At this moment, the tension rose because we did not know if the students would be able to display their real interests in front of the class. After all, from the outset the targeted objectives had been proposed by us instead of them.

Then we remembered that idiomatic expression, “the darkest hour is just before dawn”. Our doubts can also be sources of clarity in the opportunity to reflect on our decisions and actions; they show us a way to advance in our work towards a learning milieu of type 6, as is proposed by Skovsmose (2012) (bear in mind that learning milieu of type 6 is characterised by reference to real-life situations). This caused us to face a great challenge: to allow the students to move beyond explicit instructions and beyond their constant need for the approval of the teachers. On the contrary, the objective of the landscape was to guide the students to work in a cooperative process of constant negotiation, and in that way, to construct an ideal landscape.

By rethinking our purposes, we recognised, as Skovsmose (2012) proposes, that during the exploration of a landscape of investigation it is possible to move between different learning milieus. The idea is that it is not advisable to remain in one particular milieu, or to continue in one that does not contribute to the intentions of the researchers or the students.

Scene 4: Choosing!

A new lesson started, and with it the opportunity to start again with the objective of changing the style of lesson. This new space gave us an opportunity to build, along with our students, a tool that was meaningful and allowed them to interpret and transform their lives and their social contexts.

Following that plan, students were instructed to enter the blog and the forum entitled “Finally, What Payment Method Have You Decided to Use for Your Shopping?” In this forum, the students participated by expressing their decisions based on the presentations and their research. They also needed to clarify whether they had changed their mind about the payment method since the beginning of the activity. Here is an extract of what one of the students wrote: “we decided to use the Codensa card,⁹ because it gives us many benefits, and now we can do cash loans. This is easier to obtain and manage” (Gutiérrez and Rodríguez, 2015, p. 71).

Thus, each action and decision developed by the students was recorded both in the blog and in the work folder, allowing them to identify the importance of knowing the process of applying for a loan, including the way in which the amount granted by a financial institution is defined. This last aspect became the generator of the guidelines that shape Scene 5.

It is important to recognise that even the most difficult moments can provide opportunities for growth. Our experience made us realise that the presentations did not emerge from a personal need felt by our students. Nevertheless, the presentations gave them access to much information, and this information changed some of the decisions that the groups had originally made.

Scene 5: What Do I Need?

In the continuous search to know and understand the process of applying for a consumer loan, the students filled in the forms required for this application. Consequently, they started to recognise the relationship between the income of the person requesting the consumer loan and the amount of money that this person might be given. We can see evidence of that in the following remark by Julian:

Here! This is when we did it wrong because we put the salary of Stiven and his income that is 100,000 COP (Colombian Pesos) per week. If we are going to request the loan in Codensa, they said that they can give you up to four times your income. The top amount that they will loan him is 2,000,000 COP. That means that he wouldn’t be able to afford anything¹⁰ (Gutiérrez and Rodríguez, 2015, p. 77, our translation).

The students’ way of working started to please us as, without needing too much guidance, they managed to establish for themselves a cause-effect relationship between the variables of the mathematical model applied for the approval of consumer lending (Barbosa, 2006).

At this point, the tensions that we faced in the construction of the landscape of investigation eased slightly, and we were able to view the negotiation with the students with optimism. Following this thought, we could create the possibility of fostering a learning milieu in which mathematics became a tool that allowed them to understand the decisions that are taken in society and in certain situations.

As the students continued exploring the topic, they started to reflect on what happened in their own families, and related it to the work they had been developing in class.

Days like this allowed us as researchers to open up a space for reflection, a space in which we found evidence of the students’ progress towards autonomy. They oriented their interests by rooting them within their own concerns, and not within what the teachers gave them. This progress increased our confidence, and made us believe that it would be easier to advance.

Scene 6: And My Family? Are They Going to Be Indebted?

When students embarked on an examination of how their families did their shopping, the dull walls of the classroom—that seem to be designed to exclude or protect the students from the dynamics of the social world of adults (though eventually, they are going to face that world)—opened up for them. They came to recognise what happens outside the walls of the classroom and inside the walls of their homes, with respect to financial difficulties.

This recognition becomes very important because it brings back into the classroom information that comes from real-life contexts. It is modelled through mathematics, just as we saw in the forum blog in the intervention that Allison did: which are the payment means related to consumer lending that your family is currently using?

The head of the home has requested a loan from the bank ‘Finamerica’. At the beginning they gave her some advice; she needed the loan for a family meeting that could not be postponed. The request was for 2,000,000 COP and it was approved. In order to get it, she went to the bank, and they only approved 1,150,000 COP because it was her first time and at the end, they only gave her 1,007,550 COP, saying that the rest of the money was a fee for the documentation (Gutiérrez and Rodríguez, 2015, p. 78, our translation).

At the same time, we saw that, despite the uncertainty which emerged through the development of our landscape of investigation, the work should not be forced or simulated. It should be inspired by the students’ own interest in the situation and the living experience that it offers them.

