14. Landscapes of Investigation with Seniors1
© 2022 Gomes da Silva et al., CC BY-NC 4.0 https://doi.org/10.11647/OBP.0316.14
The authors discuss how seniors engage in mathematical activities when inserted into landscapes of investigation. Through a qualitative approach, the researchers collected data in two meetings with a group of seniors who were involved in landscapes of investigation in a mathematics education project aimed at that age range. The authors also highlight that the activities fostered interaction/collaboration between the seniors, causing them to change their conception of mathematics, producing knowledge and raising philosophical, aesthetic, technological and mathematical discussions.
Several countries around the world are experiencing the phenomenon of an increasingly ageing population. In Brazil, according to data from the Brazilian Institute of Geography and Statistics more than a third of the population will be aged 65 and over in three decades’ time (IBGE, 2019). This phenomenon has been one of the main reasons for governments to act in favour of ameliorating the quality of life of this growing section of the population—for example, through enhancing educational opportunities.
In the case of Brazil, such practices are expressed as a right of seniors in the Legal Rights of the Senior Citizens (Law n. 10.741 of October 2003) and in the National Education Plan (Law n. 13.005 of June 2014). Additionally, research in education—and in mathematics education in particular—has shown that the involvement of seniors in such practices can help them remain cognitively active, interact socially and learn new content. It can also help them establish the necessary knowledge to claim their rights and participate more actively, critically and creatively in their own lives, remaining active in the community (Julio and Silva, 2019; Lima, 2015; Lima et al., 2019; Scortegagna, 2010). In addition, Scagion (2018) warns that the vision of seniors related to mathematics may be connected mainly to common sense conceptions and ideas. Based on Social Representation Theory from Moscovici (2005), Scagion (2018) has interviewed seniors who have participated in mathematics activities offered by a university project. “Mathematics is everywhere” or “Mathematics is important for quality of life” were some of the social representations identified among the participants. Scagion (2018) points out that these representations end up being incorporated in their discourse in only a very superficial way. They are not able to provide details about the nature of usefulness and the contexts where mathematics could be applied. According to Scagion, involvement in educational practices related to mathematics education could expand their repertoire.
In this chapter, landscapes of investigation with seniors2 will be analysed, bringing together elements from our experiences in two meetings held in 2018 through an extension project3 aimed at the development of mathematics education for seniors, called Mathematics Conversations. These meetings were organised around two topics: the golden ratio and the Fibonacci sequence. In the first meeting, eight participated and in the last, seven participated. Table 1 introduces the seniors that participated in both activities. All names are pseudonyms.
Name |
Age (years) |
Education |
Meeting 1 |
Meeting 2 |
Lúcia |
67 |
Portuguese Teacher |
X |
X |
Marina |
65 |
Pharmacist |
X |
|
Simone |
63 |
X |
X |
|
Sandra |
65 |
Psychologist |
X |
X |
Joana |
63 |
Pharmacist |
X |
|
Selma |
64 |
Physical Education teacher |
X |
|
Lúcia |
67 |
Attorney |
X |
|
Glória |
71 |
Teacher training degree |
X |
X |
Ana |
65 |
Mathematics teacher |
X |
|
Pedro |
65 |
Electrician |
X |
|
Adelia |
65 |
Geography teacher |
X |
The research method was participant observation. Data was collected through audio recordings of the meetings and conversations with the seniors during and after the activities, and field notes. The research team transcribed and organised the data to analyse how the seniors engaged in landscapes of investigation. The processes of preparing landscapes of investigation, as well as the analysis, are based on Skovsmose (2000, 2011) and on research in mathematics education with seniors, such as Lima (2015), who provided relevant guidelines for working with this age range.
