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A model for the dynamics of affect

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This article consists of two parts. First, a model of interacting representational systems will be presented. Model combines the cognitive and affective domains under a common framework of representations. This model will be used for describing the
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  A model for the dynamics of affect  Markku S. Hannula Zusammenfassung. Dieser Artikel besteht aus zwei Teilen. Im ersten Teilwird eine Modell der wechselwirkenden Representierungssysteme vorgestellt.Die Modell fügt der kognitive und der affective Bereich unter ein gemeinsamesFramework der Representationen zusammen. Mit dieser Modell wird dieWechselwirkung zwischen Beliefs, Vorschriften, Werte, Emotionen undAktionen beschrieben. Im zweite Teil dieses Artikels wird einFalluntersuchung über zwei Mädchen vorgestellt. Die zwei Mädchen hatten ineiner gemeinsamen Klasse studiert, aber ihre Empfindungen waren ziemlichverschieden. Die Falluntersuchung wird mit der vorgestellten Modellgescrieben. Abstract. This article consists of two parts. First, a model of interactingrepresentational systems will be presented. Model combines the cognitive andaffective domains under a common framework of representations. This modelwill be used for describing the interplay between beliefs, scripts, values,emotions, and actions. The second part of the article will present a case studyof two girls who attended the same class, but experienced this quitedifferently. The case will be interpreted using the created model.   1. Introduction This article presents a model for describing the role of affective variables instudent behaviour, which also allows a description of the development of attitudes and beliefs. The model attempts to answer two important and difficultquestions: the interaction of ‘cool’ reason and ‘hot’ emotion, and theinteraction between the experiential state and more stable ‘traits’. The model is based on a notion of representations, which are divided into overt and latentones. Overt representations form the constantly changing ‘landscape of themind’; they are experienced as images, feelings, and thoughts. Latentrepresentations are the relatively stable patterns that can be activatedrepeatedly as overt representations. These are usually labelled as concepts, beliefs, emotions, schemas, and scripts.The model has been developed as a part of a PhD project, where the srcinalaim in 1996 was to look at the differences in the development of girls’ and boys’ self-confidence in mathematics. As is well documented, girls are lessconfident about their skills and talent in mathematics than boys (e.g. Fennema& Sherman, 1978; Jones & Smart, 1995; Leder, 1995; Hannula & Malmivuori,1997). However, the research project soon encountered the problem that noexisting theoretical framework on affect could provide a satisfactory basis for describing how attitudes or beliefs develop. Several others have since then alsodeclared a need to develop the theoretical background (e.g. Di Martino & Zan,2001; Furinghetti & Pehkonen, 1999; Ma and Kishor 1997; Ruffell, Mason &1  Allen, 1998). This lack of satisfactory theoretical background led to thedevelopment of a new model of affective and cognitive representations, whichwas srcinally published in 1998 (Hannula, 1998a). This is a revised version of that paper. 2. Theoretical background2.1. Attitudes, beliefs, and emotions Within mathematics education research, we can see two main approaches toaffect. One approach has been to focus on those aspects of affect that are morestable (attitude, beliefs), and the other to focus on the role of emotional statesin the process of mathematical thinking (especially problem solving).The concept of attitude has been criticized by some researchers (e.g. DiMartino & Zan, 2001; Ruffell, Mason & Allen, 1998; see also Hannula, 2002).The critics have pointed out that attitude is an ambiguous construct, it is oftenused without proper definition, and it needs to be developed theoretically. Themost obvious problem with attitudes is the discrepancy between espoused andenacted attitudes. Moreover, attitude measures seem to need substantialrefining (Ma and Kishor 1997).Di Martino and Zan (2001) distinguish two basic approaches to definingattitude towards mathematics:1) A ‘simple’ definition describes it as the degree of affect associated withmathematics; i.e. attitude is the emotional disposition toward mathematics.This kind of definition ignores the cognitive element in attitude. However,even those who use this kind of definition, often rely on paper and pencil tests,which makes it hard to distinguish emotional disposition from beliefs.2) A three-component definition distinguishes