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.

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.

Kidney Induction: control by Notch, Wnt and GDNF/Ret signalling

The permanent mammalian kidney (metanephros) develops as a result of complex reciprocal tissue interactions between a ureteric epithelium and the renal mesenchyme. The overall goal of the research in this thesis was to gain data that will eventually help in elucidating the formation of congenital renal malformations. The experiments in my thesis aimed to reveal the mechanisms by which Notch, Wnt and GDNF/Ret signalling pathways regulate the development of functional kidney. The function of Notch pathway was studied by a transgenic mouse model, where it was shown that overactivation of Notch signalling disturbs kidney development and alters the expression of Gdnf and Ret/GFRa1. This indicates that Notch signalling interplays with GDNF/Ret in the regulation of the primary ureteric budding and its subsequent branching. The data also suggested that strict spatio-temporal regulation of these two pathways is required for determination of ureteric tip-identity, which appeared to be crucial for the branch formation. The function of Wnt signalling in the ureteric morphogenesis was studied by in vivo and in vitro methods to show that a canonical pathway is required for ureteric branching. Stabilisation and deletion of the canonical pathway mediator, b-catenin specifically in the ureteric epithelium result in renal aplasia/hypodysplasia. These defects originate from severe blockage of ureteric branching due to the disrupted Ret signalling. Consequently, ureteric tip specific markers are lost and ureteric stalk identity is expanded throughout the whole epithelium. Thus, the data demonstrates that the Wnt/b-catenin pathway plays an essential role in the patterning and branching of the ureteric epithelium. A novel in vitro method was generated and utilised in nephron induction studies to reveal the mechanisms through which nephrogenesis is induced. Transient GSK3 inhibition results in stabilisation of b-catenin in the isolated renal mesenchyme, which efficiently triggers nephron formation. Also genetic stabilisation of b-catenin specifically in the mesenchyme results in spontaneous nephrogenesis. The results show that activation of the canonical Wnt pathway is sufficient to initiate nephrogenesis, and suggest that this pathway mediates the nephron induction in murine kidney mesenchymes. Taken together, this thesis demonstrates Notch and Wnt signalling pathways as novel regulators of ureteric branching morphogenesis, and that activation of the canonical Wnt pathway is sufficient for nephron induction. The studies also indicate that the Notch and Wnt pathways cross-talk with GDNF/Ret signalling in the patterning of ureteric epithelium.

Lactic acid bacteria and their antimicrobial peptides : Induction,detection,partial characterization, and potential applications

Bacteriocin-producing lactic acid bacteria and their isolated peptide bacteriocins are of value to control pathogens and spoiling microorganisms in foods and feed. Nisin is the only bacteriocin that is commonly accepted as a food preservative and has a broad spectrum of activity against Gram-positive organisms including spore forming bacteria. In this study nisin induction was studied from two perspectives, induction from inside of the cell and selection of nisin inducible strains with increased nisin induction sensitivity. The results showed that a mutation in the nisin precursor transporter NisT rendered L. lactis incapable of nisin secretion and lead to nisin accumulation inside the cells. Intracellular proteolytic activity could cleave the N-terminal leader peptide of nisin precursor, resulting in active nisin in the cells. Using a nisin sensitive GFP bioassay it could be shown, that the active intracellular nisin could function as an inducer without any detectable release from the cells. The results suggested that nisin can be inserted into the cytoplasmic membrane from inside the cell and activate NisK. This model of two-component regulation may be a general mechanism of how amphiphilic signals activate the histidine kinase sensor and would represent a novel way for a signal transduction pathway to recognize its signal. In addition, nisin induction was studied through the isolation of natural mutants of the GFPuv nisin bioassay strain L. lactis LAC275 using fl uorescence-activated cell sorting (FACS). The isolated mutant strains represent second generation of GFPuv bioassay strains which can allow the detection of nisin at lower levels. The applied aspect of this thesis was focused on the potential of bacteriocins in chicken farming. One aim was to study nisin as a potential growth promoter in chicken feed. Therefore, the lactic acid bacteria of chicken crop and the nisin sensitivity of the isolated strains were tested. It was found that in the crop Lactobacillus reuteri, L. salivarius and L. crispatus were the dominating bacteria and variation in nisin resistance level of these strains was found. This suggested that nisin may be used as growth promoter without wiping out the dominating bacterial species in the crop. As the isolated lactobacilli may serve as bacteria promoting chicken health or reducing zoonoosis and bacteriocin production is one property associated with probiotics, the isolated strains were screened for bacteriocin activity against the pathogen Campylobacter jejuni. The results showed that many of the isolated L. salivarius strains could inhibit the growth of C. jejuni. The bacteriocin of the L. salivarius LAB47 strain, with the strongest activity, was further characterized. Salivaricin 47 is heat-stable and active in pH range 3 to 8, and the molecular mass was estimated to be approximately 3.2 kDa based on tricine SDS-PAGE analysis.