The previous situation allowed the students to establish a relationship between the economic hardships that exist in their homes, caused by the use of consumer lending, and the fact that the money that should cover basic needs is used for paying debts. A student named Angie mentioned that “the situation in some homes is sometimes a bit complicated because there are many debts from loans and credit cards” (Gutiérrez and Rodríguez, 2015, p. 81, our translation).

The conversations among some of our students left us with a lot of questions about how to move forward with the activity. This situation drove the students to seek the approval of the teachers who were traditionally seen as the authority in terms of knowledge. However, it takes time for students to become truly autonomous, as Steven commented: “it takes her forever, and at the end, she provides no explanation” (Gutiérrez and Rodríguez, 2015, p. 79, our translation).

Based on what they said, we recognised almost immediately that our students still looked to us for our approval about what they thought and did. This experience of security helped them to continue with their work. The discomfort that Steven expressed is common among students, and Julian mentioned: “I get desperate because I start to look for help using other means. I can get support from one of my classmates, but in the end it is the teacher who knows. That will give me the security that it is solid” (Gutiérrez and Rodríguez, 2015, p. 80, our translation).

At that moment, we faced a challenge in the learning milieu that we wanted—and still want—to foster in the mathematics classroom. Our interest is that, in this landscape, students work autonomously, and the teacher is no longer the person that possesses absolute knowledge. Consequently, we shifted our attention towards the possibility of opening up spaces of democratic participation where the knowledge of the students is also recognised.

Scene 7: A New Look to Your Shopping

The time came when students started questioning the reality of the credit system and how much people end up paying when they use consumer lending. This concern drove students to research more about how companies determined the number of payments and the value of them.

At this stage, the internet became a very important part of the landscape, because students used it to interact with a virtual simulator. The simulator allowed them to recognise that the value to be paid depends directly on the number of instalments of the loan, something they had previously been unaware of. The students were asked to register this information in their folders (as seen in Fig. 1 and Fig. 2). In this activity, mathematics continued to be the vehicle in the process of investigation.

Fig. 1. Registration in the work folders of the students (Gutiérrez and Rodríguez, 2015).

Fig. 2. Translation of the work folder.

The exploration of the landscape then moved forward, not only at the level of our pedagogical practices, but also at the level of the objectives that we had set initially. We found that the students had developed a better comprehension of some elements of the mathematical model and the way it functions. Nevertheless, the students were still not able to fully explain the process of calculating the value of each instalment, as we can see in the following observation made by Julian:

The company Codensa must earn some money for each payment. For the customer it is more expensive because they have to pay more than what they asked. This process generally happens without the customer being fully aware of the extra money that they are paying for the item they purchased (Gutiérrez and Rodríguez, 2015, p. 86, our translation).

Scene 8: Now What Do We Do?

On this day, two of the groups that had worked on the same credit system interacted with each other to answer questions. Their concerns were about the model employed to obtain the interest for the instalments, and their relationship with the existence of exceeding value. This value is charged to the customer without them really knowing where it comes from. This can be seen in the following conversation:

Karen: Julian, look here! What else can I do? What do you think is missing?… 32,104, we are missing those 32 that they are charging us extra, but we do not know why. Look here! There is a 28%. I do not know where that came from.

Julian: We also found the 28 point something, and we calculated the formula of percentage and that could not be the price that they charged you. I would say that is 2.1276%.

Karen: And where does it come from?

Julian: From the rate… the interest rate, but I do not know, I do not know if what I am doing is right.

Karen: No, and I do not know what to do…

(Gutiérrez and Rodríguez, 2015, p. 87, our translation)

Taking into account Julian’s confusion about whether or not his procedure was correct, we had the opportunity to reflect on how his insecurity did not necessarily mean a defeat for us teachers. Actually, it gave us evidence about the process of questioning a specific model of the credit system. Unlike the previous scenes in which we were uncertain about the upcoming events, this experience allowed us to direct the work of our students more effectively. In this case, it showed that, in this space, students can reflect about the functioning of consumer lending.

At the same time, the previous situations let us understand that, through reflection about the mathematical modelling, reflexive and technical discussions tend to emerge according to Barbosa (2006). These discussions established a relationship between the interest rate expenses and the number of payment instalments that the total debt is divided into. In this landscape of investigation with all its ups and downs, one student—Lorena, who had been working with Karen—had a critical position towards the functioning of the mathematical model, and she expressed her opinion as we can see in the following extract from her work folder:

Despite many attempts and mathematical operations, there is no way to get a specific result. We did the calculations over and over again without knowing where all that amount of money they were charging came from. This means that the companies are stealing from the people who have to request consumer lending. The worst thing is that the people do not realise it. They just accept the terms and pay for the money, which they should not do in the first place. This exercise helped us a lot, because it made us realise what is in store for us (Gutiérrez and Rodríguez, 2015, p. 90, our translation).

At this point, we can see that the objective of the research scenario with respect to the process of modelling is properly oriented. We achieved this despite the fears, uncertainties, and tensions that were experienced during the whole process, not only by the students but also by us, the teachers and researchers.