1. Landscapes of Investigation with Seniors: The Mathematics Conversations Project
Lima (2015) was a pioneer in the mathematics education field involving systematic approaches to the development of pedagogical activities aimed at seniors. His research inspired us to develop the Mathematics Conversations project, aimed at seniors participating in the Third Age Open University Program at the Federal University of Alfenas. Since 2018, activities have been held weekly at this university, and include the active participation of seniors with different backgrounds—as illustrated by Table 1—who chose to participate in the project to learn more about mathematics and stimulate their minds. Two teachers and four students from the mathematics degree programme at the Federal University of Alfenas formed the project team in 2018. The team planned the activities weekly and discussed what occurred in previous meetings with the seniors. In this process, the team explored articles and experiences, and discussed the adaptation of teaching materials for working with seniors. In addition, doubts and mathematical questions raised by the seniors were used as possibilities for the creation of new proposals. The project also provided research data related to mathematics education and seniors.4
The project sought to develop activities in a learning environment that would foster the creation of landscapes of investigation, as proposed by Skovsmose (2000, 2011). The landscapes of investigation differ from activities based on the “exercise paradigm”, in which the central premise is that for each exercise or task, there is one, and only one, correct answer, and the teacher is at the centre of the educational process. A landscape of investigation favours the use of investigations in teaching and learning processes. It makes the students responsible for exploring and explaining/justifying knowledge production, and depends on their acceptance of the investigation. In a learning environment, the differences between the pedagogical work based on landscapes of investigation and the exercise paradigm can be categorised based on the articulation of these two practices and three references (Table 2).
Exercise Paradigm |
Landscapes of Investigation |
|
References to Mathematics |
(1) |
(2) |
References to Semi-Reality |
(3) |
(4) |
References to the Real World |
(5) |
(6) |
The first reference is related to questions and mathematics activities. The second, a constructed reality, is called semi-reality, and the third refers to situations in the real world. In school mathematics education, Environments 1 and 3 are the most common, as they refer to exercises formulated in the context of mathematics and semi-reality, respectively. In Environment 1, imperative technical activities predominate, such as “Solve the equation …” or “Calculate the value of the hypotenuse of the given right triangle”. In Environment 3, situations are constructed in order to train students in certain mathematical techniques. Even though Environment 5 is based on real-world data, students only use such information to solve a given closed task. Environments 2, 4 and 6 are related to the use of investigative practices based on the three references. According to Skovsmose (2000, 2011), in landscapes of investigation, questions such as “What happened if…” characterise the teacher invitation, while questions such as “Yes, what happened if…” characterise students’ acceptation of entering the investigation process.
Considering that most of the school mathematics experienced by the seniors will have been based on the exercise paradigm, which in many cases caused traumas and even an aversion to mathematics, we started with landscapes of investigation.5 This was revealed to be a good option, because it encouraged different discussions and conversations about mathematics and other subjects, captivating the participants and causing more seniors to join the project, which started with only one senior woman. From there, other seniors began to participate, losing their fear of facing mathematics-related matters.
Another aspect is that in landscapes of investigation, the process is more important than a specific result. In this process, there are many conversations about mathematics and other subjects among the participants. The encouragement and valorisation of these conversations are based on a critical conception of education (Alrø and Skovsmose, 2003; Freire, 2016), prioritising dialogue as fundamental to the production of knowledge. Alrø and Skovsmose (2003) provide us with guidelines regarding the use of dialogue with elements that constitute what they call the inquiry-cooperation model (IC-model). This model is formed by dialogic acts that emerge during the interaction between teachers and learners. In our case, with the project team and the seniors, these were namely: to establish contact, perceive, recognise, take a stand, think aloud, reformulate, challenge and evaluate. To develop the pedagogical activities, we also considered Lima’s recommendations (2015). According to him, it is recommended that we prepare the environment in advance of receiving the seniors—increase font size, use appropriate voice intonation for everyone to hear well and fully participate in the activity, and finally respect the time the senior needs to carry out the activities.
2. The “Golden Ratio” and “Fibonacci Sequence”
To discuss the engagement of the seniors in two landscapes of investigation, we begin by describing how the development of the learning environments occurred. Afterwards, we bring in elements that will enable us to raise the topic of discussion as planned.
In order to work with the “golden ratio” and the “Fibonacci sequence”, the project team encouraged the construction of learning environments based on landscapes of investigation. The matters came from a request from one of the seniors, who wanted to know what “golden ratio” meant and what its relationship was with the Fibonacci sequence.
As the matters had arisen from the interest of only one participant, the team was concerned with building learning environments so that all the seniors would accept the invitation to participate. Thus, we planned two activities, implemented in two different meetings, called “golden ratio landscape of investigation” and “Fibonacci sequence landscape of investigation”. To plan the activities, we carried out a bibliographical review on these matters.6 Thus, the team “opened the exercises” (Skovsmose, 2011)—in other words, they transformed/adapted the activities from the bibliographic review to compose the landscapes of investigation.