Topological stability through tame retractions

A smooth map is said to be stable if small perturbations of the map only differ from the original one by a smooth change of coordinates. Smoothly stable maps are generic among the proper maps between given source and target manifolds when the source and target dimensions belong to the so-called nice dimensions, but outside this range of dimensions, smooth maps cannot generally be approximated by stable maps. This leads to the definition of topologically stable maps, where the smooth coordinate changes are replaced with homeomorphisms. The topologically stable maps are generic among proper maps for any dimensions of source and target. The purpose of this thesis is to investigate methods for proving topological stability by constructing extremely tame (E-tame) retractions onto the map in question from one of its smoothly stable unfoldings. In particular, we investigate how to use E-tame retractions from stable unfoldings to find topologically ministable unfoldings for certain weighted homogeneous maps or germs. Our first results are concerned with the construction of E-tame retractions and their relation to topological stability. We study how to construct the E-tame retractions from partial or local information, and these results form our toolbox for the main constructions. In the next chapter we study the group of right-left equivalences leaving a given multigerm f invariant, and show that when the multigerm is finitely determined, the group has a maximal compact subgroup and that the corresponding quotient is contractible. This means, essentially, that the group can be replaced with a compact Lie group of symmetries without much loss of information. We also show how to split the group into a product whose components only depend on the monogerm components of f. In the final chapter we investigate representatives of the E- and Z-series of singularities, discuss their instability and use our tools to construct E-tame retractions for some of them. The construction is based on describing the geometry of the set of points where the map is not smoothly stable, discovering that by using induction and our constructional tools, we already know how to construct local E-tame retractions along the set. The local solutions can then be glued together using our knowledge about the symmetry group of the local germs. We also discuss how to generalize our method to the whole E- and Z- series.

Growth factor induction in intervertebral disc tissue: Observations in basic mechanisms of disc degeneration and rearrangement

