Mathematical problem solving through the lens of ethics and Aristotelian attitude: A case study

Citation: Marbán, J.M.; Fernández-Gago, J. Mathematical Problem Solving through the Lens of Ethics and Aristotelian Attitude: A Case Study. Mathematics 2022,10, 2565. https://doi.org/10.3390/ math10152565 Academic Editor: Jay Jahangiri Received: 19 June 2022 Accepted: 22 July 2022 Published: 23 July 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Mathematical Problem Solving through the Lens of Ethics and Aristotelian Attitude: A Case Study JoséM. Marbán1,* and Joaquín Fernández-Gago 2 1 Department of Didactics of Experimental Sciences, Social Sciences and Mathematics, University of Valladolid, 47011 Valladolid, Spain 2Department of Didactics of Social Sciences, Experimental Sciences and Mathematics, University of Malaga, 29071 Malaga, Spain; [email protected] *Correspondence: [email protected] Abstract: Attitudes are central to the affective domain in mathematics. However, we still have a long way to go to reach a clear consensus on the characterization of such construct and its operationalization, as there are a wide variety of intertwined approaches depending on the theoretical framework that accompanies each of them. In this article, we explore the idea of incorporating ethics as part of a new characterization of attitude in problem solving contexts by means of an instrumental case study consisting of a key informant 10th grade student. Data were collected via classroom observations, questionnaires, semi-structured interviews and written productions. Results were analyzed using the lens of Aristotelian Ethics and the Theory of Creative Intelligence and show how happiness, responsibility and strength are key concepts that help to describe and understand the way the student behaves when solving mathematical problems. We conclude that besides the traditional attention paid to attitudes towards math, it is worth directing our attention towards attitudes influenced by ethics and personal affect as a whole. Keywords: attitude; ethics; case study; mathematics; problem solving MSC: 97C20 1. Introduction There is a general consensus that the mathematical competencies of citizens in certain societies, including Spain, are disturbingly low. This perception, which had previously been brought to light by different local diagnostic tests at the end of the 20th century, has resurged in the last two decades after reports were published by international programs such as TIMSS and, fundamentally, PISA, causing the social and political eye to focus on the search for reasons behind this deficiency in mathematics education. A significant amount of attention has been paid to researching the latent factors behind academic achievement in mathematics education together with research into several possible interrelationships that might influence it. Some authors have focused their attention on analyzing the relationship between attitudes and other affective elements, such as anxiety within mathematics learning and, more specifically, in contexts of mathematical problem solving [ 1 – 7 ]. Despite major advances in our understanding of the interaction between the affective and cognitive domains, conclusive data on causal issues have not been found yet. The present article focuses particularly on the component of the affective domain that has received the most attention in studies related to this field: attitude. This construct is critical to the understanding and interpretation of situations and behavior in different contexts of mathematics education. Attitude is a construct closely related to emotion and belief systems and although the main objective is to analyze attitudes, this article also aims to highlight the relevance of ethics in attitude when solving mathematics problems, shining light on this question and leading the way towards a new characterization of attitude. Mathematics 2022,10, 2565. https://doi.org/10.3390/math10152565 https://www.mdpi.com/journal/mathematics Mathematics 2022,10, 2565 2 of 18 Attitude as a construct has been a focus of research in mathematics education in the last decades, and is generally associated with the affective domain, as defined by [ 1 ]: a construct of an amalgamation of attitudes, beliefs and emotions. Later, Ref. [ 8 ] would include a new component encompassing values, moral judgement, and ethical judgement all as one intervening factor in the decision-making process before problem solving. More specifically, Ref. [ 8 ] (p. 212) explain: “But what has this to do with mathematical learning and problem solving? The importance of the values/morals/ethics component of affect pertains to the individual’s feelings, tacit or overt, about learning, problem solving success, mathematic behavior, etc., as (morally desirable) virtues or values”. Using a structure in which the attitude construct does not explicitly appear, Ref. [ 9 ] (p. 137) relate the above-mentioned factor with meta-affect when affirming “Beliefs, values, and ethics also play a role in meta-affect, as they influence how an emotion functions ecologically in the individual’s personality—i.e., the purpose served by the emotional feeling”. In this structure, there is a component called mathematical integrity, which refers to the postures related to when a mathematical problem is considered “right,” when the learner’s understanding suffices, when a problem solution is considered satisfactory or when solving a mathematics problem is considered to be worth it. These studies do not mention how different ways of conceiving ethics can influence behavior of