emotional response, beliefs, and behaviour as components of attitude. This second approach seemsincompatible with the widely accepted view (e.g. McLeod, 1992, DeBellis &Goldin, 1997) of attitude, emotions and beliefs as belonging to the affectivedomain.In Hannula (2002a) it was suggested that attitude consists of four differentevaluative processes: situational emotions, expectations (memories of emotions), associations (automatic emotional reactions to certain stimuli) andvalues.In a similar vein, although much research has been done on mathematical beliefs, the experts do not agree on how to define the concept of belief (Furinghetti & Pehkonen, 1999). Consequently, there is no uniformunderstanding of the distinctive role of beliefs within the affective domain.The widely accepted view that beliefs include an affective component hassometimes led to all affective elements being included in the concept of beliefs2  (e.g. Schoenfeld, 1985), while others separate cognitive beliefs from emotions(Mc Leod, 1992; Pehkonen, 1998).Much research has also been done on the individual cognitive processes duringmathematical problem solving (e.g. Schoenfeld, 1985; Silver, 1987; Presmeg,1999). Although during problem solving people often encounter intensiveemotions of frustration and/or joy, this has usually been regarded as somewhat peripheral to the actual problem solving. Sometimes, researchers (Carlson2000, Schoenfeld 1985, 122-125) have documented emotions and managingemotional responses as characteristic of professional mathematicians’ problem-solving behaviour. The non-rational aspects of problem-solving behaviour have been explained by the beliefs the solver has. So far, the emotional aspectshave not been part of the mainstream of research around mathematical problemsolving. However, there are some important exceptions that are reviewed below.Mandler’s constructivist approach (e.g. Mandler 1982, 1989) sees emotions asinitiated by a gut reaction to a discrepancy in an expected schema, which isfollowed by a cognitive analysis. According to Mandler, emotion is alwaysexpressing some aspect of value. Although Mandler’s theory has beenimportant in the field of mathematics education, it is based on an overlysimplistic theory of emotions, and it does not properly incorporate theinfluence of less intensive emotional states.Goldin (1988, 2000) presented "affective pathways" as a framework for thedynamics of the affective domain in mathematical problem solving. These pathways are established sequences of states of feelings that interact withcognition, and suggest strategies during a problem-solving process. Affects arenot merely the 'noise' of human behaviour in problem solving, but form arepresentational system parallel to, and crucial for, cognitive processing.Goldin used the framework of representational systems, which was dividedinto subsystems. Under the cognitive representational subsystem can bedistinguished the subsystems of verbal/syntactic processing, imagistic processing, and planning and executive control. Parallel to the cognitivesubsystem is an affective representational system. Later, DeBellis and Goldin(1997) divided the affective representational system into four facets of affective states, which interact on the individual level: emotional states,attitudes, beliefs, and values/morals/ethics. Interactions with the socioculturalcontext and the affective climate of the environment are also included in thismodel.Perhaps the most promising approach to emotions in mathematical thinking ismade by a group of researchers from the University of Leuven (Op’t Eynde,De Corte, Verschaffel, 1999; 2001). They try to study the interplay of emotion,motivation, and cognition in problem solving and, furthermore, integrate thiswith a socio-constructivist approach. In their view, emotions are primarily3  grounded in and defined by the social context. However, they do not yet provide a full analytical framework or theory.To summarise, there are several important concepts in the affective domain, but not all of them are well defined. Especially ‘attitude’ and ‘belief’ are problematic concepts that need to be used carefully. In this article, belief will be defined as purely cognitive, although it has an intensive interaction with theaffective domain. Furthermore, the concept ‘attitude’ will be totally discarded, because– to my present understanding – it is not a uniform concept but aconfluence of several different processes (see, Hannula 2002). Instead, I willinclude the concept of affective schema, which is close to the narrowdefinition of attitude. Thus, the following model is built around the conceptsof emotion (emotional state), affective schema, value, and belief. 