Other situations that are worth highlighting here are the interactions among students. Karen, for example, mentions that she approached Julian’s group because he normally “did well in mathematics” (Gutiérrez and Rodríguez, 2015, p. 89, our translation). This statement shows us the empowerment that a person can experience when he or she has knowledge in this area. It establishes a power relationship that we perceive in this case as a challenge. Considering that, with respect to this landscape of investigation, it is fundamental to recognise how dialogue and teamwork contribute to citizenship education, as formulated by Silva and Kato (2012).

Based on what happened, we felt that it had been a good day for all of us. We believed that, one way or another, each participant had achieved, at their own pace, a socio-critical perspective on the class and their own reality.

Scene 9: The History of Things

The critical reflections that started to take place in the classroom made it possible for personal opinions about the credit system to emerge. This was a good opportunity to create dialogue. In this way, our students started sharing their points of view that were supported by their own explorations of the landscape of investigation. They also recognised the possibility of relating these ideas to the situations of their own families.

The satisfaction that we experienced when we saw the path that we had taken was comforting. This satisfaction allowed us to consider a family simulation, like the one that happened in Scene 1, as being a functioning educational approach. It showed that students can empathise with their families—in more colloquial terms, they can put themselves in their families’ shoes. They came to consider something that they had never thought of, neither in the classroom nor in their homes (Gutiérrez and Rodríguez, 2015, p. 94).

Additionally, the milieu that was created allowed us to retry, at some point, the processes of negotiation with the students. Through the creation of a forum, the work students developed in the classroom could be shared with families and other members of the community. This means that the landscape of investigation could open the doors of the classroom towards problematic socio-economic situations.

Nevertheless, today we have to recognise that the institutional pressure within the school has started to intensify. Due to decisions taken at the administrative level of the school with respect to the school calendar, and specific instructions for the academic end of the year, we were unable to do what we had wanted to do at the beginning of the project with respect to the creation of the forum. In the following fragment of an InterView,¹¹ we can see some indications of this:

Teacher: We have talked about a forum that was going to be established on a Saturday because there were some time issues and other institutional activities, but we couldn’t make it happen. Would it have been good to do it?

Miguel: Yes! I think that we could have done that reflection with the parents, the one that we did ourselves.

Lorena: I think it would have been useful because some adults do not know where all of this comes from. They just pay. But all this information would have been very useful.

Julian: Well, it was very interesting and very nice to make that forum. We could have brought our relatives—parents, cousins, uncles—so they could get more information about this important topic.

Jeisson: I would have taken my whole family, because many of them are not aware of this.

(Gutiérrez and Rodríguez, 2015, p. 97, our translation)

The InterView allows us to recognise the intention of moving beyond the walls of the classroom. We tried to get the families and other people from the community involved in the process of mathematical modelling from a socio-critical perspective, but that turned out to be too challenging. In this way, the reflexive discussion among the students would not be able to widen the impact of this landscape of investigation. Nevertheless, it could create greater engagement of society through the students. This observation is supported by Silva and Kato (2012).

The time to end the class had arrived, and also the time to conclude the landscape of investigation. There was only time left to thank our students for allowing us to learn with them, to recognise their interests, and to see how they gave new meaning to their realities through mathematics. Consequently, we can acknowledge that the tensions, uncertainties, and fears were worth it, just like Julian mentions in one of the InterViews:

It is important to change the space you’re in; we were locked down in a classroom for about eight hours and we continued with the same monotony… we moved to the library, the audio-visual room, the computer room—when we did that we could clear our minds for a moment and we were more relaxed; we knew that we had a problem to solve and we went to solve it. We had the opportunity of using the internet and we could find out more about what we had to do. It was important to change spaces because if we did not have all the information to support what we thought, we would not be able to change our minds (Gutiérrez and Rodríguez, 2015, p. 101, our translation).

10. Reflections

The set-up of the landscape “Let’s Go Shopping” gave us numerous possibilities and some challenges due to our aim to create a learning milieu of type 6 (Skovsmose, 2012). We considered that the interactions among the students should justify each action and each step forward in the investigative process.

However, our lack of experience and training constantly pushed us to plan and over-specify the instructions to guide the students’ work. As a consequence, to a stronger extent than we had hoped we were leaving aside the intentions, backgrounds, and foregrounds during the negotiation process, all of which might give meaning to working with the landscape of investigation.

Bringing together all the classroom participants made it possible to establish reflections about the social aspects embedded in the project, with respect to the use of consumer lending and its impact on the families’ finances.

Researching the landscape of investigation made it possible to take the mathematics lesson to other locations than the classroom. In this way, we reinforced the use of technological tools that helped students undertake research beyond the use of texts and information provided by the teacher.

As part of the exploration of the landscape of investigation, we observed a change in the students’ views. This seemed to be due to moments of reflection about social aspects that were not considered before in the mathematics class. Thus, the students’ discussions allowed us to identify understandings and insights about their role in society and the reality in which their families are immersed.

At the beginning of this project, many expectations, fears, and uncertainties emerged because it involved a negotiation process of the landscape of investigation with the students. Some of these expectations gave the project a new meaning, both for us and for the students themselves. This provided us with the starting point of a path that leads us to further explorations.

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