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  • 201.
    Johansson, Jonas
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Belov, Ilja
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    An experimental setup for validating a CFD model of a double-sided PCB in a sealed enclosure at various power configurations2005In: Thermal, Mechanical and Multi-Physics Simulation and Experiments in Micro-Electronics and Micro-Systems, 2005: EuroSimE 2005, Piscataway, NJ: IEEE Order Department , 2005, p. 127-133Conference paper (Refereed)
    Abstract [en]

    A flexible experimental setup enabling power control of a fully populated double-sided PCB has been realized, and is described in detail. A CFD model of a double-sided PCB housed in a sealed enclosure has been validated in a 19/spl deg/C environment by means of temperature and flow measurement. The difference between simulated and measured component temperatures has been within 10%. Potential errors both in the model and in the experiments have been discussed and their impact on temperatures has been numerically evaluated.

  • 202.
    Johansson, Jonas
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Belov, Ilja
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    CFD Analysis of an Avionic Module for Evaluating Power Distribution as a Thermal Management Measure for a Double-sided PCB2007In: Twenty Third Annual IEEE Semiconductor Thermal Measurement and Management Symposium: SEMI-THERM Proceedings 2007, San Jose, CA USA, March 18-22, 2007, Piscataway, N.J.: IEEE , 2007, p. 233-243Conference paper (Refereed)
    Abstract [en]

    Thermal design aspects of an avionic module including fully populated PCBs housed in a sealed enclosure have been studied by means of computational fluid dynamics. Effect of power distribution between the sides of a double-sided PCB on the case temperature of surface-mounted components has been investigated within a proposed simulation strategy. Simulation-based guidelines have been developed for thermal design of avionic modules, regarding preferable power configuration on a double-sided PCB, representing an alternative approach to thermal management, as compared to introducing additional cooling devices.

  • 203.
    Johansson, Lovisa
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Tre gånger så många eller Tre gånger fler: En kvantitativ studie kring hur elever tolkar vanliga aritmetiska jämförelser2016Independent thesis Advanced level (degree of Master (One Year)), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Mathematics should, as far as possible, be expressed unambiguously – there should be no doubt what is meant with a mathematical expression. There is no reason why arithmetic comparisons should be treated differently. However, there are suggestions that arithmetic comparisons could mean different things, depending on how they are presented.The aim of this study is to investigate how students in the grades 4-6 (age 10-13) interpret arithmetic comparisons in mathematical text assignments. Arithmetic comparisons refer to as proportional relations which describe a relationship, for example between two quantities. Do students separate comparisons as three times as many and three times more? 

    How do students interpret the comparisons double and half? 

    Do the interpretations differ between boys and girls? 

    Do the interpretations differ between students with Swedish as first language and students with Swedish as second language?

    The study is based on a survey in which 188 students participated. The analysis shows that the absolute majority of students in the study interpret times as many and times more as synonyms. Most students also make the generally accepted interpretation of the comparisons double and half.

  • 204.
    Johansson, Lovisa
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Jonsson, Josefine
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Structure sense: Ett matematikdidaktiskt begrepp som håller på att formas2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Structure sense är ett begrepp som nyligen har börjat användas inom matematikdidaktik föratt beskriva elevers förståelse för matematiska strukturer. Syftet med det härexamensarbetet är att utforska begreppet structure sense genom att jämföra olikadefinitioner av begreppet. Vidare är syftet också att jämföra structure sense med tre andraliknande begrepp: symbol sense, personal structure och awareness of mathematical pattern and structure. Litteraturstudien bygger på 13 vetenskapliga publikationer som samlades in ochanalyserades. Samtliga publikationer är skrivna av internationella forskare. I resultatetpresenteras och jämförs structure sense utifrån fyra olika definitioner. De olikadefinitionerna av begreppet är riktade mot olika nivåer av matematik men gemensamt förbeskrivningarna är att structure sense uppfattas som en eller flera förmågor som innefattarett mångfacetterat sätt att uppfatta matematisk struktur.

  • 205.
    Johansson, Malin
    Jönköping University, School of Education and Communication.
    Multiplikativt tänkande eller additivt tänkande?: En kvalitativ studie om hur elever i årskurs 3 löser multiplikativt strukturerade uppgifter2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
  • 206.
    Johansson, Tom
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research.
    Att urskilja det kritiska: En variationsteoretisk studie om undervisning med växande geometriska mönster2019Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Teaching of growing geometrical patterns should be seen as a bridge between arithmetic and algebraic thinking, that through giving the students tasks that enables generalization of arithmetic expressions. Swedish students’ results show that they perform better arithmetically than algebraically therefore, the aim of this study is to ascertain which aspects that are critical to students’ development from an arithmetic thinking to an algebraic thinking.

