En god taluppfattning är grunden till matematisk förståelse som eleven skall bygga sina kunskaper på. Utan en god grund i taluppfattning uppstår svårigheter för den undervisning som lärare vill åstadkomma i skolan. Ett vidare begrepp av taluppfattning är number sense som är en känsla för talens storlek och rimligheten i matematikuppgifter. Syftet är att undersöka hur lärare arbetar med matematiksvårigheter inom området taluppfattning för att utveckla elevens lärande. Undersökningen beskriver hur lärare identifierar elever i matematiksvårigheter med tanke på grundläggande taluppfattning. Detta gjordes genom kvalitativa undersökningar i form av intervjuer samt studier av åtgärdsprogram gällande hur lärare identifierar problemen. Informanterna är sex yrkesverksamma lärare med olika matematiska utbildningar inom grundskolans årskurser. Det framkom att den vanligaste åtgärden enligt åtgärdsprogram var arbete i liten grupp, vilket enligt forskning är en handling som inte gynnar eleven, om undervisningen är oförändrad. Däremot ansåg flesta lärare att den bästa metoden för att både upptäcka och avhjälpa svårigheter med taluppfattning var resonemang i matematik med eleven. Brist på resurser gjorde att detta var svårt att organisera. Lärare angav tidbrist som orsak till att kartläggningar av elevers behov saknades i de flesta underlag till åtgärdsprogram.

This thesis presents an analysis of a learning situation of positive and negative integers. It is considered as a case study within the more general context of the development of school learning analysis tools based on cognitive psychology. An analysis of error patterns obtained from a group of 55 secondary I level students has shown: (1) that the acquisition of the integer concept can imply several levels of integration of the sign and the numerical aspect; (2) that the set of all possible addition and subtraction items can be classified according to their possibility of being related to previous numerical knowledge; (3) that errors in addition and subtraction can be explained by either the utilisation of previous knowledge that was contrary to what was taught, or by systematic transformations of the procedures that were taught, or by problems with the integer concept. The thesis ends with a discussion of the properties of an information processing system that could generate the observed performances, and of the pedagogical implications of that kind of analysis.

It is known that almost all numbers are transcendental in the sense of Lebesgue measure. However there is no simple rule to separate transcendental numbers from algebraic numbers. Today research in this direction is about establishing new transcendence criteria for new families of transcendental numbers. By applying a recent refinement of Subspace Theorem, Boris Adamczewski and Yann Bugeaud determined new transcendence criteria for real numbers which we shall present in this thesis. Published only three years ago, their articles explore combinatorial, algorithmic and dynamic approaches in discussing the notion of complexity of both continued fraction and b-adic expansions of a certain class of real numbers. The condition on the expansions are those of being stammering and non-eventually periodic. Taking together these articles give a well-structured picture of the interrelationships between sequence characteristics of expansion (i.e. complexity, periodicity, type of generator) and algebraic characteristics of number itself (i.e. class, transcendency).

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The development of numerical skills in children with Down syndrome is an area of research that has been neglected in the literature despite overwhelming evidence of its importance, both pedagogically, and for everyday functioning. The present study examines two important sub-skills of numeracy. Twelve boys with Down syndrome were compared to 24 typically developing boys (matched on verbal mental age and on chronological age) on two novel, computerized tasks designed to measure sequential processing and counting. Boys with Down syndrome performed comparably to both groups of typically matched controls on the sequential task. However, differences emerged when boys with Down syndrome were required to point and attribute meaning to each step on the counting task. These findings offer novel insights into the development of number skills and provide important data that can aid in the creation of syndrome-specific education strategies to maximize the potential of children with Down syndrome.

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