the problem solver. Within the affective domain, attitudes play a large part in mathematics education and have generated a considerable amount of research. Ref. [ 10 ] point out that, in the world of mathematics, attitudes have been less clearly defined than in the field of psychology, where attitudes are thought of as a predisposition with a high emotional load that influences conduct; in this way, the influence attitudes have on the teaching–learning process and mathematics academic achievement is well-known in Refs. [ 11 , 12 ], along with the influence attitudes have on anxiety in mathematics [ 13 – 17 ]. Studies of this have found that students with better attitudes towards mathematics tend to have better perceptions of the usefulness of math, higher intrinsic motivation to study, better mathematic self-concept [ 18 ], higher confidence in mathematics learning [ 19 ] and, above all, they show more willingness to approach the subject [20,21]. According to [ 3 , 22 , 23 ], one can see the lack of consensus on attitude among researchers, as there are many theoretical frameworks to work from. At the same time, it is important to note the disparity between instruments used in the mediation and study of this area. In any case, most works generally observe certain aspects, tendencies and ideas that merit mentioning. One of them is the search for relationships and associations between attitude and academic achievement. The second aspect is relative to the use of dichotomic classifications of positive and negative attitudes towards mathematics. Finally, one aspect integral to the studies mentioned is the establishment of relationships and interrelationships between the constructs of attitudes, beliefs and emotions, constructs that have yet to be made “universal” not only in the context of mathematics education but also in many others, for instance, science education [ 24 ]. In this last case, the TMA model by [ 22 , 25 ], a three-part, three-dimensional model composed of emotional disposition towards math (emotion), vision of math (beliefs) and perceived competence in math (self-perception), has become an important reference to look at. As such, research on attitudes in mathematics could be divided into positivists, works that focus on looking for the relationship between attitudes and achievements, and interprets, in other words, those studies focused on the understanding of the phenomenon more than on quantitative links between attitudes and achievements. In the second group, it is important to highlight the work by [ 1 ], who reconceptualized the attitude construct by proposing a structure in which beliefs are elements that cause emotion to come up. Moreover, in this structure, attitudes are built from repeated emotional reactions, which establishes that both social context and individual experience contribute to the origin of beliefs. McLeod’s work has inspired subsequential studies on beliefs, emotions, and attitudes to have two components: one that refers to cognition and the other to stability. Mathematics 2022,10, 2565 3 of 18 Beliefs are considered to be more stable and cognitive while emotions are less stable and less cognitive. Refs. [19,26] for support, arguing that beliefs are enormously influential on the surge of emotions and that repeated emotional reactions are the origins of attitudes. Lastly, it is important to note that for McLeod, the social (cultural) context and individual experiences are a large contributing factor to the creation of beliefs. This model, despite having a large theoretical base, has received some criticism stating that the concept of attitude is a blind spot, seen as a mix of cognitive (beliefs), affective (emotions) and conative (behavior) aspects. In response, attitude is beginning to be linked to the observer and not the participant. In social psychology, on the other hand, recent theories present consensus on the multidimensionality of the attitude construct understood as a tripartite model with three components: affective, cognitive and behavioral. If this model is applied to mathematics education, attitude can, therefore, be characterized into three components: emotional disposition towards maths, beliefs about maths and behavior related to maths. The previous model tends to fit attitudes into only one general emotional disposition. Nonetheless, one limitation to the threefold notion of attitude as a combination of beliefs, emotions and behavior, is the circular reasoning pointed out by [ 27 ] (p. 346) when affirming that “we accept that beliefs influence thoughts and actions, and sometimes hidden beliefs, which operate subconsciously, are detected through how people act or think”. There are also limitations to the notion of attitude as emotional disposition, despite its advantages, as it does not include cognitive aspects. Positive attitude can be defined in terms of positive emotions, for example. Being able to define attitudes as positive or negative is critical to the development of this construct and the framework of this study. A third definition that avoids circular reasoning has been proposed by [ 28 ]. According to them, attitudes towards maths can be described as a model of beliefs and emotions associated with maths. Other authors such as [ 29 ] differentiate between attitude towards mathematics, with a clear affective component and mathematical attitudes, with a clear cognitive component. Ref. [ 30 ] establishes various types of attitudes, such as appropriate initial attitude and the enjoyment of a challenge, and two purely mathematical attitudes: the inductive attitude