    This study is a learning study where a lesson was planned and performed in three different classes in grade 5 and 6. The lessons included a pre-test and a posttest to further validate the study and the tests, combined with the lessons, contributed to the result of the study. When planning and analyzing the lessons the theory that was applied was variation theory which focuses on what is learned, what learning that takes place and what can be improved to further the learning.

    The result of this study shows that it can be critical for students to discern regularities in growing geometrical patterns and also to separate regularity from proportionality. Furthermore, the study found another critical aspect which is to discern the meaning of letters within mathematics. The result also reveals two variation patterns which enables discerning of aspects that appear critical for students.

    When teaching about growing geometrical patterns there are several aspects teachers need to make possible for students to discern. Through utilization of variation patterns that are well prepared and thought through these aspects can be visualized and consequently auxiliary advance students understanding for algebra.

  • 207.
    Johansson, Ulf
    et al.
    Jönköping University, School of Engineering, JTH, Computer Science and Informatics, JTH, Jönköping AI Lab (JAIL). Department of Information Technology, University of Borås, Sweden.
    Linusson, Henrik
    Department of Information Technology, University of Borås, Sweden.
    Löfström, Tuwe
    Jönköping University, School of Engineering, JTH, Computer Science and Informatics, JTH, Jönköping AI Lab (JAIL). Department of Information Technology, University of Borås, Sweden.
    Boström, Henrik
    School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Sweden.
    Interpretable regression trees using conformal prediction2018In: Expert systems with applications, ISSN 0957-4174, E-ISSN 1873-6793, Vol. 97, p. 394-404Article in journal (Refereed)
    Abstract [en]

    A key property of conformal predictors is that they are valid, i.e., their error rate on novel data is bounded by a preset level of confidence. For regression, this is achieved by turning the point predictions of the underlying model into prediction intervals. Thus, the most important performance metric for evaluating conformal regressors is not the error rate, but the size of the prediction intervals, where models generating smaller (more informative) intervals are said to be more efficient. State-of-the-art conformal regressors typically utilize two separate predictive models: the underlying model providing the center point of each prediction interval, and a normalization model used to scale each prediction interval according to the estimated level of difficulty for each test instance. When using a regression tree as the underlying model, this approach may cause test instances falling into a specific leaf to receive different prediction intervals. This clearly deteriorates the interpretability of a conformal regression tree compared to a standard regression tree, since the path from the root to a leaf can no longer be translated into a rule explaining all predictions in that leaf. In fact, the model cannot even be interpreted on its own, i.e., without reference to the corresponding normalization model. Current practice effectively presents two options for constructing conformal regression trees: to employ a (global) normalization model, and thereby sacrifice interpretability; or to avoid normalization, and thereby sacrifice both efficiency and individualized predictions. In this paper, two additional approaches are considered, both employing local normalization: the first approach estimates the difficulty by the standard deviation of the target values in each leaf, while the second approach employs Mondrian conformal prediction, which results in regression trees where each rule (path from root node to leaf node) is independently valid. An empirical evaluation shows that the first approach is as efficient as current state-of-the-art approaches, thus eliminating the efficiency vs. interpretability trade-off present in existing methods. Moreover, it is shown that if a validity guarantee is required for each single rule, as provided by the Mondrian approach, a penalty with respect to efficiency has to be paid, but it is only substantial at very high confidence levels.

  • 208.
    Johansson, Ulf
    et al.
    Jönköping University, School of Engineering, JTH, Computer Science and Informatics, JTH, Jönköping AI Lab (JAIL).
    Löfström, Tuve
    Jönköping University, School of Engineering, JTH, Computer Science and Informatics, JTH, Jönköping AI Lab (JAIL).
    Boström, Henrik
    School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Sweden.
    Calibrating probability estimation trees using Venn-Abers predictors2019In: SIAM International Conference on Data Mining, SDM 2019, Society for Industrial and Applied Mathematics, 2019, p. 28-36Conference paper (Refereed)
    Abstract [en]

    Class labels output by standard decision trees are not very useful for making informed decisions, e.g., when comparing the expected utility of various alternatives. In contrast, probability estimation trees (PETs) output class probability distributions rather than single class labels. It is well known that estimating class probabilities in PETs by relative frequencies often lead to extreme probability estimates, and a number of approaches to provide more well-calibrated estimates have been proposed. In this study, a recent model-agnostic calibration approach, called Venn-Abers predictors is, for the first time, considered in the context of decision trees. Results from a large-scale empirical investigation are presented, comparing the novel approach to previous calibration techniques with respect to several different performance metrics, targeting both predictive performance and reliability of the estimates. All approaches are considered both with and without Laplace correction. The results show that using Venn-Abers predictors for calibration is a highly competitive approach, significantly outperforming Platt scaling, Isotonic regression and no calibration, with respect to almost all performance metrics used, independently of whether Laplace correction is applied or not. The only exception is AUC, where using non-calibrated PETs together with Laplace correction, actually is the best option, which can be explained by the fact that AUC is not affected by the absolute, but only relative, values of the probability estimates. 

