Dynamic interplay of affect and cognition in school mathematics learning

This article concentrates on the dynamic interplay of affect and cognition in school mathematics learning. The aim of the study is to produce a systematic analysis and rich theoretical description of the functioning of affect and cognition in socio-culturally and contextually conditioned mathematics learning situations. The analysis and dynamic description are presented in close connection with the obtained research results of mathematics education and affect. The included meta-analysis or theoretical synthesis of previous research results is developed with respect to important recent conceptualizations of metacognition, self-regulation, and self-systems and to learning models applied in the scientific field of general educational psychology or within the psychological research of mathematics education. Various conceptualizations and models of affect, learning, and self-regulation are integrated in the study into a unified understanding of personal learning processes with affect and mathematics.

The basic idea of the study consists of an emphasis laid on dynamic theoretical analyses and illustrations dealing with affect and mathematics learning or performances in the school context. The dynamic argument is joined with concepts such as personal learning processes, mental processes, metalevel processes, self-regulation processes, and especially self-system processes. Self, self-systems, self-system processes, and personal agency act as the upper theoretical frame of reference for illustrating the important interplay of affect and cognition in personal mathematics learning processes. More specifically, arousal and experiences of highly influential affective responses are analyzed in regard to pupils' mathematical self-systems and interpretative or self-evaluative processes in social mathematics learning situations and contexts. Affective arousal and states are further connected with various aspects of pupils' mental structures and processes, in particular with their self-beliefs and self-belief systems. Further theoretical deepening of this personal and unique situational dynamics results in a detailed analysis of metalevel processes, personal agency, self-regulatory reflections and actions as the core of pupils' personal mathematics learning or self-system processes and their affective self-experiences with mathematics. Moreover, these personal aspects or self-system processes are considered as the core of the dynamics of affect and cognition in mathematics learning processes in a social environment. Essential qualitative distinctions in mathematics learning and affective experiences are made due to the high or low experienced personal agency, efficiency, and confidence with mathematics.

As many other sciences, mathematics has also emerged from social needs and from practical usages. However, at an early stage it was also associated with the intellectual needs felt by mathematicians to connect the mathematical aspects with logical frameworks or proof structures. Looking at mathematics means paying attention to the logic of mathematical proofs, but also to the methods used to discover certain truths. Hence, it includes as well focusing on the qualities and development of thought processes of mathematicians and the language applied (Baron, 1972). Today, increased notions and studies lean on the hypothesis that mathematics, as well as mathematics instruction, appear and develop as social constructions and in social communication. But, there have been times when mathematics has been generally considered as a path to absolute truths existing outside us and waiting to be discovered. These perspectives have turned into the understanding of mathematics as a useful and even necessary tool for dealing with and predicting other phenomena within societies. In this, the pure mathematics of the well educated and of a few nationals or philosophers has grown into a central tool for several other scientific fields. In addition, it has become a tool for the applied mathematics needed by individuals in their daily activities. At present, mathematics seems to structure our social reality unknowingly, making the surrounding reality highly mathematized (c.f., Kupari, 1994). The increased complexity of everyday life is accompanied by an enhanced significance of mathematics for all citizens, at least in well-developed countries.

The nature of school mathematics has followed the changes in and developments of scientific or academic mathematics. But more particularly, school mathematics relates to the goals of school education and instruction expressed in curriculum's that, in turn, appear to derive from the present needs of society, technology, and the economy. Traditional computational approaches have turned into more advanced mathematics and concepts with an orientation to understanding the underlying structures of mathematics (Resnick & Ford, 1981). Furthermore, the traditional goal of instruction as constituting pupils´ well-structured and basic knowledge of mathematics has turned into efforts to enable pupils to apply their knowledge to complex everyday situations e.g. through modelling (c.f., Baron, 1972; Kupari, 1994; Opetushallitus, 1991). Christiansen et al. (1986, p. 14) present different functions fulfilled by school mathematics: a) as a tool, a social necessity, b) a body of knowledge to be acquired before the next stage of education begins, c) a constituent of a general education, d) a key to advancement, and e) an obstacle course which serves to distinguish between the “able” and the rest. These categories clearly express the multitude of socio-cultural meanings and significance attached to mathematics today and also reflected in teachers´, parents´, and employers´ expectations of pupils. The related expectations and appreciations, however, often pass the idea of mathematics as possessing an inherent interest or appeal that would also essentially build up pupils´ more general knowledge structures or understanding, even representing a significant path to their personal growth.