and the precision and rigor attitude. Lastly, some authors such as [22] contemplate the idea of attitude in association with the attitude of some teachers, such as complaining in response to the uncontrollable errors of their students. As one can see, there is a clear lack of consensus among authors on the construct of attitude, which must not be interpreted as a weakness or limitation but rather a natural manifestation of the complexity of the affective domain itself in mathematics. This situation calls for a clear conceptualization of the components and constructs referenced by every author in their works to avoid possible adverse effects for the lack of consensus. Consequently, the present authors wish to precisely describe the starting point of this study, based on the synthesis given insofar in which the main focal points influencing the theoretical framework of this paper have been stated along with those that have influenced the configuration of the affective domain components are taken into consideration here. Thus, concerning the relationship in the first place that the attitude construct has with emotions and beliefs, we maintain that there are self-perception beliefs and beliefs about mathematics, as proposed by [ 22 , 25 ], that influence the behavior of students. Because the tripartite model is in use, one must implicitly accept that when the student’s attitude is being studied, one must also look into their emotional disposition and beliefs. Moreover, in the present study, one must consider that students’ own beliefs with respect to values/morals/ethics can explain their actions. In the second place, this study steps away from qualifying attitudes as positive or negative, because, as [ 31 ] proports, one can have a positive or negative attitude relative to the point of view from which the task is evaluated, and both are driven by the person’s values. In the third place, in line with [ 30 ], the present study marks two different aspects of attitude. The first is psychological, which is related to the enjoyment of a challenge, and Mathematics 2022,10, 2565 4 of 18 the other is related to mathematics in practice. In the fourth place, this characterization is based on the voluntary disposition of the individual, which sometimes goes unnoticed in other characterizations. Lastly, comparable to [ 32 ], this paper extends the characterization of attitude to attitude towards mathematics problems. Finally, we think it is worth mentioning [ 33 ] although it is not concerned with attitudes as it has somehow influenced the authors when facing this study. In sum, this study takes into account all the critical aspects as mentioned and commented on in this section. Using a particular theoretical framework which includes the attitude construct, the behavior of one student playing the role of an instrumental case study is analyzed through interpretative lens, with comments and a complimentary analysis supported by the previously mentioned theoretical framework and taking into account the behavior of students in other studies, such as those of [ 32 , 33 ], when faced with mathematics problems. As such, the case is thought as a contribution to deepen the topic of attitude and to provide useful information to refine already existing theories by means of conclusions that might lead to starting points for the statement of certain hypothesis or to the design of new case studies. Hence, the case plays a secondary and supporting role in arriving at the formulation of statements about the object of study and, in particular, to provide preliminary tentative answers to the following research questions: is it feasible to create/extend existing characterizations of attitude by including Aristotelian Ethics; and, subsequently, is it relevant to use this characterization to analyze how some students perform in mathematical problem solving? Next, the theoretical framework of this study is presented, followed by the methodological design and the presentation and discussion of the results derived mainly from interviews and writings associated with mathematical problem solving. The paper ends with conclusions and includes some comments on the limitations of the study as well as possible future lines of research. 2. Theoretical Framework Aristotelian Ethics and the Theory of Creative Intelligence [ 34 ] together make up the bases of the theoretical framework of the present study. Hence, now we proceed to first explaining Aristotelian Ethics for later evaluating said ethics through Marina’s work. It is important to begin with the definition of ethics and morals used in this article. Here, morals are understood as customs or norms that a society accepts and follows and that are associated with certain values. On the other hand, the word “ethics” is understood in the way that [ 35 ] (p. 1058) explained “ . . . ethical historians have limited their studies to those ideas of moral character with a philosophical basis, that is to say, that instead of being supposed, ethics are examined to their core; in other words, they are philosophically justified”. Therefore, ethics includes the principles we base ourselves on and that determine our actions within what is moral. As the participating student rationally reflected on if her actions to resolve the problems were linked to being happier or not, the present study is on the level of ethics. One question that comes up at this point is why one must consider values/morals/ethics when understanding the actions taken to resolve a problem. One possible response has been provided earlier through our reference to [ 8 ] and will also be detailed here. Other authors who have considered ethics in mathematics education are [ 36 – 38 ]. We could also include here [ 39 ], even though he does not speak explicitly about ethics in his decisionmaking theory, as the principles exposed are part of the directions that a teacher can follow, which are deeply related to ethics. So, there are two approaches to dealing with ethics in mathematics education: on one side, emphasis on the teacher’s decisions and, on the other side, emphasis on the decisions that students make. Our work is of the second type. Ref. [ 9 ] establishes relationships between the set conformed by beliefs, values and ethics, on the one side, and individual meta-affect, on the other. Unlike [ 9 ], the authors of the present work consider that taking ethics/moral/values into account does not constitute by itself having meta-affective awareness. Moreover, certain convictions about this Mathematics 2022,10, 2565 5 of 18 ethics/morals/values “mixed bag,” some of which are potentially beliefs, might model and influence in behavior when solving problems and thus involve meta-affectivity as well as how they can be a directing force behind meta-affectivity. Under these premises, since ethics is the most inclusive construct of the above-mentioned “mixed bag”, from now on we will refer only to ethics. Aristotle begins Nichomachean Ethics [ 40 ] writing about happiness and rationalizing the existence of a supreme being, a being that is eudaimonia or happiness. The problem is in what one considers to be happiness. For Aristotle, there are three different types of lives: the voluptuous life, the political life and the theoretical life. So, those who choose the first orbit identify happiness with pleasure. For those who choose, as Aristotle puts it, the voluptuous life, any other type of satisfaction would be ephemeral. The second type of life as categorized by Aristotle is one that is political, with happiness rooted in honor. Aristotle treats this one with more respect because recognition of another man’s honor and deserving honor was morally relevant yet had degraded to confusion with fame. While fame comes from a mistaken evaluation or ridicule, the essence of honor comes from a moral evaluation. Even so, Aristotle did not consider honor but rather the power to reason as the maximum level of life to be reached. Although Aristotelian notions included in the present study are described below, it is important to add that the present authors have not carried out exhaustive research into all the notions contemplated by Aristotle. In Book II, Aristotle speaks of moral virtue as a state of rational choice when deciding the best actions to take that balance the relationship between pleasure and pain. Aristotle affirms: “Hence the importance, as Plato points out, of having been definitely trained from childhood to like and dislike the proper things; this is what good education means” (p. 93). In this way, if the student manages to confront a problem despite the discomfort of not understanding it or not knowing how to start, then he/she either rejects the temptation to give up or the student accepts that learning can be beneficial, all despite negative emotions that can come up in the process, presenting coherency with Aristotle’s ethics. In Book III, the definitions of voluntarily and involuntarily appear. The involuntary act is due to a first principle extrinsic to man, such as force or ignorance, leaving the rational choice more restricted than the voluntary act in the sense that the act is within our power and is fruit of deliberation. The idea of attitude when confronted with a mathematics problem is within rational choice and, as we will see, is a choice of events before a challenge. Book III focuses on courage as the term between fear and confidence and it affirms, as well, that the brave are action-takers. This could be the same in the case of some students who, despite feeling fear when they cannot resolve a problem, decide to act. Apart from the authors who have studied the construct of attitude in Mathematics Education, some of whom have already been mentioned previously, the present study is fundamentally based on the works of Marina [ 34 , 41 ]. This author affirms that there is an intelligence responsible for receiving information, elaborating upon it, and producing responses, and it is called computational intelligence. There is another intelligence, executive intelligence, in charge of creating ideas based on the information produced by computational intelligence. By Marina’s definition, a person can access, control, and direct their mental activity. Moreover, any intellectual task requires the same mental activities according for this author, from writing a poem to solving a mathematics problem. The way of doing so is what will determine if the person is more intelligent or less intelligent. Thus, for the author, intelligence does not exist as an independent ability. One cannot perceive, remember, imagine, compare, conceptualize, decide and, apart from that, be intelligent. One must use an adjective to indicate that intelligence has to do with the way of using one’s faculties, for example: there is an intelligent way of watching, intelligent memory, an intelligent way of imagining, intelligent motivation, intelligent language and so on with other aspects. To use intelligence, one must invent projects. The present authors will focus on the project, invented or not, by our student to solve mathematics problems and investigate whether that project serves a greater project. Mathematics 2022,10, 2565 6 of 18 Ref. [ 34 ] (p. 184) considers that the voluntary construct is not in opposition to motivation, and affirms “Because of this, it is important to highlight that drive is the ability to give oneself orders and obeying meditated values and not only feelings.”