A model of human mind (Hannula 2004, p. 51) 

Contexts in source publication

Context 1

... Hähkiöniemi in his study aiming to acquire information on how pupils think. Representation is one of the central concepts in Hähkiöniemi’s research (2006). He considers representations not only as tools for expressing our thoughts, but also as tools to think with. Further, representations are not seen only as symbolic, but also as graphical and kinesthetic, and there are invisible internal sides in them as well as visible external sides (e.g. gestures). These sides are inseparable. Different representations can enrich pupils’ mathematical thinking, and “[t]he object of thinking is constructed through using different representations” (ibid, p. 15). For a more general framework, Hähkiöniemi (2006) uses Tall’s theory of three worlds of mathematics (e.g. Tall 2004). These three worlds are the symbolic world where symbols act dually as processes and concepts, the embodied world of visuo- spatial images, and the formal world of properties. From these, Hähkiöniemi studies pupils’ use of different representations in the embodied and symbolic worlds. He concentrates on what kind of procedural and conceptual knowledge pupils are using and how they consider derivative as a process and as an object. In the theoretical framework, or in the results, we could not find any mentioning of affective factors, only the pupils’ cognitive activity was studied. Affective factors and how they influence the learning is not discussed even in the discussion chapter. All the five dissertations discussed above were strongly connected to (secondary school) pupils’ mathematical thinking. In the work of Hannula (2004), the focus was more on affect, and on the role of affective factors in mathematical thinking. Both affect and mathematical thinking are concepts that are widely used in mathematics education research, and with either of the concepts there is no common agreement on their definitions. Where Hannula (2004) aimed at clarifying and refining the definition of affect, Joutsenlahti (2005) built a new model for mathematical thinking resting on numerous previous definitions. The remaining three, Hihnala (2005), Merenluoto (2001), and Hähkiöniemi (2006) did not define mathematical thinking explicitly. It seems like they refer to mathematical thinking as thinking about mathematics, where mathematical thinking appears when the pupils calculate something, explain their understanding of a mathematical situation, or thinking is interpreted from the pupils’ written solutions to different tasks. In the studies presented, mathematical thinking is viewed through mathematical or information processes, conceptual change, or representations. Also problem solving among many other approaches to mathematical thinking is mentioned. This shows how complex it is to describe human thinking and how it can be viewed from many different starting points. This is also the case with affect, as Hannula (2004) shows in his theory review. Only Hannula (2004) and Joutsenlahti (2005) clearly deal with affect in their studies. Hannula defines affect through self-regulation where emotion, cognition and motivation are central concepts. Beliefs, values, and attitudes are seen as mixtures of motivational, emotional and cognitive processes. Joutsenlahti (2005) consider pupils’ view of mathematics as influential to mathematical thinking, and beliefs, attitudes, and emotions are studied. This view can be connected to Hannula’s model in future studies. In Merenluoto’s (2001) research, concepts of emotions in certainty judgements, beliefs that influence our search of knowledge, and prior experiences are mentioned. From these, only emotions as feelings of certainty are studied. Hihnala (2005) and Hähkiöniemi (2006) do not mention clearly any affective factors in their theoretical framework. Hihnala (2005), however, mentions motivation in his discussion in the conclusion part of his dissertation, as it might explain some of the variation of pupils’ grades through time. Hähkiöniemi (2006) continues to interpret the theory and results strictly through cognition. One interesting thing that we notice from the dissertations is the notions of metacognition. When Hihnala (2005) talks about problem solving as one way to look into pupils’ mathematical thinking he mentions how the pupils’ metacognitive skills are important in problem solving. This is argued also in Joutsenlahti (2005), where metacognitions are additionally considered as part of information processes. In the work of Merenluoto (2001), more closely in her theory review, metacognitive awareness is mentioned as something that is missing from novice’s explanations, in comparison to experts’ explanations. Metacognition is not part of the self-regulation system in Hannula’s work (2004). However, Hannula does define four aspects that constitute ‘the meta-level of mind,’ and metacognition is part of that structure together with cognitive emotions, emotional cognition, and meta-emotions. So, from Hannula’s model of human mind (figure 1), where cognition and emotion are important, beliefs and values are something that are important to planning and executing something (in Viitala’s study that something is solving PISA-problems). From meta-level of mind, metacognitions are something that influence the study of mathematical thinking essentially and should be recognised. Metacognitions and pupils’ metacognitive skills are essential also for Viitala’s study where interviewing pupils about their solutions for PISA-tasks will be central in the data collection. These five dissertations are written within six years in Finland. So, are the researchers referring to each others’ work? Merenluoto (2001) is the first of the five dissertations and the most cited one. Joutsenlahti (2005) referred to Merenluoto in his review on previous research on mathematical knowledge and skills in upper secondary school, Hihnala (2005) referred to Merenluoto in his review on mathematical concept knowledge, and Hähkiöniemi (2006) on his chapter of limiting processes inherent in the derivative. Hannula’s (2004) categorization on meta-level of mind is cited by Joutsenlahti, and Hihnala cited Joutsenlahti while he discussed about mathematical thinking in problem solving and in information processing. One interesting point is also that all researchers who focused mainly on mathematical thinking (Merenluoto 2001, Joutsenlahti 2005, Hihnala 2005, and Hähkiöniemi 2006) refer to Sfard’s work, more specifically, to her reification theory on mathematical thinking. Finally, even though there is a strong line of research on affect in Finland (Hannula 2007), many times it does not reach the research on mathematical thinking. Mathematical thinking is seen as a cognitive function, and the definitions of knowledge are important in these studies. From the dissertations we studied, only Joutsenlahti (2005) and Hannula (2004) took affective components into account in their studies explicitely and they both utilized McLeod's classification of affect (emotions, beliefs and ...

