<p>Studien syftar till att granska ett urval av pedagogiska datorprogram avsedda för matematik för att få reda på vad det är för typ av matematikuppgifter användaren (eleven) kan möta. </p><p>Den teoretiska referensramen behandlar tre områden. Dessa är matematik, olika sätt att kategorisera matematikuppgifter samt olika sätt att kategorisera datorprogram. </p><p>Sammanfattningsvis visar resultatet att det är svårt att kategorisera matematikuppgifter strikt. Beroende på val av program kan eleven möta uppgifter där det matematiska innehållet innefattar allt från ett upp till sex olika områden av grundskolans matematik. Av eleven krävs främst fakta- och färdighetskunskaper för att lösa de olika uppgifterna. Förståelsekunskapen finns med i fyra av de fem granskade programmen. Innehållet i de olika matematikuppgifterna hör i huvudsak hemma i ett vardagligt eller matematiskt sammanhang men uppgifter med en kontext av annat slag finns. Svårighetsgraden varierar beroende på val av program. Här spelar även individuella faktorer en stor roll.</p>

Detta arbete handlar om elevers kunskaper i att översätta en matematisk problemtext till en ekvation. Jag har dels studerat tidigare forskning inom området och dels gjort en egen studie. Huvudsyftet med arbetet är att ta reda på om det finns problem för elever i årskurs nio att finna en ekvation som kan lösa en bestämd uppgift och i så fall vilka är svårigheterna för eleverna. För att uppfylla syftet med detta arbete har jag valt att göra en litteraturstudie, en kvantitativ studie samt en mindre kvalitativ studie. I litteraturstudien tar jag bland annat upp vad algebra och ekvationer är för något, algebrans betydelse i skolan och vad tidigare undersökningar säger i det område jag undersöker. I min undersökning har 49 elever i årskurs nio deltagit i den skriftliga undersökningen och tre elever gjorde en gruppintervju. Litteraturstudien och min studie avslutas med en diskussion. I diskussionen dras det slutsatser mellan min undersökning och litteraturstudien Några slutsatser som har framkommit av undersökningen och litteraturstudien är att elever i årskurs nio har svårt med att översätta ett problem till en ekvation. Det är framförallt förståelsen för bokstäver och likhetstecknets funktion i ekvationer som är svårt för eleverna. Avslutningsvis tar jag upp några didaktiska perspektiv på ekvationer som kan vara till hjälp för lärarna i undervisningen.

Studien syftar till att granska ett urval av pedagogiska datorprogram avsedda för matematik för att få reda på vad det är för typ av matematikuppgifter användaren (eleven) kan möta. Den teoretiska referensramen behandlar tre områden. Dessa är matematik, olika sätt att kategorisera matematikuppgifter samt olika sätt att kategorisera datorprogram. Sammanfattningsvis visar resultatet att det är svårt att kategorisera matematikuppgifter strikt. Beroende på val av program kan eleven möta uppgifter där det matematiska innehållet innefattar allt från ett upp till sex olika områden av grundskolans matematik. Av eleven krävs främst fakta- och färdighetskunskaper för att lösa de olika uppgifterna. Förståelsekunskapen finns med i fyra av de fem granskade programmen. Innehållet i de olika matematikuppgifterna hör i huvudsak hemma i ett vardagligt eller matematiskt sammanhang men uppgifter med en kontext av annat slag finns. Svårighetsgraden varierar beroende på val av program. Här spelar även individuella faktorer en stor roll.

An introduction of computer software into mathematics classrooms makes the didactical situation more complex compared with previous learning environments (Blomhøj, 2005). A technological tool becoming a mathematic work tool in the hands of the students is a process that has turned up unexpectedly complex (Artigue, 2002). In addition to this problem, the teachers as the users of the tool go through the same process, while, at the same time, trying to integrate the tool into their teaching activities in a meaningful way. For these reasons it seems important to contribute to the research focused on the learning and teaching conditions in environments, where computer software is newly introduced, in order to better understand impacts of the introduction of different software in mathematics classrooms. In this study the dynamic mathematical software GeoGebra was used. GeoGebra is freely available for a number of platforms and has drawn much attention during the last years with growing user communities (www.GeoGebra.org). However, being generally available just recently, there are, comparatively, few studies on the use of GeoGebra in classroom settings. In this thesis the introduction and integration of GeoGebra was investigated in two studies with different perspectives. In the first study students’ work with GeoGebra in their mathematical activities related to the integral concept has been researched. In the second study teachers’ utilization of the didactical potential has been investigated. The results of the two studies show that GeoGebra as a mathematical tool in the hands of the students and the teachers can have a significant role in supporting their mathematical work if exploited in a, from a didactical perspective, adequate way. A learning and teaching environment based on GeoGebra bring with it a possibility to work with mathematical concepts in a broader way compared with blackboard based classrooms. GeoGebra’s facilities makes it possible to communicate mathematics in different ways and expressing mathematical concepts in different representations in a more direct way than in non dynamical environments. Communicating mathematics in different ways and expressing mathematics knowledge through different representations is of significant importance for students, not least in relation to the new curriculum for mathematics in Sweden (The Swedish National Agency for Education, 2011), where these aspects are explicitly named as aims for students to work towards. On the other hand, the investigations also showed that the introduction and the integration of GeoGebrawas a complex process for both the students and the teachers in this research. The introduction and integration of the software in the students’ mathematical activities made the didactical situation more complex and a differentiation of students’ work with the software was observed. For some students the use of the software seemingly supported their mathematical work, and at the same time for some students the result was the opposite; the use of the software was seen as a disturbing factor in their mathematical activities. When it comes to the study of teachers’ work with GeoGebra the investigations revealed that they encountered different types of obstacles that prevented them from utilizing the full didactical potential of the software in their teaching of mathematics. Three different types of obstacles were identified: technical - a teacher is not able to operate the software in the intended way; epistemological - a teacher is not aware of the didactical potential of GeoGebra and howto exploit it in in a way that supports students’ learning of integrals; didactical - a teacher is not aware of the complexity of technology based environments or he/she is aware of this aspect, but not comfortable with his/her competence in carrying out the process of integration of the software into his/her teaching without external help and support. Even if it is difficult to see the software detached from the context in this research, it seems that many of the obstacles perceived by the teachers in the experimental group, as well as difficulties students perceived in their work with the software, were related to the fact that they were inexperienced with the software and, consequently, lacked in knowledge in how to exploit its features in their mathematical activities. As it seems, the teachers would encounter the same obstacles every time they try to integrate a new, to them unfamiliar, software into their teaching practice. Also many of the students would experience same difficulties if they are not adequately supported in this process. Based on this, there are reasons to believe that problems with integration of GeoGebra into mathematics classrooms identified in this research would be similar in relation to integration of other dynamic mathematic software into mathematics classrooms, or even broader, other types of software as e.g. Computer Algebra Systems (CAS), as long as the integration considers the use of an unfamiliar software.