. How the student in this study talks to herself in her project will be present in the analysis. Ref. [ 34 ] (p. 140) establishes failures in intelligence and how they can be tied to beliefs, specifically beliefs related to emotions or feelings, affirming that “Under apparently spontaneous and original feelings there are fundamentally implicit beliefs taking place.”. For [ 41 ] (p. 163) attitudes are more mediated by intelligence than emotions. In the tension that exists between ideas/occurrences being produced by computational intelligence and then becoming actions, one can try to control the intelligence where such ideas/occurrences are produced. Thus, Marina characterizes attitude in the following way: It has to do with the voluntary disposition of intelligence, which can access, control, and manage mental operations to achieve a determined style of occurrences. This differs from character, which is also a fixed style of occurrences, because it is voluntary. This differs from habits because it is not automatic. The importance of attitude within vital economy is derived from the fact that it is the point where intelligence negotiates with feelings. If a person begins to feel emotion when presented with a problem, tension or conflict between the desire to feel good and the desire to resolve the problem is created, which can create contradictory ideas such as either to abandon the situation, try to manage emotions and resolve the problem or even justify whether carrying out the problem, such as a challenge, makes sense or not. From the TMA point of view, the tripartite system used by the present authors would describe the actions taken, the attitude component, in terms of beliefs and emotions, while, for Marina, part of this description would also include the voluntary decision to manage these tensions and contradictions. Furthermore, passive attitudes could be present when there is certain stimulation and we decide that executive functions do not go beyond setting a goal of non-action, and active attitude when one decides to act after being confronted with certain information. An attitude of negotiation could also be present when faced with contradictory occurrences and taking sides. A flexible attitude could be present if one manages different plans and a defensive attitude when one tries to support their argument. An Aristotelian attitude is one in which a person, when confronted with difficulties, adopts an attitude based on happiness, moral virtue and strength in the Aristotelian sense. Attitude can be derived, or not, from mathematics attitudes (see Figure 1) such as inductive attitude or attitude of precision and rigor, as [30] proposes. Mathematics 2022, 10, x FOR PEER REVIEW 6 of 18 one’s faculties, for example: there is an intelligent way of watching, intelligent memory, an intelligent way of imagining, intelligent motivation, intelligent language and so on with other aspects. To use intelligence, one must invent projects. The present authors will focus on the project, invented or not, by our student to solve mathematics problems and investigate whether that project serves a greater project. Ref. [34] (p. 184) considers that the voluntary construct is not in opposition to motivation, and affirms “Because of this, it is important to highlight that drive is the ability to give oneself orders and obeying meditated values and not only feelings.”. How the student in this study talks to herself in her project will be present in the analysis. Ref. [34] (p. 140) establishes failures in intelligence and how they can be tied to beliefs, specifically beliefs related to emotions or feelings, affirming that “Under apparently spontaneous and original feelings there are fundamentally implicit beliefs taking place.”. For [41] (p. 163) attitudes are more mediated by intelligence than emotions. In the tension that exists between ideas/occurrences being produced by computational intelligence and then becoming actions, one can try to control the intelligence where such ideas/occurrences are produced. Thus, Marina characterizes attitude in the following way: It has to do with the voluntary disposition of intelligence, which can access, control, and manage mental operations to achieve a determined style of occurrences. This differs from character, which is also a fixed style of occurrences, because it is voluntary. This differs from habits because it is not automatic. The importance of attitude within vital economy is derived from the fact that it is the point where intelligence negotiates with feelings. If a person begins to feel emotion when presented with a problem, tension or conflict between the desire to feel good and the desire to resolve the problem is created, which can create contradictory ideas such as either to abandon the situation, try to manage emotions and resolve the problem or even justify whether carrying out the problem, such as a challenge, makes sense or not. From the TMA point of view, the tripartite system used by the present authors would describe the actions taken, the attitude component, in terms of beliefs and emotions, while, for Marina, part of this description would also include the voluntary decision to manage these tensions and contradictions. Furthermore, passive attitudes could be present when there is certain stimulation and we decide