The golden ratio landscape of investigation was developed with the seniors as follows. First, we showed them several images: Leonardo da Vinci’s Mona Lisa; the Notre Dame cathedral in Paris; a shell; the Apple company symbol; an ear; a sunflower; a credit card, and the Vitruvian Man by Leonardo da Vinci (Fig. 1). Then, we asked the seniors to say what they thought the images had in common. The intention was to invite them to explore and raise questions.
|
|
|
|
|
|
|
|
Following that, we presented the same figures with golden spirals inscribed in rectangles, without telling them at once what that meant (Fig. 2).
|
|
|
|
|
|
|
|
We then proposed a discussion about the golden ratio, addressing its historical context and the mathematical elements involved. We started by inviting a senior woman to read a text,7 which highlighted the use of the golden ratio in the Renaissance. The text featured a picture of the Vitruvian Man, showing a man of golden proportions inscribed on a circumference and a square. After the questions, comments and hypotheses, we proposed that the seniors should use their own body measurements to explore the golden ratio. In the last step, the seniors constructed the golden rectangle with a ruler and a compass, and then calculated the golden ratio of the rectangle constructed. To conclude, the seniors were given other images, such as swirls, snails, milk glass flowers and the symbols of Toyota, Pepsi and the National Geographic television channel (Fig. 3). In these symbols, the seniors could identify the golden ratio, or else verify that its proportions referred to it.
|
|
|
|
The Fibonacci sequence landscape of investigation sought to make the seniors discuss the functioning of a mathematical sequence. Specifically, it aimed to help them identify a relationship between the mathematical contents handled in the golden ratio scenario with the Fibonacci sequence. As the seniors were eager to know the historical context, the project team chose to introduce the subject by reading a text8 about Leonardo de Pisa (or Fibonacci), highlighting his importance in the introduction of the numbering system we use currently—the Indo-Arabic numbering system or decimal numbering—and the rabbit problem, which allows one to establish the Fibonacci sequence. The rabbit problem, adapted from Lívio (2008), is formulated as: “A man sets a couple of rabbits in a fenced place. How many couples of rabbits can be generated by this couple in a year if we assume that each month each couple generates a new couple that will start to breed from the second month of life?” After presenting the rabbit problem, the seniors were invited to work on it collaboratively, using a cardboard table and cardboard pictures of adult and young rabbits they could manipulate. The seniors were first required to familiarise themselves with the proposal, which they did by exploring and asking questions, and by establishing negotiations and strategies to solve the problem.
To promote discussion about the results obtained, the seniors completed a table individually by registering the number of couples of adult rabbits, young rabbits and the total number of rabbits for each month. The objective was to establish the relationship between the elements of the sequences formed in the columns of the table to determine the characteristics of the Fibonacci sequence. Then, to relate the Fibonacci sequence to the golden ratio, the team invited the seniors to divide the9 terms of this sequence and register the results in their table, to investigate what was happening with those ratios.
The seniors were given images to observe, such as the family tree of a drone (male bee), the configuration of sunflower seeds, the number of petals of a daisy flower and the growth of plant branches—all of which can serve to illustrate the Fibonacci sequence. Afterward, the team proposed as homework the resolution of some questions that had been taken from textbooks that dealt with numerical sequences from words and shapes (Fig. 4).
|
|
|
|
3. Seniors’ Engagement in a Landscape of Investigation
The first step in discussing the involvement of seniors inserted in a landscape of investigation concerns their acceptance to participate. According to Skovsmose (2011), this is the first (and perhaps most important) moment of a mathematical investigation in an educational context. The analysis of the data produced in this study indicates that the seniors accepted both proposals. For example, as soon as the team introduced the first scenario, even though they did not find a common characteristic among the images presented, the seniors tried to find/perceive regularities by asking questions, and they took a stand, thinking aloud.
Ana tried to identify geometric figures in common, Pedro observed that the figures were centralised and somewhat symmetrical, and Lúcia and Marina said that the curves were the characteristic the figures had in common. Lúcia identified a square, remembering a previous meeting where the group discussed magic squares, and asked whether the activity would be about curves. This finding was confirmed when the team showed them the same images, but this time with golden rectangles and spirals inserted, and the seniors identified the insertions and made statements such as: “A proportional division was made. […] It’s like a shape… […] It’s like a template” (Pedro), and “the smallest part [of the spiral] is distributed in the whole” (Sandra).