  • 209.
    Johansson, Ulf
    et al.
    Jönköping University, School of Engineering, JTH, Computer Science and Informatics, JTH, Jönköping AI Lab (JAIL).
    Löfström, Tuve
    Jönköping University, School of Engineering, JTH, Computer Science and Informatics, JTH, Jönköping AI Lab (JAIL).
    Linusson, Henrik
    Högskolan i Borås, Department of Information Technology, Borås, Sweden.
    Boström, Henrik
    The Royal Institute of Technology (KTH), School of Electrical Engineering and Computer Science, Stockholm, Sweden.
    Efficient Venn Predictors using Random Forests2019In: Machine Learning, ISSN 0885-6125, E-ISSN 1573-0565, Vol. 108, no 3, p. 535-550Article in journal (Refereed)
    Abstract [en]

    Successful use of probabilistic classification requires well-calibrated probability estimates, i.e., the predicted class probabilities must correspond to the true probabilities. In addition, a probabilistic classifier must, of course, also be as accurate as possible. In this paper, Venn predictors, and its special case Venn-Abers predictors, are evaluated for probabilistic classification, using random forests as the underlying models. Venn predictors output multiple probabilities for each label, i.e., the predicted label is associated with a probability interval. Since all Venn predictors are valid in the long run, the size of the probability intervals is very important, with tighter intervals being more informative. The standard solution when calibrating a classifier is to employ an additional step, transforming the outputs from a classifier into probability estimates, using a labeled data set not employed for training of the models. For random forests, and other bagged ensembles, it is, however, possible to use the out-of-bag instances for calibration, making all training data available for both model learning and calibration. This procedure has previously been successfully applied to conformal prediction, but was here evaluated for the first time for Venn predictors. The empirical investigation, using 22 publicly available data sets, showed that all four versions of the Venn predictors were better calibrated than both the raw estimates from the random forest, and the standard techniques Platt scaling and isotonic regression. Regarding both informativeness and accuracy, the standard Venn predictor calibrated on out-of-bag instances was the best setup evaluated. Most importantly, calibrating on out-of-bag instances, instead of using a separate calibration set, resulted in tighter intervals and more accurate models on every data set, for both the Venn predictors and the Venn-Abers predictors.

  • 210.
    Jonsson, Denice
    Jönköping University, School of Education and Communication.
    Matematikläromedels erbjudande av symmetri: En kvalitativ och kvantitativ studie anpassad för årskurs 1-3.2018Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The mathematics teaching material is an important part of the Swedish education. It has been proven that mathematics teaching material are been used as an replacement of the curriculum in school. But the publisher does not need to include every part of the curriculum for mathematics. A basic part in geometry is symmetry. Symmetry is described as an own section in the curriculum, but it has not always been so. Therefore, this study is about mathematics teaching materials offers of symmetry. The aim of this study is to examine mathematics teaching materials offer in symmetry for pupils in grade 1-3 and how they match with the curriculum in mathematics. To this study, three research questions have been produced. The first one: Witch symmetry offers in mathematics teaching materials for grade 1-3? The second one: In how big extent presents symmetry in mathematics teaching materials? The third one: In witch way does mathematics teaching material offers pupils to, see, discover, construct and describe symmetry in pictures and in nature? The method that has been used in the study is a quantitative content analysis and a qualitative document analysis, with an affordance theory. With a total of nine series of mathematics teaching materials grade in 1-3 in Swedish school. The analysis shows that pupils are appointed with the possibility to work with mirrorsymmetry, translation, tesselering, symmetric and asymmetric tasks. The analysis also shows that the extent of symmetry task varies in different offers in teaching materials. The analysis also shows that pupils offers to see, discover and construct symmetries. However, it is uncommon that pupils get the opportunity to describe different types of symmetry.