Bakgrund: Granskar man TIMSS 2011 rapport (Skolverket, 2012) ser man att elevernas resultat inom matematik jämfört med andra länder sjunker när det gäller elevernas matematiska förmåga. Av det fyra områdena inom matematik i årskurs fyra är det främst inom Taluppfattning och aritmetik samt Geometri. Sedan TIMSS 2011 publicerades har det förts en debatt kring om man ska undervisa i standardalgoritmer eller i skriftlig huvudräkning. Syfte: Syftet med denna studie är att undersöka på vilket sätt elevernas eget användande av beräkningsstrategier vid lösning av numeriska uppgifter i addition påverkas av lärarens undervisning i skriftlig huvudräkning. Metod: I studien har vi använt oss av både kvalitativa och kvantitativa datainsamlingsmetoder. Den kvalitativa datainsamlingsmetoden bestod av lärarintervjuer och den kvantitativa datainsamlingsmetoden bestod av ett kunskapsprov gällande matematikuppgifter som eleverna skulle besvara. Resultaten för de båda datainsamlingarna analyserades sedan både kvalitativt och kvalitativt. Resultat: Resultatet visar att i utvecklandet av elevernas matematiska förmåga är det viktigare att undervisningen byggs på en begreppslig förståelse än vilka beräkningsstrategier läraren undervisar i.

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In this thesis we investigate the double obstacle problem for p-harmonic functions on metric spaces. We minimize the p-energy integral among all functions which have prescribed boundary values and lie between two given obstacles. This is a generalization of the Dirichlet problem for p-harmonic functions, in which case the obstacles are —∞ and ∞. We show the existence and uniqueness of solutions, and their continuity when the obstacles are continuous. Moreover we show that the continuous solution is p-harmonic in the open set where it does not touch the continuous obstacles. If the obstacles are not continuous, but satisfy a Wiener type regularity condition, we prove that the solution is still continuous. The Hölder continuity for solutions is shown, when the obstacles are Hölder continuous. Boundary regularity of the solutions is also studied. Furthermore we study two kinds of convergence problems for the solutions. First we let the obstacles and the boundary values vary and show the convergence of the solutions. We also consider generalized solutions for insoluble obstacle problems, using the convergence results. Moreover we show that for soluble obstacle problems the generalized solution coincides, locally, with the standard solution. Second we consider an increasing sequence of open sets, with union Ω, and fix the obstacles and the boundary values. We show that the solutions of the obstacle problems in these sets converge to the solution of the corresponding problem in Ω.

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Many testing, estimation and confidence interval procedures discussed in the multivariate statistical literature are based on the assumption that the observation vectors are independent and normally distributed. The main reason for this is that often sets of multivariate observations are, at least approximately, normally distributed. Normally distributed data can be modeled entirely in terms of their means and variances/covariances. Estimating the mean and the covariance matrix is therefore a problem of great interest in statistics and it is of great significance to consider the correct statistical model. The estimator for the covariance matrix is important since inference on the mean parameters strongly depends on the estimated covariance matrix and the dispersion matrix for the estimator of the mean is a function of it. In this thesis the problem of estimating parameters for a matrix normal distribution with different patterned covariance matrices, i.e., different statistical models, is studied. A p-dimensional random vector is considered for a banded covariance structure reflecting m-dependence. A simple non-iterative estimation procedure is suggested which gives an explicit, unbiased and consistent estimator of the mean and an explicit and consistent estimator of the covariance matrix for arbitrary p and m. Estimation of parameters in the classical Growth Curve model when the covariance matrix has some specific linear structure is considered. In our examples maximum likelihood estimators can not be obtained explicitly and must rely on numerical optimization algorithms. Therefore explicit estimators are obtained as alternatives to the maximum likelihood estimators. From a discussion about residuals, a simple non-iterative estimation procedure is suggested which gives explicit and consistent estimators of both the mean and the linearly structured covariance matrix. This thesis also deals with the problem of estimating the Kronecker product structure. The sample observation matrix is assumed to follow a matrix normal distribution with a separable covariance matrix, in other words it can be written as a Kronecker product of two positive definite matrices. The proposed estimators are used to derive a likelihood ratio test for spatial independence. Two cases are considered, when the temporal covariance is known and when it is unknown. When the temporal covariance is known, the maximum likelihood estimates are computed and the asymptotic null distribution is given. In the case when the temporal covariance is unknown the maximum likelihood estimates of the parameters are found by an iterative alternating algorithm and the null distribution for the likelihood ratio statistic is discussed.