In the second landscape of investigation, the team considered that the seniors agreed to join the activity because of the warm discussions that emerged when they explored the rabbit problem, in which they tried not only to understand it but also to establish an action plan. At first, those discussions caused the group to fragment, because Joana was so convinced that her conjectures were correct that she decided to solve the problem by herself, while the other seniors remained in groups, working collaboratively.
Despite the participants’ acceptance of the invitation to join the landscape of investigation, some seemed to feel unmotivated to continue exploring the situation. The reading of the text about the use of golden ratio in the Renaissance, and the discussion on questions and assumptions raised at the beginning of the golden ratio landscape of investigation, proved to be motivating for the introduction of the measurement of seniors’ bodies. However, Ana, for instance, was not interested in the subject during this first stage. She engaged in the activity only when we started the actual body measurement. She began calculating in order to compare the measurements and ratios made with the golden ratio. At this stage, the project team measured some seniors, while some of the seniors measured each other. What we want to draw attention to here is the importance of the variation of activities in a landscape of investigation addressed to seniors. Invitations for certain activities may not be accepted by all participants, as happened with Ana, who became involved only when the practical activity did begin.
Besides, Lima (2015) states that the organisers of the activities must prepare for unexpected situations, such as the seniors’ lack of interest or criticism toward the activity. In this way, this study reiterated, based on Julio and Oliveira (2018), that more than preparing an activity, it is paramount to prepare for it, thinking about possibilities of producing meanings, putting oneself in the place of the others (the seniors), and being open to interaction.
At different moments in both landscapes of investigation, the seniors asked questions. An example was in the golden ratio landscape of investigation, in which the reading made it possible to raise questions such as: “Is there any relationship between the lower and upper limbs?” (Marina), and “does this work for everyone?” (Sandra). Glória commented that certain proportions might be related to European people, but that for other ethnic groups, such as the Chinese or Japanese, they would not be valid. Another example happened in the Fibonacci sequence scenario, when Lúcia commented that a sequence would be constructed, as she realised what would happen in the first months of the rabbit table. Regarding the elaboration of strategies in the Fibonacci sequence landscape of investigation, the very statement of the rabbit problem caused many disagreements between the seniors. It really was a challenge for them, and it was necessary to carry out a long process of negotiating strategies.
Alrø and Skovsmose (2003) highlight the importance of evaluating the perspective of the participants during the investigative process in a landscape of investigation, to understand their points of view about the problem and reach a common purpose. In the case of the Fibonacci sequence landscape of investigation, the rabbit problem was characterised by being a fictitious situation about the rabbits’ breeding, which the seniors found difficult to detach from a real situation. Thus, the team needed to intervene in the discussion to clarify this issue, and to discuss in the group the objective of the problem, restructuring it from a perspective that was common to all—for example, considering the couple of rabbits from the first month as a young couple.
The participants were offered a large table which they should fill in together. They chose to remain seated, and as they completed the column of months, the person closest to the row referring to a given month filled it with young and adult rabbits (Fig. 5). Thus, everyone was able to help to construct the table, following what was being done, while some filled in their own tables at the same time.
As much as a table-filling strategy had already been established with the seniors, some of them were still having difficulties about what they should do. Thus, a break was needed to discuss the strategy in use with the group again. Only after all agreed and understood, were they able to continue. The strategy the seniors used to distribute rabbits each month was always to count the number of rabbits from the previous month that would be adults the next month and distribute them. Then, they checked the number of rabbit couples from the previous month that could have young couples next month, and distributed them to their respective parents. For example, in the eighth month, there were twenty-one rabbit couples: thirteen adult couples, and eight young couples (Fig. 6). Thus, to complete the ninth month, first, they placed twenty-one couples of adult rabbits and, as in the eighth month they had thirteen adult couples, only these couples would generate new bunnies in the ninth month so, the participants placed thirteen young couples so that each couple was beneath an adult couple. Thus, in the ninth month, thirty-four couples had been placed on the table.