  • 211. Jonsson, Denice
    et al.
    Artursson, Andrea
    Elevers förståelse av symmetri: en litteraturstudie2017Independent thesis Basic level (university diploma), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Dagens matematik är uppbyggd av flera arbetsområden, ett av de här områden är geometri. Inom området geometri finns det flera viktiga begrepp elever ska få möjlighet att utveckla   kunskaper om. Det här arbetet är inriktat mot begreppet symmetri, ett begrepp som barn börjar utveckla förståelse av i tidiga år och som de flesta barn förväntas veta innebörden av. Syftet med den här litteraturstudien är att undersöka hur matematikdidaktisk forskning beskriver barns förståelse av det matematiska begreppet symmetri.

    För att kunna besvara syftet har två forskningsfrågor tagits fram: vilka kunskaper visar barn förståelse av inom symmetri och vilka delar av begreppet visar barn svårigheter för? Därefter har en systematisk litteraturstudie genomförts. Det insamlade materialet har valts ut från databassökningar och kedjesökningar. Materialet består av åtta internationella artiklar (sju tidskriftsartiklar och ett konferensbidrag), publicerade mellan 1984 och 2015.

    Urvalskriterier har använts för att hitta artiklar som svarade på våra forskningsfrågor. Materialet som valts ut ska vara vetenskapligt, inkludera begreppet symmetri och matematikdidaktik i de tidiga skolåren. Resultatet visar att barn har god förståelse av spegelsymmetri när de utför konkreta arbetsuppgifter. Det matematiska språket har en avgörande roll för elever när de ska visa förståelse för begreppet. Missuppfattningar beror vanligtvis på en bristande terminologi. Eleverna visar även svårigheter för oregelbundna geometriska former och när former roteras för att skapa mönster.

  • 212.
    Jonsson, Isabell
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Elevers olika erfarenheter kring subtraktionstecknet: En kvalitativ studie om nio elevers olika användande och förståelse av subtraktionstecknet2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The aim of this study is to investigate how nine pupils in schoolyear 5 experiences the subtraction sign. The study will present and reveal the pupils’ different perceptions and difficulties regarding the subtraction sign. This study has been performed through qualitative interviews by nine pupils describing how they comprehend the subtraction sign in different contexts. The theoretical point of view in this study is life-world phenomenology. When analyzing the study’s datasets, a number of themes has been formed that later are addressed in the result. The result of this study shows that the pupils considers the subtraction sign as to take something away or to decrease a number. The result also shows that the pupils considers subtraction to be difficult, and errors often occurs because of multiple steps during the calculations. Furthermore, the result reveal that the pupils are not aware of their choice regarding methods and strategies when making calculations in subtraction.

  • 213.
    Jonsson, Josefine
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Att strukturera och beräkna matematiska uttryck: En studie om hur elever i årskurs 5 hanterar utvecklade aritmetiska uttryck2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Some of the difficulties students experience in algebra can be due to lack of understanding of the structure in mathematical expressions. Structure, in this context, refers to how a mathematical entity consists of its parts, and how these parts are related to each other. Previous studies also indicate that students’ difficulties in algebra devolve upon a lack of arithmetical knowledge. In arithmetic, students can manage by using informal methods, while algebraic activities require a greater awareness of mathematical structures. It has therefore been argued that students’ difficulties with algebraic expressions are caused by a lack of knowledge of the structure in arithmetic expressions. 

    The purpose of this study is to investigate how 5th grade students calculate and structure longer arithmetic expressions, meaning numerical expressions with several operations, for example, 5 · 6 + 4 · 2 · 3. This study covers numerical expressions with three or four operations. 

    The study includes 116 students from three different schools. The analysis is based on data from solutions of tasks on a written worksheet. The worksheet consisted of ten arithmetic calculation assignments that the students worked with individually. The analysis of the data revealed different approaches that students used to structure and calculate the arithmetic expressions, particularly four methods were used in several tasks. Through the different approaches that students used to calculate mathematical expressions, different ways to create structure could be discovered. Many students based their calculations on the surface structure of an expression and only a few students seemed to be able to identify the hidden structure of an expression. 