Context 2

... occurs when mathematical tasks or problems are solved. Affective components for us follow the work of McLeod (1994) who classified affective components into emotions, beliefs and attitudes and DeBellis and Goldin (1997) who developed this classification further by adding values to it. Emotions are mostly affective and the least stable of these, whereas beliefs are mostly cognitive and the most stable. Attitudes and values belong somewhere in between these two. The four components are different from each other, but they are interacting so that the study of one component cannot be completely separated from the three other components. Mathematical thinking is a broad area to study and some serious selection is needed in exploring the connection between mathematical thinking and affective factors. Because data in Viitala’s intended study on pupils’ mathematical thinking is gathered from Finnish pupils, we concentrate on Finnish research. In Finland there exists high level research on affect, and a review concerning it has been published by Hannula (2007). Thus, in this paper the focus is on finding connections between mathematical thinking and affective components in various Finnish studies. We want to explore larger studies on mathematical thinking, and Finnish doctoral dissertations serve as a good starting point. When we investigate the dissertations, the focus is not on the results but on how mathematical thinking and affect are presented in the theory. There are too many doctoral studies on mathematical thinking to report on in this paper, so we limit the exploration to research done in secondary school in Finland, where Viitala is collecting her data from. Finally, we limit the discussion to five dissertations from the past 10 years where mathematical thinking is a central concept. In many cases there are further reports and development on theory published, however, because of the limitation of pages for this paper we concentrate only on the five ‘original’ reports from Hannula (2004) and Hihnala (2005) from lower secondary school, and Joutsenlahti (2005), Merenluoto (2001), and Hähkiöniemi (2006) from upper secondary school. We will now describe what has been said about mathematical thinking and affect in the five dissertations, one by one. Two of the dissertations are based on data from lower secondary school and three of them from upper secondary. We start with the two from lower secondary school, and then proceed to the remaining three. The aim of the study by Markku S. Hannula (2004) is to “increase the coherence of the theoretical foundation for the role of affect in mathematical thinking and learning” (ibid, p. 4). The dissertation includes theoretical and empirical work, and three research tasks are set: to make an analysis of the concepts used for describing affect in mathematics education and, if necessary, refine the definitions, to describe the role of affect in mathematical thinking and learning, and to describe how experiences influence the development of affect (pp. 36-37). From these, some results of the first two questions are presented below. Hannula studies affect in mathematics education research from the same starting point as we do. He starts with McLeod’s (1994) classification of affect dividing it into beliefs, attitudes and emotions, and adds values to this categorization following DeBellis and Goldin (1997). However, as Hannula notes, these four concepts do not cover the whole field of affect, and from other concepts used in literature, he adds motivation. Hannula shows how affective components are viewed from different theoretical frameworks (e.g. from cognitive or social dimensions). Hence, in pursuing to construct a holistic framework of the human mind, he includes physiological, psychological, and social views into his search. After reviewing the literature and re-evaluating the concepts used in them, Hannula ends up with cognition, motivation, and emotion which all belong to the individual’s self-regulative system. In this system, cognition and emotion are viewed as representational systems which require motivation as an energizing system. Cognition codes information about self and environment, and emotions about progress towards personal goals. Motivation originates from human needs. They all are deeply intertwined and each regulates the others to some extent. From the original four concepts only emotion fits into the framework Hannula introduces, when the other three (beliefs, attitudes, and values) are seen as mixtures of motivational, emotional and cognitive processes. This framework is built to clarify the role of affect in processes of the human mind. Emotion, cognition and motivation are described as “fundamentally different kinds of processes that together constitute the human mind” (ibid, p. 20). Attitudes, beliefs, values, and even emotions are defined in ways that include motivational, emotional, and cognitive processes. The model of human mind is presented