1 INFINITE AND UNBOUNDED SETS: A PRAGMATIC PERSPECTIVE Cristina Bardelle University of Eastern Piedmont A. Avogadro, Alessandria, Italy This paper explores some of the ambiguities inherent in the notions of finite/infinite sets and bounded/unbounded sets for what concern Euclidean spaces. The study, carried out with seven mathematics students, shows that wrong attitude towards mathematical language is a major source of misconceptions about cardinality and boundedness. In particular, everyday use of mathematical terms combined with a lack of coordination of different representations proves to be a hindrance in the conceptualization of basic properties of subsets of metric spaces. INTRODUCTION In the last years research in mathematics education has pointed out the central role played by language in the learning of mathematics and has addressed this topic from a variety of different perspectives. The complexity of mathematical language does not refer to the symbolic component only, but it depends on all semiotic representations used in mathematics such as symbolic notations, diagrams, figures and so on, including the verbal component. From the point of view of semiotics many studies stressed the importance of treatment within the same semiotic system and of the translation between different representations of mathematical objects as well (Janvier et al., 1987; Duval, 1993). From the functional linguistic viewpoint as many studies developed to highlight the role of context in the learning of mathematics. Some constructs and ideas from pragmatics were exploited by authors (see e.g. Ferrari, 001, 004; Morgan, 1998; Pimm, 1987) in order to provide theoretical frameworks to interpret difficulties in the learning of mathematics. For example Ferrari (004, p. 384) claimed that students competence in ordinary language and in the specific languages used in mathematics are other sources of troubles. In particular some studies pointed out difficulties arising from the overlapping of everyday language and mathematical language (see e.g. Bardelle, 010; Cornu, 1981; Ferrari, 004; Kim et al., 005; Mason & Pimm, 1984; Tall 1977). But not enough attention has been paid to the fact that mathematical terms are often borrowed from everyday language and used with meanings different from everyday-life usage. The aim of this paper is to further investigate this subject. In particular here we deal with the concept of boundedness and infiniteness of subsets of Euclidean spaces. The study of metric spaces and their properties is basic in the curricula of mathematics undergraduate students and the
2 concept of boundedness is fundamental for the learning of other topological properties such as compactness. Research questions The presented research investigates some typical students behaviours concerning the recognition of bounded or unbounded subsets of n, endowed with the Euclidean distance. For example, some students seemingly justify that a set is unbounded because it has infinite elements. In particular the study explores the interplay of pragmatic aspects with the flexibility in switching between different representations of bounded/unbounded sets and finite/infinite sets. Do the everyday meanings of the words finite/infinite and bounded/unbounded influence students behaviour in the resolution of problems involving such words? Which is the role of the formal definition of bounded/unbounded set and finite/infinite set? What are the elements (problem formulation, system of representation of the set, context, etc.) that could influence the meaning of bounded/unbounded set and finite/infinite set used by students? THEORETICAL FRAMEWORK Functional approach Functional linguistics is a part of linguistics that studies language in relation to its functions rather than to its form, and pragmatics is a general functional (i.e. cognitive, social and cultural) perspective on language and language use, aimed at the investigation of processes of dynamic and negotiated meaning generation in interaction. Here I adopt Halliday s definition of register (Halliday 1985), which has been thoroughly discussed by Leckie-Tarry (1995). A register denotes a linguistic variety based on use that is a conventional pattern or configuration of language that corresponds to a variety of situations or contexts. Ferrari (004), following Leckie- Tarry (1995), distinguishes between colloquial registers and literate registers. The first ones refer to linguistic resources adopted in spoken communications prevalently but also in informal written communication such as sms messages, s, etc. whereas the second ones refer to written-for-others texts mainly such as books but also to formal spoken communication such as in academic lessons. Ferrari (004, p.387) argues that the registers customarily adopted in advanced mathematics share a number of features with literate registers and may be regarded as extreme forms of them.