2.2. Interacting representational subsystems To describe the changes in beliefs and attitudes a framework of (inner)representations is used. An individual receives information through the senses.This information is transformed to represent the srcinal source. For example,were a teacher to write 'p=2πr' on the blackboard and ask the pupils to copy theformula in their notebooks, many pupils would create a verbal representationin their minds and utter to themselves "p equals two" while writing it. Thenthey would look at the blackboard again and try to create a mental image of the letter π and draw it. Some might sigh: "Oh no! Another boring formula. How am I going to remember it?"  The different inner representations cover every piece of information that individuals have about their environment. The physical, social and cultural environments are represented in the inner world.Social relationships, self-concept and values are there, as well as memoriesand traumas from childhood.It is useful to divide the human representational system into subsystems. Under the cognitive representational subsystem can be distinguished the subsystemsof verbal/syntactic processing, imagistic processing, and planning andexecutive control. Parallel to the cognitive subsystem is an affectiverepresentational system. (Goldin, 1987; 1988; 2000).The human representational system is so integrated, that the division intosubsystems is always somewhat problematic. Goldin's division is a 'top-level'view that is based on the different systems we are able to reach throughintrospection. Other ways of splitting the whole could be equally justified. A'bottom-level' view would begin from different sensory systems and thenintegrate these into more complex systems. Some parts of the brain arespecialized for processing visual information, others for auditive information.Although the different subsystems may be functionally and physiologicallydifferentiated from each other, they continuously interact and the borderlinesmay be fuzzy.4  The affective representational system has a physiologically different basisfrom the cognitive system. Affect is represented partly chemically. Emotionsregulate the levels of hormones in the blood and, vice versa, certain chemicalsregulate emotions. (See, for example, Power & Dalgleish, 1997; Buck, 1999)Many psychological theories have attempted to explain the human mind either as a primarily cognitive or affective system. The first view assumes thatcognition always precedes affection (Eagley & Chaiken, 1993, p. 422). Thecontrary view has dominated in The Freudian tradition. During the lastdecades, several researchers have been constructing integrated affect-cognitionmodels (Goldin, 1987; 1988; 2000; Leventhal, 1982; Power & Dalgleish,1997; Toskala, 1991; 1993; Zajonc, Pietromonaco & Bargh, 1982; see alsoEagly & Chaiken, 1993, p. 423). The structure of the model that I present hereowes much to a model of 'interacting cognitive subsystems' (Barnard &Teasdale, 1991, in Toskala, Happonen & Mäntynen, 1996). 2.3. Overt and latent representations The 'landscape of the mind' is continuously changing. Each representationalsystem is all the time receiving new information through the senses and/or from other representational systems, and the representations change in reactionto this information. Such dynamically changing representations are in this paper called overt. Each representational system also has some kind of memory system, where information can be stored and from where it can beretrieved as overt representations. These representations that potentially may be activated are called latent representations.Latent representations are not stored anywhere in the usual sense. Rather, theyexist as potential overt representations within the platform that represents the presently active overt representation. This platform – the human body – andhence all representations, are partially genetically determined.The overt representation in each subsystem is determined by three factors: (a)the information to be processed, (b) the latent representations of thesubsystem, and (c) the way in which the subsystems are organized into asystem (Barnard & Teasdale, 1991). 2.4. Affect in the framework of representational systems We can divide the human representational system into cognitive and affectivesubsystems. We can also separate the overt and latent representations. Now itis possible to build a model that can be used for explaining the changes inattitudes (Figure 1). This model is not empirically tested nor based on my ownempirical research. It is a theoretical construction that has been built to fit thereported findings in this field. The model is simplified by leaving out somemechanisms that are not related to beliefs or attitudes, for example, thegenetically determined transformation process from situation to sensory5

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