History and the situation today show that mathematics has had a very special and essential impact on the development of cultures. This central impact, together with the mysteriousness attached to it, has sustained strong ideas and attitudes about mathematics for long time. The special nature of mathematics as a discipline and as being difficult ultimately to clarify, has been suggested as a reason for the significance traditionally given to it within societies, as well as for the traditional appearance of negative attitudes towards mathematics. The most quoted statement of the nature of pure mathematics, made by Bertrand Russell (see Baron, 1972, p. 31), may well be used to illustrate this special nature:

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we say is true.

The abstractness of mathematics and the differences in the symbol systems and concepts used in mathematics language, as compared to other scientific fields or school subjects, form the basis of mathematics as a special discipline (Kilmister, 1972). These aspects of mathematics set high demands on cognitive processes and learning as well as detach mathematics from the context and experience of normal everyday life (Fennema & Loef, 1992; Resnick & Ford, 1981). These demands are suggested as reasons for the appearance of negative experiences with mathematics and the development of negative views, attitudes, or affect regarding mathematics. Viewing mathematics as a difficult and demanding subject has caused it to be highly regarded and has been generally used to measure academic abilities. Accordingly, mathematics has had a ritual value in societies (Christiansen et al., 1986). Another significant aspect of mathematics as a special discipline relates to the traditional understanding of mathematics as a male domain. In turn, this sets essential restrictions for girls´ mathematical attainments and attitudes toward mathematics, measured in gender differences in studies of mathematics education during several decades.

Research results show that, unlike many other school subjects, mathematics represents a subject towards which pupils´ attitudes are developed very early in the school learning context. These long and cross-culturally measured attitudes relate mostly to the importance, difficulty, and like or dislike of mathematics. Moreover, these attitudes are found to significantly relate to or explain mathematics achievements. In addition, mathematics attitudes have been found to change from positive to negative during the secondary school grades and to depend on some personality features or on a sociocultural background. More recent research results indicate a decrease in the generally found gender differences in mathematics achievements or skills and mathematics attitudes, as well as a decrease in the positive connection between mathematics attitudes and achievements. This may be due to the improved measurements, changed achievement test, and treatment of negative attitudes toward mathematics (Friedman, 1989; Frost et al., 1994; Kupari, 1994; Ma & Kishor, 1997). More importantly, these reseach results have shown a decrease in the importance attached to mathematics by pupils. On the other hand, the increased significance of mathematics in societies and the recent essential changes taking place in overall mathematics instruction create new challenges to learning and the use of mathematics, in which affect is playing a significant role in mathematics instructional contexts. The effects of these affective aspects are most commonly attached to differences in pupils´ motivation, especially in respect to mathematical attainment and participation and the found gender differences in these.

With the new challenges, recent research in education has revealed that the traditional emphasis on purely cognitive or generally applicable cognitive factors and achievements with mathematics has been complemented by significances placed on the qualities and functioning of socio-culturally reflected or personally meaningful constructions related to mathematics in the school learning context. How pupils view and approach mathematics and mathematics learning situations will determine their goals and modes of understanding, responding, and behavior in doing and learning mathematics. Pupils make sense of and approach the contents and contexts of mathematics in individual ways, yet always on the basis of those personal and environmental frameworks and understanding dominating their mathematics learning and performances in instructional settings. These have important connections to pupils´ past experiences and their

individual mental structures, including their goals and attitudes toward mathematics, but these derive also from the wider socio-cultural views, goals, patterns, and features attached to school mathematics and learning situations. These socio-cultural aspects are sustained, supported, and reflected by teachers, parents, school practices, mathematical communities, educational communities, or by whole societies. Mathematical beliefs represent the most common concepts attached to these kinds of influential individual or socio-cultural views and structures in recent mathematics education research. Significant beliefs are viewed to act behind pupils´ or teachers´ responses and behavior during mathematics instruction or performances, as well as behind the development of pupils´ mathematical knowledge and skills.