  • 214.
    Josefsson, Caroline
    Jönköping University, School of Education and Communication.
    Huvudräkning: En kvalitativ studie om lärares syn på huvudräkningsstrategier2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
  • 215.
    Josefsson, Elin
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Källberg, Paulina
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Elevers och lärares möjligheter samt hinder i arbetet med representationsformer i matematikundervisningen2016Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Under vår verksamhetsförlagda utbildning uppmärksammade vi att eleverna i årskurs 4-6 uppvisade kunskapsluckor i användandet av olika representationsformer. Den representationsform som eleverna föredrog att arbeta med var symboler. Det väcktes ett intresse att undersöka varför eleverna i huvudsak valde att använda en representationsform vid beräkningar. Litteraturstudiens syfte är dels att öka förståelse för elevers och lärares arbete med olika representationsformer, dels att belysa på vilka sätt de olika formerna av representationer kan vara till nytta för elever i årskurserna 4-6 i matematikundervisningen. Studien fokuserar även på att synliggöra eventuella problem som kan uppstå vid användandet av representationsformer och hur lärare på ett effektivt sätt kan motivera elever att använda olika representationsformer. Arbetet baseras på vetenskapliga artiklar samt litteratur som hittats via informationssökningar. Resultatet av litteraturstudien visar att med hjälp av representationsformer kan elever öka sin matematiska förståelse samt utveckla sitt matematiska tänkande. Forskningen redovisade att elever ofta upplever svårigheter gällande växlingen mellan representationsformer men poängterade även att lärare upplever svårigheter med representationsformer på grund av bristande kunskap. Vidare visar resultatet att läraren har en viktig uppgift i att skapa och upprätthålla en positiv inställning till arbetet med representationer. Centralt i matematikundervisningen måste också vara att kommunikation mellan elev-lärare och elev-elev kring representationsformer sker för att gemensamt utveckla och befästa kunskaperna.

  • 216.
    Jönsson, Lukas
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Matematik "på riktigt": En kvalitativ studie av lärares uppfattningar om relationen mellan verkligheten och matematikundervisningen2018Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Mathematics has been used for several thousands of years. Even if the mathematics progresses there is one thing that consist: we still use it in the real world. To work with the connection between the mathematics and the reality is one of the mathematics teachers’ tasks. Previous research is pointing out that the education of mathematics in lower years should aim to prepare the students for higher education studies and an increasingly more demanding society where mathematical knowledge, among them mathematical modeling, is crucial to achieve a deeper connection between the real life and the mathematics. Since the mathematics is so important on several levels and teachers need to be able to motivate their students to work with the subjects that are situated in schools, does the teachers’ own view and attitude towards real-life connected mathematics play a big part.

    The study emanates from the teaching theory Realistic Mathematics Education and aims to contribute with knowledge about how mathematic teachers, who teaches students aged 9-13, can perceive the reality linked to their teaching. This is done by six semi-structured interviews with teachers who are active in teaching mathematics for students aged 9-13. To answer the aim of the study, it emanates from four questions:

    • What do teachers regard as the role of mathematic education in school?

    • How do mathematic teachers use the reality in their teaching?

    • How do mathematic teachers design a task linked to the reality, for their students?

    • How do mathematic teachers reason about a mathematic modeling-task?

    The results shows that teachers are positive towards the use of a real-life connected education in mathematics and that they believe it benefits the students. Several of the teachers in the study associated the term real-life connected teaching in mathematics to larger tasks or projects with the students and could see difficulties in finding the time to plan and execute said activities. When working with tasks that are connected to the reality, the teachers focus more on the process than the final answer. The discussion around the tasks is also something that is also important according the teachers

  • 217.
    Karlsson, Peter
    Jönköping University, Jönköping International Business School, JIBS, Economics.
    On the Asymptotics of Increasing Dimension Models: Methods for Complete or Incomplete Data2009Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    In some multivariate contexts there is a close relation between the number of parameters (p) and the number of observations (n). In a situation where p grows with n it frequently happens that the statistic does not converge to its true parameter. An additional issue is if the data set also contain missing observations and , for example, as p grows with n so does the number of missing observations. This situation arises in the context of empirical applications of the Arbitrage Pricing Theory model, where the data is incomplete due to the nature of stocks leaving or entering the stock exchange. In this thesis two situations of increasing dimension are considered: firstly, the case of complete data sets where the statistic of interest is the inverse covariance matrix where three types of shrinkage estimators of the inverse covariance matrix are investigated, particularly as an ingredient of a composite estimator, specifically Zellners seemingly unrelated regression  models and the Mahalanobis distance. Secondly, the case arising in empirical application of the APT model where the data set is incomplete and the interest is to model the underlying covariance structure among the variables by a few factors. Two possible solutions to the problem are considered and a case study using the Swedish OMX data is conducted for demonstration.

  • 218.
    Karlsson, Peter S.
    Jönköping University, Jönköping International Business School, JIBS, Economics, Finance and Statistics.
    Issues of incompleteness, outliers and asymptotics in high dimensional data2011Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of four individual essays and an introduction chapter. The essays are in the field of multivariate statistical analysis of High dimensional data. The first essay presents the issue of estimating the inverse covariance matrix alone and when it is used within the Mahalanobis distance in High-dimensional data. Three types of ridge-shrinkage estimators of the inverse covariance matrix are suggested and evaluated through Monte Carlo simulations. The second essay deals with incomplete observations in empirical applications of the Arbitrage Pricing Theory model and the interest is to model the underlying covariance structure among the variables by a few common factors. Two possible solutions to the problem are considered and a

    case study using the Swedish OMX data is conducted for demonstration. In the third essay the issue of outlier detection in High-dimensional data is treated. A number of point estimators of the Mahalanobis distance are suggested and their properties are evaluated. In the fourth and last essay the relation between the second central moment of a distribution to its first raw moment is considered in an financial context. Three possible estimators are considered and it is shown that they are consistent even when the dimension increases proportionally to the number of observations.