Joana was working on her own during this task. When she saw that she had made miscalculations, she returned to the group, following the setting up of the table and helping to complete the tenth-month row. Even when filling in the table, Joana used a different strategy than the other seniors: she counted how many couples they had in each row in the previous month and distributed them in the following month. After she did this for all rows, she analysed which rows had adult couples in the previous month, and distributed the young couples in the following month. For example, in the ninth month, there were thirty-four couples of rabbits: twenty-one adult couples and thirteen young couples (Fig. 7). To complete the tenth month, Joana took the same amount of couples of rabbits as she had in the first row of the ninth month and placed them in the tenth month. As these eight couples were already adults in the ninth month, they would generate young couples in the tenth month. Therefore, Joana placed eight young couples corresponding to each adult couple above in the second row. After that, she took the same number of couples of rabbits from the second row of the ninth month and placed them in the third row of the tenth month. Since those couples were young in the ninth month, they would not generate young couples in the tenth month. Joana followed the same procedure for all the rows until she finished the tenth month.
In the beginning, the other participants did not understand Joana’s strategy, and, when asked, Joana explained it until everyone understood. Some women commented that “changing the order of factors does not change the product”. We noticed that this sharing encouraged the seniors to discuss the topic during the development of the activity. As already mentioned, in a landscape of investigation, the investigation process often becomes more important than the result itself (Skovsmose, 2011).
As in Lima et al. (2019), this study showed that the engagement of the seniors with landscapes of investigation allows them to develop creativity through the exposure of ideas and sharing of conclusions, leading to the emergence of different solutions to a problem and production of knowledge. In both meetings, the seniors interacted/collaborated actively (Fig. 5). They decided in the group about the strategies for filling in the table by presenting different arguments, and they worked collaboratively to fill the table and discuss the number of rabbits. In Figure 8, which illustrates the measurement of bodies, we noticed a more significant interaction between the seniors and the project team.
|
|
|
Also, as in Lima et al. (2019), the analysis indicated that the seniors’ engagement in a landscape of investigation fostered the production of mathematical knowledge. For example, when the seniors drew the golden rectangle with a ruler and a compass in the first activity, they did not find it hard to use them to construct the rectangle nor to calculate the golden ratio, using two different strategies: while some seniors calculated the ratio between the measurement of the larger side (AE) and the smaller side (AD), others calculated the ratio between the measurement of segments AB and BE, as shown in Figure 9. Then, each participant shared how they calculated, easing the interaction with each other, and showing they understood the subject.
In the Fibonacci sequence landscape of investigation, after the seniors filled out the cardboard table, the team proposed that they filled in their own tables. To conclude the investigation, we discussed the possible relationships between the numbers of the sequence formed in the column that represented the total of rabbits. In this way, the seniors could share their ideas, reasoning, and justification about the topic. The team that conducted the activity was in the background, trying to foster dialogue amongst the participants. At the beginning of the discussion, Sandra, Selma and Glória reported that they had already realised that each term, starting with the third, was obtained by adding the two previous terms (Fig. 10). They also found that this relationship occurred in the other columns (adult couples and young couples). The discussions they raised helped others to identify this pattern as well.
At this point, the research team chose to inform the seniors that the sequence formed was known as the Fibonacci sequence. The seniors continued to express thoughts and ideas about the relationships in the table out loud, realising that the sequence was formed in the other columns, starting in different rows. Sandra noticed another relationship between the elements in the table (Fig. 10), relating all the columns—that is, the sum of the total (of the first row) and the number of couples of young rabbits of the second row gave, as a result, the number of couples of adult rabbits of the third row; the sum of the total (of the second row) and the number of couples of young rabbits in the third row gave, as a result, the number of couples of adult rabbits in the fourth row, and so on. She raised this point and wanted to discuss it with the other seniors. Glória commented “the table breeds like rabbits”.
Sandra established regularities that the team had not noticed during the preparation of the activity. Other assumptions also surprised team members when carrying out the activities. For example, Joana tried to relate the total number of rabbits in the table to a percentage and questioned the existence of a growth graph of the situation explored. She also asked how we use this sequence and whether it is valid for other animals’ breeding. We consider that, at that moment, the senior sought to produce/relate mathematical knowledge through the landscape of investigation. Also, the activities carried out provided the seniors with the ability to establish relationships between the activities carried out in the project. During the Fibonacci sequence landscape of investigation, the seniors had to fill a column in the individual table with the values of the ratio between the terms of the Fibonacci sequence. Using calculators, pencils, paper and mental calculation, right after the first divisions, they realised that the value was getting closer to the golden mean with which we had worked a few weeks before. Joana asked whether it would be possible to build a graph with those values. The team took the opportunity and presented a graph from Belini (2015) on the digital lab whiteboard with the values found in the divisions, aiming to corroborate the seniors’ argument regarding obtaining the golden ratio (Fig. 11).
