Truth, Proof and Gödelian Arguments : A Defence of Tarskian Truth in Mathematics

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established to be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion.

Achievement and Social Strategies and the Cumulation of Positive and Negative Experiences During Young Adulthood

These longitudinal studies focused on investigating young adults during transition into a new educational environment. The aims were to examine: (1) what kinds of achievement and social strategies young adults deploy, (2) whether the deployment of these strategies predicts people's success in their studies, their life events, their peer relationships, and their well-being, (3) whether young adults' success in dealing with educational transition (e.g. success in studies, life events, peer relationships and well-being) predict changes in their strategies and well-being, and (4) the associations between young adults' social strategies, interpersonal behaviour, person perception, and their peer relationships and satisfaction with them. The participants were students from Helsinki university and from two vocational institutes (the numbers ranging between 92 and 303). The results revealed that achievement and social strategies contributed to individuals' success in dealing with both the academic and interpersonal challenges of a new environment. Social strategies were also associated with online interpersonal behaviour and person perception, which mediated their impact on peer relationships. Achievement and social strategies changed as a result of environmental feedback. However, they also showed high stability, forming reciprocal and cumulative associations with the feedback the individuals received about their success in dealing with educational transition: the use of functional strategies, such as optimistic, defensive-pessimistic and planning-oriented strategies, increased their success, which in turn enhanced their well-being and further deployment of functional strategies. The opposite was true in the case of dysfunctional strategies, such as self-handicapping and avoidance. Key words : Achievement strategies, social strategies, transition, young adults, life events, sociometric status, social behaviour, person perception, well-being.

Assessing university students' prior knowledge

The aim of this dissertation was to explore how different types of prior knowledge influence student achievement and how different assessment methods influence the observed effect of prior knowledge. The project started by creating a model of prior knowledge which was tested in various science disciplines. Study I explored the contribution of different components of prior knowledge on student achievement in two different mathematics courses. The results showed that the procedural knowledge components which require higher-order cognitive skills predicted the final grades best and were also highly related to previous study success. The same pattern regarding the influence of prior knowledge was also seen in Study III which was a longitudinal study of the accumulation of prior knowledge in the context of pharmacy. The study analysed how prior knowledge from previous courses was related to student achievement in the target course. The results implied that students who possessed higher-level prior knowledge, that is, procedural knowledge, from previous courses also obtained higher grades in the more advanced target course. Study IV explored the impact of different types of prior knowledge on students’ readiness to drop out from the course, on the pace of completing the course and on the final grade. The study was conducted in the context of chemistry. The results revealed again that students who performed well in the procedural prior-knowledge tasks were also likely to complete the course in pre-scheduled time and get higher final grades. On the other hand, students whose performance was weak in the procedural prior-knowledge tasks were more likely to drop out or take a longer time to complete the course. Study II explored the issue of prior knowledge from another perspective. Study II aimed to analyse the interrelations between academic self-beliefs, prior knowledge and student achievement in the context of mathematics. The results revealed that prior knowledge was more predictive of student achievement than were other variables included in the study. Self-beliefs were also strongly related to student achievement, but the predictive power of prior knowledge overruled the influence of self-beliefs when they were included in the same model. There was also a strong correlation between academic self-beliefs and prior-knowledge performance. The results of all the four studies were consistent with each other indicating that the model of prior knowledge may be used as a potential tool for prior knowledge assessment. It is useful to make a distinction between different types of prior knowledge in assessment since the type of prior knowledge students possess appears to make a difference. The results implied that there indeed is variation between students’ prior knowledge and academic self-beliefs which influences student achievement. This should be taken into account in instruction.

Becoming a teacher: emerging teacher identity in mathematics teacher education

This research examines three aspects of becoming a teacher, teacher identity formation in mathematics teacher education: the cognitive and affective aspect, the image of an ideal teacher directing the developmental process, and as an on-going process. The formation of emerging teacher identity was approached in a social psychological framework, in which individual development takes place in social interaction with the context through various experiences. Formation of teacher identity is seen as a dynamic, on-going developmental process, in which an individual intentionally aspires after the ideal image of being a teacher by developing his/her own competence as a teacher. The starting-point was that it is possible to examine formation of teacher identity through conceptualisation of observations that the individual and others have about teacher identity in different situations. The research uses the qualitative case study approach to formation of emerging teacher identity, the individual developmental process and the socially constructed image of an ideal mathematics teacher. Two student cases, John and Mary, and the collective case of teacher educators representing socially shared views of becoming and being a mathematics teacher are presented. The development of each student was examined based on three semi-structured interviews supplemented with written products. The data-gathering took place during the 2005 2006 academic year. The collective case about the ideal image provided during the programme was composed of separate case displays of each teacher educator, which were mainly based on semi-structured interviews in spring term 2006. The intentions and aims set for students were of special interest in the interviews with teacher educators. The interview data was analysed following the modified idea of analytic induction. The formation of teacher identity is elaborated through three themes emerging from theoretical considerations and the cases. First, the profile of one s present state as a teacher may be scrutinised through separate affective and cognitive aspects associated with the teaching profession. The differences between individuals arise through dif-ferent emphasis on these aspects. Similarly, the socially constructed image of an ideal teacher may be profiled through a combination of aspects associated with the teaching profession. Second, the ideal image directing the individual developmental process is the level at which individual and social processes meet. Third, formation of teacher identity is about becoming a teacher both in the eyes of the individual self as well as of others in the context. It is a challenge in academic mathematics teacher education to support the various cognitive and affective aspects associated with being a teacher in a way that being a professional and further development could have a coherent starting-point that an individual can internalise.