  • 219.
    Karlsson, Peter S.
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    Point estimators of the Mahalanobis distance in high-dimensional dataManuscript (preprint) (Other academic)
    Abstract [en]

    This paper concerns the problem of estimating the Mahalanobis distance when the dimension of the data matrix is comparable to the sample size. Two different ridge-shrinkage estimators are considered and estimators of related risk functions are derived. The properties of these point estimators are investigated in terms of excess risk and bias relative to the traditional estimator.

  • 220.
    Karlsson, Peter S.
    et al.
    Departments of Economics and Statistics, Linnaeus University, Växjö, Sweden .
    Behrenz, Lars
    Departments of Economics and Statistics, Linnaeus University, Växjö, Sweden .
    Shukur, Ghazi
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    Performances of Model Selection Criteria When Variables are Ill Conditioned2019In: Computational Economics, ISSN 0927-7099, E-ISSN 1572-9974, Vol. 54, no 1, p. 77-98Article in journal (Refereed)
    Abstract [en]

    Model selection criteria are often used to find a “proper” model for the data under investigation when building models in cases in which the dependent or explained variables are assumed to be functions of several independent or explanatory variables. For this purpose, researchers have suggested using a large number of such criteria. These criteria have been shown to act differently, under the same or different conditions, when trying to select the “correct” number of explanatory variables to be included in a given model; this, unfortunately, leads to severe problems and confusion for researchers. In this paper, using Monte Carlo methods, we investigate the properties of four of the most common criteria under a number of realistic situations. These criteria are the adjusted coefficient of determination ((Formula presented.)-adj), Akaike’s information criterion (AIC), the Hannan–Quinn information criterion (HQC) and the Bayesian information criterion (BIC). The results from this investigation indicate that the HQC outperforms the BIC, the AIC and the (Formula presented.)-adj under specific circumstances. None of them perform satisfactorily, however, when the degree of multicollinearity is high, the sample sizes are small or when the fit of the model is poor (i.e., there is a low (Formula presented.). In the presence of all these factors, the criteria perform very badly and are not very useful. In these cases, the criteria are often not able to select the true model. 

  • 221.
    Karlsson, Rebecca
    et al.
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research.
    Kollberg, Olivia
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research.
    Fingrar som verktyg i matematiken under de tidiga skolåren: En litteraturstudie om fingeranvändningens för- och nackdelar2019Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    I denna litteraturstudie belyser vi olika forskares syn gällande fingeranvändningen i matematiken. Syftet med studien var att få en bild av vad forskning lyfter fram om fingeranvändning i matematikundervisningen samt användningens konsekvenser. I studien framkommer både för- och nackdelar med fingeranvändningen gällande elevers talförståelse och aritmetik. Vid datainsamlingen användes databaserna ERIC, MathEduc, Google Scholar, SwePub och Primo för att få fram vetenskapliga publikationer.  För att få fram relevanta publikationer som berör vårt område använde vi oss av inkluderings- och exkluderingskriterier. I resultatdelen jämförs, analyseras och kopplas de olika publikationerna samman. Studien resulterade i att vi fann att fingeranvändningen kan vara både fördelaktig och begränsande. En ståndpunkt som kommit fram genom studien är att fokus ska ligga på hur fingrarna används eftersom det finns både mer och mindre effektiva tillvägagångssätt. Några fördelar som har framkommit är att fingrarna kan ses som ett konkret material samtidigt som det utvecklar del och helhetsrelationer. Fingrarna kan även vara begränsande eftersom de blir otillräckliga när talen överstiger tio och förlorar därmed sin funktion. Forskare är oense om fingeranvändningen har en positiv eller negativ inverkan för elevers matematiska utveckling.

  • 222.
    Karlsson, Rebecka
    Jönköping University, School of Education and Communication.
    Vi hör ihop: Hur elever beräknar numeriska uttryck med sina egenskapade räkneregler.2019Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Two common rules of arithmetic that students learn about in education are the order of operations and the counting from left to right. However, previous research has shown that students also use made-up rules which are not usually used in mathematics. The aim of this study is to investigate the rules of arithmetic created by the students themselves. The aim is achieved by examine what kind of less-known rules of arithmetic that students apply and also how consistent students are in their use of a type of rule.  