The analysis considered this situation to highlight that the seniors had understood the mathematical elements which the team had dealt with in that meeting and could relate them to what was being discussed during the Fibonacci sequence landscape of investigation. This corroborates that conducting mathematical investigations can provide opportunities for the seniors to use and rescue their mathematical knowledge, expanding their capacities and learning further (Lima, 2015; Lima et al., 2019). Another point to reinforce this perception came from one of Glória’s statements: she was pleased to be able to talk with her family about those activities, citing as an example that she had told them that the golden ratio gives us the shape of a credit card. This can be read as an inclusive practice into family conversations for Glória, as she talked about subjects that they did not previously know about and they respected her knowledge, even those whom she considers as “experts in mathematics”.
Seniors have a vast knowledge of the world. During activities developed throughout the Mathematics Conversations Project, they established relationships between mathematical approaches and people’s daily practices and cultures. Thus, we consider that the open characteristic of landscapes of investigation linked to this knowledge of the world can foster the development of important moments of production of mathematical knowledge and different kinds of discussions. For the teacher or team who is in charge of activities, this can give rise to unforeseen moments, referred to in the literature as risk zones, characterised by the unpredictability and teachers losing control of the situation, which is even more likely to happen in a landscape of investigation (Penteado and Skovsmose, 2008; Silva and Penteado, 2013; Skovsmose, 2011; Penteado, 2001). These moments provided opportunities for the seniors and the executive team to discuss and undertake research into the topics. In other words, this research work verified that risks bring possibilities (Penteado and Skovsmose, 2008).
One of the characteristics of a landscape of investigation concerns the exploration of the different paths that may arise during the development of activities. As Skovsmose (2011) points out, the trails of the landscapes of investigation bring different modes of exploration. According to the author, there are moments to proceed slowly and cautiously, and others to throw yourself totally into the situation and see what happens. Those trails can often have an uncertain outcome. In the case of the landscapes of investigation covered in this chapter, there were several discussions—for example, about philosophy, aesthetics, technology and mathematical applicability. We consider these discussions as indicative of a senior’s engagement in a landscape of investigation.
Philosophical discussions, for instance, were moments when the activities provided reflections concerning the very nature of mathematics. For example, during the golden ratio landscape of investigation, Sandra asked “and the inventor of this [golden ratio], did he discover it?”, which raised the question among the seniors whether the golden ratio was a human creation or a discovery. During this discussion, the strand of human discovery prevailed. In addition, this discussion reappeared at the end of the meeting, when images related to the golden mean were displayed, causing the belief among the seniors that “we discover” things done by a creator—in this case, God—to prevail. This is exemplified in Sandra’s speech: “I get angry at those atheists who think that everything came about by chance. How can they not believe in a superior being with such intelligence to create so many things that, for example, have the same proportion?”
Ana initiated an example of an aesthetic discussion. In her words, “this story of the golden ratio representing beauty is silly”. For this senior woman, besides thinking that da Vinci’s Mona Lisa is not beautiful, she did not use the golden ratio in her painting classes and did not think that its use could make the paintings more attractive. Other seniors also commented that the standard of beauty of the time could be quite different from today’s because, if da Vinci made Mona Lisa using the golden ratio, it was probably considered beautiful at the time but, in their view, it would not be so today.
While carrying out the landscapes of investigation, the group discussed the use of technology both specifically and generally. As a specific aspect, the team observed that some of the seniors refused to use a calculator. This happened while calculating the proportions related to the golden ratio. One of the senior women explained that she does not use a calculator in her daily life, whereas another revealed that she wanted to do the mathematics by hand to remember the division’s algorithm. In the Fibonacci sequence landscape of investigation, the ratios between the terms of the Fibonacci sequence required more complex calculations done with pencil and paper only. At this time, the seniors, except for one who refused to use the calculator, eventually discovered how easy it was to use such a device. According to Doll et al. (2016), the refusal to use the new technology, as occurred with the calculator, is an attitude expected from the seniors. They are generally not against the use of technology, but they do offer some resistance against it.