that executive functions do not go beyond setting a goal of non-action, and active attitude when one decides to act after being confronted with certain information. An attitude of negotiation could also be present when faced with contradictory occurrences and taking sides. A flexible attitude could be present if one manages different plans and a defensive attitude when one tries to support their argument. An Aristotelian attitude is one in which a person, when confronted with difficulties, adopts an attitude based on happiness, moral virtue and strength in the Aristotelian sense. Attitude can be derived, or not, from mathematics attitudes (see Figure 1) such as inductive attitude or attitude of precision and rigor, as [30] proposes. ATITUDE MATHEMATICS ATTITUDE INDUCTIVE OF RIGOUR INFLUENCED BY ETHICS ARISTOTELIC PASIVE FLEXIBLE Figure 1. Attitude influenced by ethics and mathematics attitude; designed by the authors. The theoretical framework of the present study, as presented in Ref. [ 42 ] includes computational and executive intelligence, as established by Marina (Figure 2). However, executive intelligence is defined here by the functions proposed by [ 43 ]: goal making, planning, carrying out a goal-oriented plan and the effective execution of such. For some Mathematics 2022,10, 2565 7 of 18 authors, attitude has three dimensions: beliefs, emotions and behavior, which the present model takes into account while not being characterized in its entirety by them. Mathematics 2022, 10, x FOR PEER REVIEW 7 of 18 Figure 1. Attitude influenced by ethics and mathematics attitude; designed by the authors. The theoretical framework of the present study, as presented in Ref. [42] includes computational and executive intelligence, as established by Marina (Figure 2). However, executive intelligence is defined here by the functions proposed by [43]: goal making, planning, carrying out a goal-oriented plan and the effective execution of such. For some authors, attitude has three dimensions: beliefs, emotions and behavior, which the present model takes into account while not being characterized in its entirety by them. Figure 2. Attitude is a voluntary disposition of intelligence; designed by the authors. The term “beliefs” refers to judgements, both conscious and unconscious, that we conform with because we are pledged to them. Beliefs act in groups or conglomerates and make up a system, as [44] establish. Beliefs that influence behavior when solving mathematics problems are, for example, beliefs about mathematics teaching, mathematics learning, about maths in general, and beliefs about the social context. Feelings are integrated in blocks of information that include judgements by the subject involved in the situation and the predisposition of the subject to act. Emotions are transient feelings that appear abruptly with physical, conscious manifestations. As the balance of the situation is on a continuum, emotions can change quickly and tend to be connected in groups, showing shifts of emotions within the same group or across groups. For [45] (p. 47), one feeling can become another: “feelings can give way to other feelings”. In this study, the Aristotelian ideas of ethics has been filtered using Marina’s ideas on ethics. In Ref. [34] (p. 17): “human intelligence is like an intelligence transfigured for freedom”, and in that freedom a human being can “invent” two projects, one being happiness and the other perfection. From the Aristotelian Ethics point of view, part of being human, and thus part of what goes into being happy, is reasoning. Ref. [34] (p. 62) says: “what is good for man is an active life of a being who can reason and, on one hand, obey that reasoning, and on the other had own it and think it”. Here is where Marina expands upon the notion of happiness to include the notion of intelligence. Referring to the idea above, [34] (p. 162) reaffirms: “What this means is that, in reality, happiness consists of living intelligently. And as man has the intelligence to create, it all comes down to living creatively…”. This idea INTELLIGENCE computational attitude executive can be seen as Figure 2. Attitude is a voluntary disposition of intelligence; designed by the authors. The term “beliefs” refers to judgements, both conscious and unconscious, that we conform with because we are pledged to them. Beliefs act in groups or conglomerates and make up a system, as [ 44 ] establish. Beliefs that influence behavior when solving mathematics problems are, for example, beliefs about mathematics teaching, mathematics learning, about maths in general, and beliefs about the social context. Feelings are integrated in blocks of information that include judgements by the subject involved in the situation and the predisposition of the subject to act. Emotions are transient feelings that appear abruptly with physical, conscious manifestations. As the balance of the situation is on a continuum, emotions can change quickly and tend to be connected in groups, showing shifts of emotions within the same group or across groups. For [ 45 ] (p. 47), one feeling can become another: “feelings can give way to other feelings”. In this study, the Aristotelian ideas of ethics has been filtered using Marina’s ideas on ethics. In Ref. [ 34 ] (p. 17): “human intelligence is like an intelligence transfigured for freedom”, and in that freedom a human being can “invent” two projects, one being happiness and the other perfection. From the Aristotelian Ethics point of view, part of being human, and thus part of what goes into being happy, is reasoning. Ref. [ 34 ] (p. 