3 Semiotic representations in the concept image and concept definition As mentioned before, registers used in mathematics are highly literate; in fact, the development of mathematics but also its teaching and learning requires the introduction of a variety of representations such as symbolic notations, verbal texts, geometrical figures, diagrams and so on. According to Duval (1993) there cannot be noésis without sémiosis, where sémiosis denotes the production of a semiotic representation and noésis denotes the conceptual learning of an object. Moreover Duval states that the cognitive functioning of human thought needs multiple semiotic systems and he applied his ideas to the learning of mathematics where very different representations occur. Duval makes a distinction between treatment of a representation, which is a transformation (manipulation) within the same semiotic system, and conversion (translation) between different semiotic systems 1. For example, computing the sum of two fractions is an example of treatment, whereas translating a fraction into an equivalent decimal expansion is an example of conversion. A good coordination of semiotic representations is, in my opinion, fundamental for the development of a proper concept image (Tall & Vinner, 1981). This term is used to describe the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes (Tall & Vinner, 1981, p. 15). The generation of the concept image of an individual can be influenced also by the concept definition that is a form of words used to specify that concept (Tall & Vinner, 1981, p. 15). The concept definition can be different from the formal concept definition, i.e. a definition accepted by the mathematical community at large. Moreover Tall and Vinner introduced the evoked concept image as the portion of the concept image which is activated at a particular time. The concept image may have conflicting aspects that may be evoked at different times. When such conflicting aspects are evoked simultaneously they cause a cognitive conflict. My claim is that the ability in the use of mathematical language in particular for what concerns the understanding of its purposes and a good coordination of semiotic representations is fundamental for a proper coherent generation of the concept image without conflicting factors. THE EXPERIMENT The research involved seven second year undergraduate mathematics students at the University of Eastern Piedmont in Italy. The students were attending a Geometry course focused on point set topology and on introductory algebraic topology. Students already encountered the concept of finite/infinite set and of 1 Duval used the term register referring to a semiotic system; here register is used with its pragmatic meaning only.
4 bounded/unbounded set in n, endowed with the Euclidean metric, during the first year. In this second year course the definition of finite/infinite set is considered well known, whereas the definition of boundedness of a set is generalized to an arbitrary metric space, paying particular attention for n with Euclidean metric. This research focused on Euclidean spaces only. The data were collected from individual interviews. The interviews were semi-structured and based on two different kinds of questions. Q1-questions n The first kind of questions were aimed to recognize if a given set in with Euclidean metric is unbounded and infinite. The sets were given by their graphical representation or by symbolic notations and in this case, whenever it was useful and possible, students were encouraged to represent it in the Cartesian coordinate system. Some examples of given sets are 1 nn, 0, ( x, y) : xy 1, 3 ( x, y, z) : x y 1, z 3, etc. Q-questions ( x, y) : x y 1, ( x, y) : y x, ( x, y) : y sin x, 3 ( x, y, z) : x y 3, The second kind of questions were aimed to understand the meaning of finite/infinite set and bounded/unbounded set used to answer to Q1-questions. Students were asked to say what is an infinite and unbounded set or to produce some examples. The interviews were carried out in order to help students to achieve a proper knowledge about the topics as well as to identify their behaviours and their concept image. In particular, Q-questions investigated their concept definition and Q1- questions were aimed to study the influence of the symbolic and graphical representation on their concept image. SELECTED FINDINGS Just two students (students A, D) out of seven showed no kind of problems dealing with the topic of this study. The remaining students presented difficulties that seem, as we shall see shortly, to be due to their lack of coordination of representations of subsets of n and to the adoption of colloquial registers. In what follows only the responses about the sets ( x, y) : x y 1 and 1 nn, 0 are presented. This choice is due to the fact that the students are familiar with them and the conversion in their graphical representation is within their reach and without requiring a previous treatment of the symbolic representation, which would be beyond the purpose of this study. Some results concerning more