Much credit for the notions of mathematical beliefs can be given to mathematical problem-solving studies, and more generally to the recent emphasis on the problem solving approach to mathematics. Obvious cognitive actions have proved to be the results of conscious or unconscious beliefs associated with the mathematical task at hand, the involved social environment, or with problem-solvers´ perceptions of themselves (Schoenfeld, 1983). The socio-cultural research base for mathematical beliefs has to do with an increased interest in social constructions and cultural symbol systems acting in mathematics learning and instructional settings. This is derived from an ethnographic research orientation or applied anthropological conceptualizations and studies. In turn, these beliefs have been studied more traditionally under the title of mathematics attitudes, reflecting pupils´ or others´ personal views or responses toward mathematics. Personally and socio-culturally held beliefs about mathematics, about mathematics learning situations, and about the self can be viewed to represent significant interpretative guidelines or a basis for pupils´ cognitive actions. Even more important, these beliefs tend to give rise to, embellish, and direct their daily personal learning intentions, actions, goals, and affective responses to mathematics, that is, the essential qualities of their personal learning processes and experiences with mathematics.

The overwhelming philosophical tendency or research paradigm of constructivism, stressed in recent research in educational as well as in mathematics education studies, can be well understood to support these perspectives. Efficient mathematics teaching or instruction is no longer considered merely as the transmission of differentiated and unchanged bits of mathematical knowledge from teacher to pupils nor learning as pupils´

passive adoption and repetition of that knowledge. Instead, mathematics learning is viewed more as based on activity and on the gradual development of and active application of personal mathematical constructions and understanding in social interactions and discussions between teacher and pupils or between pupils within a socially, contextually, and situationally determined classroom culture. Essential questions in education research are: what kinds of interactions and factors support, direct, and regulate the involved personal and socio-cultural constructive processes, and finally, how do these interactions and factors hinder or help pupils´ learning of and doing mathematics. Moreover, by emphasizing pupils´ individual and active constructive processes and personalized use of knowledge in any learning situation, the constructivist paradigm gives more room for pupils´ personal and unique affective experiences, as well as for the significance of their personal and unique situational mathematics learning processes and the self-regulatory activity intertwined with these. Accordingly, new questions or research and perspectives concerning emotional components of learning have emerged in education and in mathematics education research. In this study, we will emphasize these personal constructive and unique aspects of affective experiences in mathematics learning as related to the social environment. We will speak of pupils´ mathematical beliefs as well as their personal or unique situational interpretations, affective responses, and subsequent learning or self-regulation processes with

mathematics.

Various kinds of mathematical beliefs, attitudes, or affective responses have been discerned in studies of mathematics education that seem to have significant effects on the qualities of pupils´ performances, learning, and experiences with mathematics. But the underlying mechanisms, reasons, or processes behind these effects or research results have remained unclear and lack detailed theoretical considerations. The aim of this study is to offer an interpretative and holistic theoretical framework for understanding these underlying or mediating aspects and processes of affect in learning in real school mathematics contexts and performance situations. Accordingly, we will focus on unique mathematics learning situations within a social environment and consider pupils´ personal constructions, states, and processes in these situations. We connect these personal processes and constructions especially with their self-understanding regarding mathematics, their affective responses and experiences with mathematics, as well as with the self-regulation of their own mathematical learning and affective experiences. For this dynamic examination of affect, cognition, and social environment, we will first consider in detail the different aspects or concepts involved in our theoretical study and interpretations of pupils´ personal learning processes and affective experiences with mathematics. We will apply and further develop various conceptualizations compared to research results of pupils´ mathematical beliefs and affective responses. Then, we will move forward and consider these aspects or components in relation to the qualities of pupils´ personal mental systems or processes with affect and of their involved selfinterpretations in school mathematics learning environment and situations. 

Finally, we will deepen our view of these dynamic and affective aspects of personal mathematics learning processes by examining closely the components and qualities of pupils´ self regulatory activities and involved motivational dynamics in their mathematics learning. As the main theoretical interest and starting point of this study arises from questions  regarding the nature and role of affect in mathematics learning, we will first describe the conceptualizations and understandings attached to the affective domain of personality in education research in general, as well as studies in mathematics education. This will offer a review of the perspectives generally applied in related research.