62) says: “what is good for man is an active life of a being who can reason and, on one hand, obey that reasoning, and on the other had own it and think it”. Here is where Marina expands upon the notion of happiness to include the notion of intelligence. Referring to the idea above, [ 34 ] (p. 162) reaffirms: “What this means is that, in reality, happiness consists of living intelligently. And as man has the intelligence to create, it all comes down to living creatively . . . ”. This idea of happiness applies to the present study in the following questions: is solving these problems necessary to the progression of my academic life; will I be happier solving this problem or unhappy if I try and fail; and, if I am happier, what does that happiness mean? Mathematics education has indeed dealt somehow and sometimes with difficulties in a sense that somehow connects with the notion of happiness [32]. We try now to clarify how the group of notions appearing here make up the present study overall: Mathematics 2022,10, 2565 8 of 18 • Attitude, as defined by both Aristotle and Marina, involves voluntary acts. Marina, moreover, affirms that it is a choice between those presented by computational intelligence, and it is carried out by executive intelligence. When confronting a difficulty, different attitudes come into play such as defensive attitudes, passive attitudes, and Aristotelian attitudes. • Ethics, both for Aristotle and Marina, are closely connected to happiness. The difference between them is that, for Aristotle, reasoning is an innate human ability and thus necessary for happiness to exist, while for Marina it is “living creatively”. • Attitudes can be affected by beliefs and emotions. The present study considers beliefs related to “living creatively”. 3. Research Methodology As mentioned in the Introduction, the main purpose behind this article is to illustrate through a direct and particular approximation to the reality of a student facing mathematical problems how a new characterization of the attitude construct which includes an ethical component, a characterization that makes up our theoretical framework, can improve the understanding and interpretation of realities such as the one presented. We are interested in the depths of the human being that can be unique and individual, not in those that represent a pattern that can be adjusted to an aesthetic of a stable reality of which we cannot even be sure of its existence. Therefore, this research focuses on a case study in the sense given by [ 46 ] for whom the wholistic, empirical, interpretive and empathetic character of a qualitative paradigm on the one hand, along with the uniqueness of the selected case study itself, allow for a deep understanding of an activity in concrete circumstances as well as a close relationship between researchers and the object of study, making the methodological procedure proposed here one of the most adequate for the purpose of this research. Thus, there is no need for any kind of statistical hypothesis. At the same time, in line with the classification given by [ 47 ], the type of case study that best fits with the purposes of this research is an instrumental case study. In this sense, [ 48 ] says that the studied reality will therefore focus on one case, justifying the usefulness of such according to the following intentions: widen the knowledge base, recognize the unique character of the context, and make a contribution with it since similar studies have yet to be carried out. 3.1. Participants The case under study consisted of a single participant, who we will now refer to as Claudia, a fictitious name to protect her anonymity. Claudia was a student from 4th ESO (10th grade in high school) and she was selected based on her qualities as a key informant, shown in previous years, in addition to the fact that the year before Claudia had a teacher who used problem solving as a teaching method and as part of the content as well and because in a previous questionnaire filled out by 60 students from 4th ESO (10th grade in high school), she showed her closeness to attitudes related to Aristotelian Ethics. An important point about Claudia is that in the beginning of 3rd ESO, she presented resistance to problem solving as a teaching model, that she managed to overcome, in such a way that she even obtained good marks in mathematics. In [ 42 ] the authors investigate a case study whose data could be labeled as attitude far from Aristotelian Ethics, being this work useful for our purpose in this study. The methodological focus of this study was carried out in the following way: at the start of a teaching unit, a problem was presented which could be solved using heuristics and without being familiar with the specific knowledge contained in that unit. Then such knowledge was taught to solve a similar problem to the first in the end. The 2nd ESO (8th grade) teacher was actually in favor of a much more traditional, fundamentally expositive, teaching method and homework that required low cognitive demand, which, all in all, provided interesting experiential baggage for the purposes of this research. Mathematics 2022,10, 2565 9 of 18 3.2. Data Collection Instruments The instruments used for data collection were questionnaires, interviews and observation of the participant and a written production of three problems. The questionnaire was designed first. In it there were mostly multiple-choice questions as well as questions on a Likert-type scale. To elaborate on the questionnaire that Claudia would fill out in the end, a pilot experiment with an initial design