At the end of the meeting about the golden ratio, the team observed more general and critical comments on the use of technology. Selma commented on how technology interferes with the relationships between younger people, because nowadays, they prefer to chat through messaging. She also said that even straightforward attitudes such as saying good morning are undervalued today, showing a change in family upbringing. She noted that, in her teenage years, people needed to rely on their memory much more to store information such as phone numbers and birthdays, and that technology is replacing that today. According to Selma, this is a negative point of technology, as it can impact on the cognitive performance of young people concerning memorisation when they get old. During this activity, Selma also commented on the seniors’ ability to discover and create things without today’s technology resources, considering them more intelligent, and saying that “all those seniors left us as knowledge, we only added to and developed” (Selma).
Mathematics applicability in daily life was also a discussion the seniors brought into the activity. During the golden ratio landscape of investigation, it was necessary to bring more examples related to the determination of proportions between geometric shapes, which had not previously been thought of by the team. Thus, they also worked with the concept of similarity of shapes, which led Selma to ask: “What is the applicability of it? How important is that in life?” The research team answered that the golden ratio is used, for example, in some constructions, in arts and in the development of logos of some brands, because some people consider the golden rectangle a harmonious shape. In this sense, the landscape of investigation created routes for participants to go beyond mathematical concepts. They were able to talk about how mathematics can be seen in everyday life.
These discussions are exemplary in the sense of corroborating the claim of Skovsmose (2011) about the different paths that can arise in an investigative process. The project team took those discussions seriously, trying to incorporate them into activities to offer the seniors a chance to see mathematics differently and become more eager to learn it. They ended up showing interest, as the team could see. In one evaluation meeting, Pedro commented that he found the meeting very interesting, that it was additional knowledge for his life, and that he found it very practical. Sandra commented: “It is additional knowledge […]. I loved talking about constants. […] I’m thinking about enrolling in a mathematics undergraduate program in the next year; I was traumatised with mathematics because I failed the second grade, but I have been reading books since some years ago, and I got involved with mathematics”. Simone, for example, pointed out: “I even learned to understand mathematics in nature. […] Culturally, it was very interesting, wonderful.” Glória said: “I loved it, I learned a lot, but for me, as the oldest participant, it was too much information for a single class”.
4. Final Remarks
In this chapter we discussed the engagement of seniors in landscapes of investigation. This study reveals that the participation of seniors initially occurred after them accepting the invitation to get involved in the activities. It was possible to observe that, in some moments, the seniors explored problems and situations by formulating questions, developing tests and strategies, and defending ideas. The activities were pervaded by questions and discussions, while the seniors were seeking to understand the subjects and the mathematical concepts involved, showing their interest. Another essential aspect of engagement was the interaction amongst them in carrying out the activities and working with the project team’s members.
As evidenced throughout this chapter, the landscapes of investigation built to deal with the golden ratio and Fibonacci sequence opened up possibilities for the seniors in discussing various topics. They produced mathematical knowledge. This activity can be seen as an opportunity to acquire a more positive view of mathematics and to learn new things—an essential aspect of the life of a senior. During the activities, the project team tried to create an environment where the participants felt free to express themselves on any subject, related (or not) to the theme. The results discussed in this chapter are aligned with studies such as Lima (2015), Lima et al. (2019), Grossi (2014) and Scagion (2018), since this type of educational work can foster seniors’ self-esteem and provide opportunities for intergenerational dialogue by sharing their knowledge and experiences. In addition, the attitudes of the participants regarding asking questions, taking notes and carrying out some of the activities in their homes also evidences their commitment and willingness to learn.
References
Alrø, H. & Skovsmose, O. (2003). Dialogue and learning in mathematics education: Intention, reflection, critique. Kluwer Academic Publishers.
Belini, M. M. (2015). A razão áurea e a sequência de Fibonacci. Master’s thesis. Universidade de São Paulo (USP).
Doll, J. (2008). Educação e envelhecimento: fundamentos e perspectivas. A terceira Idade: Estudos sobre Envelhecimento, 19(43), 7–26.
Doll, J., Machado, L. R. & Cachioni, M. (2016). O idoso e as novas tecnologias. In E. V. d. Freitas & L. Py (Eds), Tratado de geriatria e gerontologia (pp. 1654–1663). Guanabara Koogan.
Freire, P. (2016). Pedagogia do oprimido (60th edition). Paz e Terra.
Garcia, V. C., Serres, F. F., Magro, J. Z. & Azevedo, B. d. (sn). O número de ouro como instrumento de aprendizagem significativa no estudo dos números irracionais. Apostila de graduação. Universidade Federal do Rio Grande do Sul.