Is a Majority Enough? Psychological Well-being and Its Relation to Academic and Prosocial Motivation, Self-regulation and Achievement at school

This cross-sectional study analyzed psychological well-being at school using the Self-Determination theory as a theoretical frame-work. The study explored basic psychological needs fulfillment (BPNS), academic (SRQ-A), prosocial self-regulation (SRQ-P) and motivation, and their relationship with achievement in general, special and selective education (N=786, 444 boys, 345 girls, mean age 12 yrs 8 mths). Motivation starts behavior which becomes guided by self-regulation. The perceived locus of control (PLOC) affects how self-determined this behavior will be; in other words, to what extent it is autonomously regulated. In order learn and thus to be able to accept external goals, a student has to feel emotionally safe and have sufficient ego-flexibility—all of which builds on satisfied psychological needs. In this study those conditions were explored. In addition to traditional methods Self-organizing maps (SOM), was used in order to cluster the students according to their well-being, self-regulation, motivation and achievement scores. The main impacts of this research were: a presentation of the theory based alternative of studying psychological well-being at school and usage of both the variable and person-oriented approach. In this Finnish sample the results showed that the majority of students felt well, but the well-being varied by group. Overall about for 11–15% the basic needs were deprived depending on the educational group. Age and educational group were the most effective factors; gender was important in relation to prosocial identified behavior. Although the person-oriented SOM-approach, was in a large extent confirming what was no-ticed by using comparison of the variables: the SEN groups had lower levels of basic needs fulfillment and less autonomous self-regulation, interesting deviations of that rule appeared. Some of the SEL- and GEN-group members ended up in the more unfavorable SOM-clusters, and not all SEN-group members belonged to the poorest clusters (although not to the best either). This evidence refines the well-being and self-regulation picture, and may re-direct intervention plans, and turn our focus also on students who might otherwise remain unnoticed. On the other hand, these results imply simultaneously that in special education groups the average is not the whole truth. On the basis of theoretical and empirical considerations an intervention model was sug-gested. The aim of the model was to shift amotivation or external motivation in a more intrinsic direction. According to the theoretical and empirical evidence this can be achieved first by studying the self-concept a student has, and then trying to affect both inner and environmental factors—including a consideration of the basic psychological needs. Keywords: academic self-regulation, prosocial self-regulation, basic psychological needs, moti-vation, achievement

Self-Concept and School Achievement of Pupils with Cleft Lip, Cleft Palate or Both : A longitudinal study

The goal of this research was to survey the self-concept and school achievement of pupils with cleft lip, cleft palate or both from juvenile age to adolescence. Longitudinal researches of self-concept and school achievement among pupils with cleft lip, cleft palate or both are uncommon. This research was the first longitudinal research ever conducted in Finland among this population. This research can be considered to be a special educational study because of the target group involved. Self-concept consists of the person s entire personality. Personality is biological and deterministic. Self-concept includes concepts, attitudes and feelings that the person has about him or her qualities, abilities and relations to the environment. The individual associates experiences to this personality with earlier observations through the social interaction. The individual will have the consciousness of the person s existence and action. The target group in this study consisted of Finnish children with clefts, who were comprised of four different age groups. The questionnaire was sent to all subjects (N1 = 419) both times. A total of 74 % of children returned the questionnaire in 1988 (N2=305). 48 % of children returned the questionnaire in 1993 (N3=203). 42% of children returned the questionnaire both times (N4=175) . These 175 children formed the research subjects. The survey was conducted in 1988, and again in 1993. In 1988, the pupils surveyed were 9 to 12 years of age, while in 1993 they were between 14 and 17 years old. The data was collected through the use of a questionnaire, which consisted of common questions and a personality inventory test that was developed for Finnish students by professor Maija-Liisa Rauste-von Wright. Quantitative analysis methods were used to examine the structure of self-concept and school achievement. Structures found in this research were observed in relation to disorder, gender and maturation. According to these results, structures of self-concepts and school achievement are in fact stable. Basic self-concept elements are seen to be formed at an early age. The developmental aspects of self-concept following puberty are observed as the stability of self-concept and as the forming of a general self. The level of school achievement is stable, but the structure of school achievement changes. From these results, it is possible to state that the gender of the child has a statistical significance regarding self-concept and school achievement. However, the experienced disorder does not have statistical significance as regards to self-concept and school achievement. Results of self-concept support the research of self-concept conducted earlier in Finland.