    In the study, 55 students did a worksheet consisting of five tasks. In total, 16 of the 55 students used some kind of rule where numbers in the numerical expressions were paired in some way. Furthermore, 13 of the 16 students were interviewed about their way of thinking when solving the tasks. The data therefore consists of the students’ worksheets and transcriptions from the interviews. 

    The study shows that, in addition to the usual conventions left-to-right and order of operations, students use three different kinds of rules of arithmetic. The three rules of arithmetic are based on the principle that numbers are paired in one way or another. Despite that almost none of the 13 students had been taught the conventional rules of arithmetic, most students use own rules that follow logical structures. In addition, the study shows that most students are not particularly consistent when it comes to choosing strategy. Many students choose to use different kind of rules of arithmetic when they are calculating expressions that are structured in almost the same way.

  • 223.
    Karlsson, Rebecka
    et al.
    Jönköping University, School of Education and Communication.
    Linder, Sanna
    Jönköping University, School of Education and Communication.
    Hur elever tillämpar räkneregler och räknelagar på numeriska uttryck2018Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Vi har uppmärksammat att elever har svårt att lösa problem med enkla metoder och strategier. Därmed ser vi ett behov av att undersöka elevers beräkningar av numeriska uttryck eftersom att det ligger till grund för att kunna lösa matematiska problem. Syftet med den här litteraturstudien är att kartlägga hur elever hanterar och beräknar numeriska uttryck genom att besvara följande forskningsfrågor: Vilka uppfattningar finns om hur numeriska uttryck hanteras av elever? Vilka missuppfattningar kan elever ha om de fyra räknesättens egenskaper? I läroplanen (Lgr11) uttrycks det att elever ska få kännedom om grundläggande matematiska begrepp och metoder för att sedan kunna ta dessa i bruk.

    Data som i vårt fall består av vetenskapligt granskad forskning har samlats in genom informationssökning där vetenskapliga texter valdes ut och granskades. Forskningen i studien har visat att det finns svårigheter hos elever om hur numeriska uttryck ska beräknas. Det finns missuppfattningar om hur räknelagar samt räkneregler ska hanteras. Elever har visat sig ha bristande kunskaper om strukturer för numeriska uttryck. Genom en systematisk litteraturstudie kan vi dra slutsatsen att det finns alternativa sätt att undervisa om prioriteringsregeln genom att använda exempelvis bilder. För att eleverna ska få bättre förståelse för räknelagarna bör de även få kännedom om alternativa metoder att beräkna numeriska uttryck med hjälp av räknelagarna. I slutsatsen har vi dragit kopplingar till det sociokulturella perspektivet. 

  • 224.
    Khalaf, Ghadban
    et al.
    Department of Mathematics, King Khalid University, Abha, Saudi Arabia.
    Månsson, Kristofer
    Jönköping University, Jönköping International Business School, JIBS, Economics, Finance and Statistics.
    Shukur, Ghazi
    Jönköping University, Jönköping International Business School, JIBS, Economics, Finance and Statistics.
    Modified Ridge Regression Estimators2013In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 42, no 8, p. 1476-1487Article in journal (Refereed)
    Abstract [en]

    Ridge regression is a variant of ordinary multiple linear regression whose goal is to circumvent the problem of predictors collinearity. It gives up the Ordinary Least Squares (OLS) estimator as a method for estimating the parameters [] of the multiple linear regression model [] . Different methods of specifying the ridge parameter k were proposed and evaluated in terms of Mean Square Error (MSE) by simulation techniques. Comparison is made with other ridge-type estimators evaluated elsewhere. The new estimators of the ridge parameters are shown to have very good MSE properties compared with the other estimators of the ridge parameter and the OLS estimator. Based on our results from the simulation study, we may recommend the new ridge parameters to practitioners.

  • 225.
    Kibria, B. M. Golam
    et al.
    Department of Mathematics and Statistics, Florida International University, Miami, Florida, USA.
    Shukur, Ghazi
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    Månsson, Kristofer
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    A simulation study of some biasing parameters for the ridge type estimation of Poisson regression2015In: Communications in statistics. Simulation and computation, ISSN 0361-0918, E-ISSN 1532-4141, Vol. 44, no 4, p. 943-957Article in journal (Refereed)
    Abstract [en]

    This paper proposes several estimators for estimating the ridge parameter k based for Poisson ridge regression (RR) model. These estimators have been evaluated by means of Monte Carlo simulations. As performance criteria, we have calculated the mean squared error (MSE), the mean value and the standard deviation of k. The first criterion is commonly used, while the other two have never been used when analyzing Poisson RR. However, these performance criterion are very informative because, if several estimators have an equal estimated MSE then those with low average value and standard deviation of k should be preferred. Based on the simulated results we may recommend some biasing parameters which may be useful for the practitioners in the field of health, social and physical sciences.