was first carried out on a group of students from 1st Bachillerato (11th grade) social science. After, the results of the experience were debated and analyzed by a discussion group made up of experts in mathematics education at the University of Huelva. These experts agreed that there was a need to include a category about attitudes and values, together with another two categories called sense of the activities and about evaluation. This group suggested that the following dimensions should be considered as part of the category about attitudes and values: •Sense/feeling of challenge; •Sense of mistake; •Sense of management in the long term; •Sense of doubt; •Sense of ethics in actions. Here, one can see three questions chosen from that questionnaire as to provide a better understanding of this instrument and its purpose: B.10. In class the teacher has brought in an activity that seemed very interesting to me but when I had to respond to what it asked me, I had a lot of questions. Mark below what you think about this class: •I would rather not have the class because it causes stress and insecurity. •It is worth it because it is more interesting and motivating. •It is worth it because if I have questions that means I can learn. •If I doubt then the teacher is doing something wrong. In this question, participants are asked to write 1, 2, 3 or 4 next to each answer option with 1 meaning they slightly agree, 4 meaning that they strongly agree and 2 and 3 being intermediate states. This question would provide information on the dimension sense of challenge, among others, within the category of the feeling of attitudes and values. If a 4 is put on the second or third option, then we would consider them closer to Aristotelian Ethics and manages difficulties and emotions such as doubt. B.12. During the first trimester, the teacher has worked on curious, interesting problems with which he/she challenges the students. In those challenges, the student must make conjectures/hypotheses about what helps them solve the problems. What would you say to other classmates to convince them to work in this way? •That they should try the conjectures, so they could solve the problems little by little. • That it is a better way, because you learn that solving problems happens in phases, steps, and that not everything is about simply knowing or not. •I prefer to repeat something that the teacher has already completed. This question would provide information on the sense of challenge dimension, among others, found within the category of feeling of attitudes and values. The first and second options are closer to Aristotelian Ethics. C.1. If you happen to make a mistake on a problem, what goes through your mind? What do you do? • I feel frustrated, but I go back and look for another way and keep the parts that are correct. •I feel frustrated, but I go back and look for another way. • I get overwhelmed and go in circles looking for ideas but don’t come up with anything. •I skip it. Mathematics 2022,10, 2565 16 of 18 Data Availability Statement: The data that support the findings of this study are available on request from author E.R. The data are not publicly available in order to respect the privacy of research participants. Acknowledgments: The authors would like to express their sincere gratitude to JoséCarrillo Yáñez. sadly deceased, since his involvement with his creative intelligence, which includes a good dose of generosity, was essential for us to write down this paper. Conflicts of Interest: The authors declare no conflict of interest. Appendix A THE MAGIC FLUTE Gumersindo Pelaez has a flute measuring 26.8 cm, which he wants to store in a shoebox with dimensions 10, 15 and 20 cm. He says it will fit, while his friend Antoñito says it will not. Which of the two is right? Use a formula to generalize the previous result. SQUARES AND MATCHES II How many matches are needed to build n 2 squares of side 1 forming another larger square, as in the following sequence? Mathematics 2022, 10, x FOR PEER REVIEW 16 of 18 Funding: This research received no external funding. Institutional Review Board Statement: The study was conducted according to the guidelines of the Declaration of Helsinki and the University of Málaga ethical standards for research. Informed Consent Statement: Informed consent was obtained from all subjects involved in the study. Data Availability Statement: The data that support the findings of this study are available on request from author E.R. The data are not publicly available in order to respect the privacy of research participants. Acknowledgments: The authors would like to express their sincere gratitude to José Carrillo Yáñez. sadly deceased, since his involvement with his creative intelligence, which includes a good dose of generosity, was essential for us to write down this paper. Conflicts of Interest: The authors declare no conflict of interest. Appendix A THE MAGIC FLUTE Gumersindo Pelaez has a flute measuring 26.8 cm, which he wants to store in a shoebox with dimensions 10, 15 and 20 cm. He says it will fit, while his friend Antoñito says it will not. Which of the two is right? Use a formula to generalize the previous result. SQUARES AND MATCHES II How many matches are needed to build n2 squares of side 1 forming another larger square, as in the following sequence? Figure A1. First two cases for the sequence of squares built with matches. Source: authors. TANGENT OF THE COMPLEMENTARY The two angels of a right triangle are called complementary because their sum is 90 degrees. 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