5 examples of sets, which were provided ad hoc to some of the students in order to better grasp their concept image, are also presented. Table 1 summarizes the students answers concerning the boundedness of the set ( x, y) : x y 1. Student ( x, y) : x y 1 Explanation A bounded ( x, y) : x y 1 1,1 1,1 B unbounded x y 1 has infinite solutions, therefore the set is unbounded C bounded It is a circumference, it is bounded D bounded It is a circumference and hence it is bounded E bounded ( x, y) : x y 1 1,1 1,1 F bounded It is a circumference, it is bounded G bounded It is a circumference, it is bounded Table 1: Answers to the question on the boundedness of x, y : x y 1 The first thing to say is that all students, except B, recognized at a first glance that ( x, y) : x y 1 is a circumference in the plane and were able to represent it graphically. Students A and E showed a literate use of mathematical language (literate register) and preferred the symbolic representation. The remaining students, except B, were not able to explain why the set is bounded. Therefore such students, among the others, were encouraged to explain when a set is bounded/unbounded. The set 1 nn, 0 revealed itself to be more troublesome for students. Table summarizes the answers about its boundedness. Students 1, n, n 0 n Explanation A bounded 1, n, n 0 0,1 n B unbounded It is unbounded because n varies in C unbounded It is unbounded on the right the naturals are infinite and hence there is no end for this set
6 D unbounded It is unbounded because there are infinite n E bounded It is contained in 0,1 F bounded The set is bounded from above and below G unbounded It is not bounded because you get infinite fractions Table : Answers to the question on the boundedness of 1 n, n, n 0 Only three students answered correctly. Since the explanation of student F, even if correct, seemed to be not so usual as a topologic argument, some investigations were conducted in his interview (see below). The remaining three students, among the others, were encouraged also after this question to give the definition of bounded/unbounded set and of finite/infinite set. In this case some students evoked a concept definition which was different from the one evoked after the first question. Table 3 summarized the responses of students about their concept definition of the boundedness of a set. Student What is a bounded/unbounded set in A A set X is bounded if there exists a ball B such that X B B She was not able to give a definition but she gave the closed interval 0,1 as an example of bounded set in unbounded set in n? and the interval 0,1 as an example of. She could not come up with an example in C A set X is bounded if there exists a ball B such that X B D She answered An unbounded set is an infinite set when she referred to sets that she could not visualize She answered A bounded set is a set placed in a limited space when she referred to sets that she could visualize E A set X is bounded if there exists a ball B such that X B F G A set X is bounded if there exists a ball B such that X Bor a product of intervals that contains it She answered An unbounded set is an infinite set when she referred to sets that she could not visualize She answered A set is bounded when, from the drawing, it is in a narrow space when she referred to sets that she could visualize Table 3: Answers to the question on the concept definition of bounded/unbounded set
7 For what concern the concept of finite/infinite set all students grasped its correct meaning. All students did not give the formal definition but used a more colloquial (colloquial register) but effective argument as showed in Table 4. Student What is a finite/infinite set in all n? All students answered that a finite set is a set with a finite number of points (or elements) Table 4: Answers to the question on the concept definition of finite/infinite set Data shows that students use an everyday meaning for the boundedness of a set even if their concept definition is correct and given with a literate register. Such a meaning is sufficient to give an answer to problems where the set is already given in the Cartesian coordinate system or students can sketch its graphical representation. In this case the definition of bounded used in the everyday meaning in Italian language that is something that has limits referring to space or time clearly fits with the concept definition of students D and G. Moreover, it seems that also students C, F adopted this colloquial meaning since they are not able, at least apparently, to provide an explanation to the boundedness of the circumference (Table 1) even if their concept definition coincide with the formal one (Table 3). One has to highlight that this everyday meaning is not more sufficient in order to answer to problem 1 n, n, n 0 (Table ). In this case all the students who gave an incorrect explanation (B, C, D, G) did not think to sketch the set but worked only with the symbolic system. The symbolic aspect of the representation of this set seemed to evoke in these students the idea of infinite elements connected to the set of natural numbers, indeed a set that tends to infinity and hence with no upper bound. Here the matter is not that students did not understand the concept of infinite set but that the first evoked meaning of infinite in the context of the symbolic representation of this set that they could not visualize at a first glance is a colloquial one that is something that never ends with reference to space and time indeed something which is unbounded. The problem is not even that they could not represent it graphically. Indeed, the students were asked to sketch it and they did it correctly (sometimes with some help). After that they recognized that they were dealing with a bounded set. Notice that student B applied this reasoning also for the circumference since she could not sketch it (Table 1). During the interview students were asked to explain their apparently incoherent behaviour. Some examples of responses are: I used infinite in order to justify that a set was unbounded because I imagined an infinite set as a set stretching to infinity (student G). I think more to the idea of infinity in order to decide about the boundedness of a set (student C).