in figure 1. The relationship between affect and mathematical thinking, as Hannula (2004) describes it, is: The second dissertation on mathematical thinking, to which the empirical data was collected in lower secondary school, is from Kauko Hihnala (2005). The aim for his research is to describe the development of mathematical thinking when shifting from arithmetic to algebra. Hihnala (2005) approaches mathematical thinking through van Hiele’s (1986) theory. Hihnala does acknowledge that van Hiele’s five level theory for mathematical thinking was developed to describe the levels of geometrical thinking, however, he adapts parts of the theory to describe the levels of algebraic thinking. In the study, mathematical thinking is studied through algebraic thinking. When doing a literature review on mathematical thinking, Hihnala identifies four lines of research that are often connected to mathematical thinking: problem solving (when pupils’ metacognitive skills are emphasized), reasoning, conceptual change in knowledge inquiry, and understanding (where processes are important). In categorizing previous research, he bases mostly on Finnish studies. In his own study he claims to examine mathematical thinking through knowledge processing. The data is collected mainly by analysing solutions of tasks that pupils gave on paper. Knowledge is examined through problem solving and it is divided into procedural knowledge and conceptual knowledge. Here Hihnala refers to the work of Hiebert and Lefevre (1986), Kieran (1992), and Sfard (1991). In his study Hihnala analysed the procedural knowledge used in tasks but acknowledges that it is the procedures that change the conceptual knowledge into perceivable form (Hiebert & Lefevre 1986). When Hihnala is constructing the theoretical framework for his study, he does not mention affective factors. He is only investigating the tasks and what tools he needs in analysing them. In the discussion of the results, however, he talks about motivation when he considers reasons for possible changes in pupils’ grades as they move forward in their studies. Also teachers’ task in motivating pupils to study is mentioned in the discussion. In the remaining three dissertations, the empirical data were collected in upper secondary school. The first dissertation we are exploring is from Jorma Joutsenlahti (2005). While Hannula (2004) did his most significant work in clarifying and refining the definitions of concepts in the affective domain, Joutsenlahti (2005) does profound work in the domain of mathematical thinking. Joutsenlahti’s dissertation includes also theoretical and empirical work. He examines different approaches to mathematical thinking and makes his own model for the concepts. Although his main problem in the study is to describe features of the pupils’ test-oriented mathematical thinking, as before, we are concentrating on the theoretical framework he is constructing and the role of affect in his theory. In Finnish curriculum one aim is to develop pupils’ mathematical thinking. This was the starting point for Hihnala’s (2005) work (in lower secondary school), as it is for Joutsenlahti (in upper secondary school). However, as Joutsenlahti highlights, mathematical thinking is something that cannot be observed directly. He introduces five central starting points for studying pupils’ 1 mathematical thinking that can impact essentially on the thinking process, or by which mathematical thinking can be understood or described. These starting points are beliefs, culture, mathematical abilities, information processing, and problem solving. Joutsenlahti places these starting points into five different approaches to mathematical thinking following Sternberg’s work (1996). These approaches are the psychometric approach (mathematical abilities), the anthropological approach (culture, beliefs), the pedagogical approach (beliefs, problem solving), the mathematics as science approach (problem solving, information processes), and the information process approach (information processes, problem solving). From the listed approaches, Joutsenlahti uses the information process approach. Here, the concept of knowledge is emphasized instead of viewing thinking as computer-like manipulation of symbols. As problem solving is part of that approach, also pupils’ metacognitions, beliefs, attitudes, and emotions (as part of the belief system that is directed to mathematics and learning mathematics) play an essential role in Joutsenlahti’s research. These all are considered to belong to the philosophical epistemology and are connected to strategies that are part of strategic knowledge. Knowledge in Joutsenlahti’s study is divided into procedural knowledge (includes mastery of skills), conceptual knowledge (includes also knowledge that is understood), and strategic knowledge. These categories of knowledge belong to the psychological epistemology and they are all connected to problem solving. Knowledge is linked to ...