The intervertebral disc is composed of concentrically arranged components: annulus fibrosus, the transition zone, and central nucleus pulposus. The major disc cell type differs in various parts of the intervertebral disc. In annulus fibrosus a spindle shaped fibroblast-like cell mainly dominates, whereas in central nucleus pulposus the more rounded chondrocyte-like disc cell is the major cell type. At birth the intervertebral disc is well vascularized, but during childhood and adolescence blood vessels become smaller and less numerous. The adult intervertebral disc is avascular and is nourished via the cartilage endplates. On the other hand, degenerated and prolapsed intervertebral discs are again vascularized, and show many changes compared to normal discs, including: nerve ingrowth, change in collagen turnover, and change in water content. Furthermore, the prolapsed intervertebral disc tissue has a tendency to decrease in size over time. Growth factors are polypeptides which regulate cell growth, extracellular matrix protease activity, and vascularization. Oncoproteins c-Fos and c-Jun heterodimerize, forming the AP-1 transcription factor which is expressed in activated cells. In this thesis the differences of growth factor expression in normal intervertebral disc, the degenerated intervertebral disc and herniated intervertebral disc were analyzed. Growth factors of particular interest were basic fibroblast growth factor (bFGF or FGF-2), platelet-derived growth factor (PDGF), vascular endothelial growth factor (VEGF), and transforming growth factor beta (TGFβ). Cell activation was visualized by the expression of the AP-1 transcription promoters c-Fos and c-Jun. The expression was shown with either mono- or polyclonal antibodies by indirect avidin-biotin-peroxidase immunohistochemical staining method. The normal control material was collected from a tissue bank of five organ donors. The degenerated disc material was from twelve patients operated on for painful degenerative disc disease, and herniated disc tissue material was obtained from 115 patients operated on for sciatica. Normal control discs showed only TGFβ immunopositivity. All other factors studied were immunonegative in the control material. Prolapsed disc material was immunopositive for all factors studied, and this positivity was located either in the disc cells or in blood vessels. Furthermore, neovascularization was noted. Disc cell immunoreaction was shown in chondrocyte-like disc cells or in fibroblast-like disc cells, the former being expressed especially in conglomerates (clusters of disc cells). TGFβ receptor induction was prominent in prolapsed intervertebral disc tissue. In degenerated disc material, the expression of growth factors was analyzed in greater detail in various parts of the disc: nucleus pulposus, anterior annulus fibrosus and posterior annulus fibrosus. PDGF did not show any immunoreactivity, whereas all other studied growth factors were localized either in chondrocyte-like disc cells, often forming clusters, in fibroblast-like disc cells, or in small capillaries. Many of the studied degenerated discs showed tears in the posterior region of annulus fibrosus, but expression of immunopositive growth factors was detected throughout the entire disc. Furthermore, there was a difference in immunopositive cell types for different growth factors. The main conclusion of the thesis, supported by all substudies, is the occurrence of growth factors in disc cells. They may be actively participating in a network regulating disc cell growth, proliferation, extracellular matrix turnover, and neovascularization. Chondrocyte-like disc cells, in particular, expressed growth factors and oncoproteins, highlighting the importance of this cell type in the basic pathophysiologic events involved in disc degeneration and disc rearrangement. The thesis proposes a hypothesis for cellular remodelling in intervertebral disc tissue. In summary, the model presents an activation pattern of different growth factors at different intervertebral disc stages, mechanisms leading to neovascularization of the intervertebral disc in pathological conditions, and alteration of disc cell shape, especially in annulus fibrosus. Chondrocyte-like disc cells become more numerous, and these cells are capable of forming clusters, which appear to be regionally active within the disc. The alteration of the phenotype of disc cells expressing growth factors from fibroblast-like disc cells to chondrocyte-like cells in annulus fibrosus, and the numerous expression of growth factor expressing disc cells in nucleus pulposus, may be a key element both during pathological degeneration of the intervertebral disc, and during the healing process after trauma.

Theoretical modeling of ionospheric electrodynamics including induction effects

This thesis deals with theoretical modeling of the electrodynamics of auroral ionospheres. In the five research articles forming the main part of the thesis we have concentrated on two main themes: Development of new data-analysis techniques and study of inductive phenomena in the ionospheric electrodynamics. The introductory part of the thesis provides a background for these new results and places them in the wider context of ionospheric research. In this thesis we have developed a new tool (called 1D SECS) for analysing ground based magnetic measurements from a 1-dimensional magnetometer chain (usually aligned in the North-South direction) and a new method for obtaining ionospheric electric field from combined ground based magnetic measurements and estimated ionospheric electric conductance. Both these methods are based on earlier work, but contain important new features: 1D SECS respects the spherical geometry of large scale ionospheric electrojet systems and due to an innovative way of implementing boundary conditions the new method for obtaining electric fields can be applied also at local scale studies. These new calculation methods have been tested using both simulated and real data. The tests indicate that the new methods are more reliable than the previous techniques. Inductive phenomena are intimately related to temporal changes in electric currents. As the large scale ionospheric current systems change relatively slowly, in time scales of several minutes or hours, inductive effects are usually assumed to be negligible. However, during the past ten years, it has been realised that induction can play an important part in some ionospheric phenomena. In this thesis we have studied the role of inductive electric fields and currents in ionospheric electrodynamics. We have formulated the induction problem so that only ionospheric electric parameters are used in the calculations. This is in contrast to previous studies, which require knowledge of the magnetospheric-ionosphere coupling. We have applied our technique to several realistic models of typical auroral phenomena. The results indicate that inductive electric fields and currents are locally important during the most dynamical phenomena (like the westward travelling surge, WTS). In these situations induction may locally contribute up to 20-30% of the total ionospheric electric field and currents. Inductive phenomena do also change the field-aligned currents flowing between the ionosphere and magnetosphere, thus modifying the coupling between the two regions.