Grossi, F. C. D. P. (2014). Os diferentes “lugares” que a escola, a leitura, a escrita e a aula de matemática têm na vida dos alunos que estão na terceira idade. Master’s thesis. Universidade Federal de São João del Rey.
IBGE. (2019). Instituto Brasileiro de Geografia e Estatística. Síntese de Indicadores sociais: uma análise das condições de vida da população brasileira―2019 (Vol. 40). Coordenação de População e Indicadores Sociais.
Julio, R. S. & Oliveira, V. C. A. (2018). Estranhamento e descentramento na prática de formação de professores de Matemática. Boletim GEPEM, 72, 112–123.
Julio, R. S. & Silva, G. H. G. d. (2019). Educação Matemática, inclusão social e pessoas idosas: uma análise do projeto conversas matemáticas no âmbito do Programa Universidade Aberta à Pessoa Idosa. Educação Matemática em Revista, 24(64), 52–70.
Law n. 10.741 of October 2003 (2003).
Law n. 13.005 of June 2014 (2014).
Leopoldino, K. S. M. (2016). Sequências de Fibonacci e a Razão Áurea: aplicações no ensino básico. Master’s thesis. Universidade Federal do Rio Grande do Norte.
Lima, L. F. d. (2015). Conversas sobre matemática com pessoas idosas viabilizadas por uma ação de extensão universitária. Doctoral dissertation. Universidade Estadual Paulista (Unesp).
Lima, L. F. d., Penteado, M. G. & Silva, G. H. G. d. (2019). Há sempre o que ensinar, há sempre o que aprender: como e por que educação matemática na terceira idade? Boletim de Educação Matemática (BOLEMA), 33(65), 1331–1356.
Lívio, M. (2008). Razão Áurea: a história de fi, um número surpreendente. Record.
Moscovici, S. (2005). Representações sociais: investigações em psicologia social. Editora Vozes.
Penteado, M. G. (2001). Computer-based learning environments: Risks and uncertainties for teacher. Ways of Knowing Journal, 1(2), 23–35.
Penteado, M. G. & Skovsmose, O. (2008). Riscos trazem possibilidades. In O. Skovsmose (Ed.), Desafios da reflexão em Educação Matemática Crítica (pp. 41–50). Papirus.
Queiroz, R. M. (2007). Razão áurea: a beleza de uma razão surpreendente. In U. E. d. Londrina (Ed.), Professional development program. Universidade Estadual de Londrina (UEL).
Scagion, M. P. (2018). Representações sociais de pessoas idosas sobre matemática. Master’s thesis. Universidade Estadual Paulista (Unesp).
Scortegagna, P. A. (2010). Políticas públicas e a educação para a terceira idade: contornos, controvérsias e possibilidades. Master’s thesis. Universidade Estadual de Ponta Grossa.
Silva, G. H. G. & Penteado, M. G. (2013). Geometria dinâmica na sala de aula: o desenvolvimento do futuro professor de matemática diante da imprevisibilidade. Ciência & Educação, 19(2), 13.
Silva, N. d. (2020). Educação Matemática a partir de um projeto de extensão direcionado a pessoas idosas: Contribuições para a formação inicial de professores de matemática. Master’s thesis. Universidade Federal de Alfenas.
Silva, R. N. d., Silva, G. H. G. d. & Julio, R. S. (2019). Educação matemática e atividades com pessoas idosas. Revista Debate em Educação, 9(1), 560–587.
Skovsmose, O. (2000). Cenários para investigação. Boletim de Educação Matemática (BOLEMA) (14), 66–91.
Skovsmose, O. (2011). An invitation to critical mathematics education. Sense Publishers.
1 This chapter is adapted from Silva et al. (2019).
2 In Brazil, a person considered to be “senior” is over sixty years old.
3 Extension Projects are understood as educational, cultural, and scientific processes that articulate the scientific knowledge with the community outside the university.
4 For example, Silva (2020), and Julio and Silva (2019).
5 See the discussion in Julio and Silva (2019).
6 For the study on the theme and elaboration of the golden ratio scenario, the team used works by Garcia et al. (sn) and Queiroz (2007). For the Fibonacci sequence landscape of investigation, works by Lívio (2008), Belini (2015), and Leopoldino (2016) were used.
7 Based on Queiroz (2007).
8 The elaboration of the text was based on Lívio (2008).
9 The division between the terms was carried out as follows: FnFn-1.