8 The research proves also the everyday usage of the term finite by student D as one can see from the following transcript: 1 I: Is ( x, y) : x y 1 unbounded? D: It is a circumference [she draws it]... it is bounded 3 I: It is infinite? 4 D: It is closed, not infinite. 5 I: What does it mean that a set is finite? 6 D: It is limited in space. 7 I: Is the set [in its graphical representation] y x unbounded? 8 D: Yes, it is 9 I: It is infinite/finite? 10 D: It is finite 11 I: Can you find points belonging to it? 1 D: yes, for example 1,1, 1 3,1 3, etcetera, 1 n,1 n..no, then it s infinite! 13 I: Why did you answer finite before? 14 D: Because it seems finite from the drawing. Finally the research highlights another kind of behaviour due to a misleading interpretation of mathematical language. Student F showed no kind of problems with sets like in Table 1,. Problems arose with sets like ( x, y) : y sin x, ( x, y) : xy 1, ( x, y) : y arctg x, etc. where functions are involved. In this case the recognition of well known functions (both graphical and symbolic representations) evoked a meaning of boundedness related to functions rather than sets. The following transcript shows this fact: 1 I: Is ( x, y) : y sin x bounded or unbounded? D: bounded because it can assume values between -1 and 1 After some explanation about his wrong answer he declared I think that the set x y [ ( x, y) : y sin x ] was in.. if I see something simple like 8 8 or y sin x I don t see the other particulars. Finally the formal concept definition of bounded/unbounded set was showed to students (B, D, G) that did not manage to provide it and they were asked if they
9 remembered it. Student B answered that she could not remember it and the other two that they knew it but they did not think to use it. DISCUSSION AND TEACHING IMPLICATIONS This paper provides an example of a wrong use of mathematical language by some Italian mathematics undergraduate students. In particular, these students do not recognize the importance of mathematical definitions. Mathematical terms, such as bounded/unbounded and finite/infinite, evoke different meanings at different times. This is a typical thinking habit that concerns the everyday-life language. In mathematical language, as well as scientific languages in general, terms are usually coined with one meaning only in order to avoid interpretative problems. This purpose should be shared with the students in order to prevent improper use of mathematical language. The argument.a goal of mathematics education as concerns language is promoting flexibility in the use of registers and awareness of the relationships between linguistic forms and context and purposes. This is by no means spontaneous but must be carefully promoted (Ferrari, 00, p.354) applies to the outcomes of this research. Moreover since data showed that different systems of representation evoke different concept images another goal of mathematics education should be promoting flexibility in switching from one representation to another. In my opinion a good practice of teaching in order to promote an appropriate concept image of the boundedness of sets should include, besides to the formal definition, examples of sets satisfying the definition as well as examples of sets that do not verify it. The choice of examples has to be done in order to evoke possible conflicts between the concept image and the concept definition, for examples such those presented in this paper. Moreover, when possible, sets have to be presented in at least two kinds of semiotic representations. Finally, a good practice should include the assignment of exercises that require, as a first step, the description of the concept definition in order to help students in focusing on the correct meaning of the concept. REFERENCES Bardelle, C. (010). Interpreting monotonicity of functions: a semiotic perspective. In Pinto, M. F. & Kawasaki, T. F. (Eds.), Proc. of the 34th Conf. of the Int. Group for the Psychology of Mathematics Education: Vol. (pp ). Belo Horizonte, Brazil: PME. Cornu, B. (1981). Apprentissage de la notion de limite: modèles spontanés et modèles propres. Actes du Cinquième Colloque du Groupe Internationale PME (pp. 3-36). Grenoble, France. Duval, R. (1993). Registres de représentations sémiotique et fonctionnement cognitif de la pensée. Ann. de Didactique et de Sciences Cognitives, ULP, IREM Strasbourg. 5,
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