  • 226.
    Kim Karlsson, Hyunjoo
    et al.
    The Linnaeus University, Växjö, Sweden.
    Karlsson, Peter
    Jönköping University, Jönköping International Business School, JIBS, Statistics. The Linnaeus University, Växjö, Sweden .
    Månsson, Kristofer
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    Sjölander, Pär
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    Wavelet quantile analysis of asymmetric pricing on the Swedish power market2017In: Empirica, ISSN 0340-8744, E-ISSN 1573-6911, Vol. 44, no 2, p. 249-260Article in journal (Refereed)
    Abstract [en]

    In this article we investigate if the Swedish consumer prices for electricity are adjusted equally fast regardless of whether the NordPool power market prices are decreased or increased. Due to relatively moderate variations in the variables, we have applied quantile regression, since it is mainly the large changes (above the median) that essentially tend to have a considerable effect on the consumer prices. Moreover, in order to adjust for stochastic- and deterministic trends, autocorrelation, structural breaks as well as to measure APT effects in the short- and in the medium-run, we apply a wavelet decomposition approach. Our results show evidence that significantly positive asymmetric price transmission (APT) effects exist in this market. More specifically, in the short-run (based on the wavelet decomposition D1 for 1–2 months cycles), we find that that there is a higher propensity to rapidly and systematically increase the consumer prices subsequently to an increase in the NordPool market price, compared with the propensity to decrease their customers prices subsequently to a corresponding drop in the NordPool market prices. However, no significant APT effects were detected in the medium- or in the long-run (i.e. the asymmetric price transmission effects are observed only in the short-run). In summary, we could isolate significant APT effects in the short-run (1–2 months decomposition cycles), and for large changes in the dependent variable (percentiles = 0.9). Therefore, only large changes in the NordPool prices lead to feedback effects in the form of asymmetric price transmission effects. Our evidence supports the notion of firms’ downward stickiness of retail prices for maximizing profit, which are not expected to be found on a fully efficient market. Although our finding shows that the price inefficiency is short-lived, these large temporal inefficiencies are still costly for the consumers. It should be noted that blunt traditional powerless methods do not detect these APT effects, while our wavelet quantile methods are powerful and make a significant contribution in the literature by providing new empirical evidence.

  • 227.
    Klarbring, Anders
    et al.
    Division of Solid Mechanics, Linköping University, Linköping, Sweden.
    Torstenfelt, Bo
    Division of Solid Mechanics, Linköping University, Linköping, Sweden.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Optimal design of fibre reinforced membrane structures2017In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 56, no 4, p. 781-789Article in journal (Refereed)
    Abstract [en]

    A design problem of finding an optimally stiff membrane structure by selecting one–dimensional fiber reinforcements is formulated and solved. The membrane model is derived in a novel manner from a particular three-dimensional linear elastic orthotropic model by appropriate assumptions. The design problem is given in the form of two minimization statements. After finite element discretization, the separate treatment of each of the two statements follows from classical results and methods of structural optimization: the stiffest orientation of reinforcing fibers coincides with principal stresses and the separate selection of density of fibers is a convex problem that can be solved by optimality criteria iterations. Numerical solutions are shown for two particular configurations. The first for a statically determined structure and the second for a statically undetermined one. The latter shows related but non-unique solutions. 

  • 228.
    Klingberg, Ellen
    et al.
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research.
    Andersson, Madeleine
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research.
    Gester - ett verktyg i matematikundervisning: En litteraturstudie2019Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Litteraturstudiens syfte är att genom didaktisk forskning beskriva gesters roll för matematiklärande. För att besvara syftet har två frågeställningar tagits fram: På vilka sätt kan gester användas i matematikundervisning? Hur kan gester hjälpa elever att utveckla förståelse för matematik? 

    Litteraturstudien är genomförd med hjälp av en litteraturanalys av tio vetenskapliga artiklar som sållats fram i en systematisk databassökning. 

    Studierna visar att gester används på olika sätt i klassrummet. Resultatet tyder på att elever tar till sig kunskap oavsett om instruktioner med gester sker av lärare i ett klassrum, är videoinspelade eller förmedlas med avatarer. Flera av studierna förespråkar att använda gester vid inlärning men att urskilja olika gester och använda dem medvetet.

    Slutsatsen är att använda gester kan vara fördelaktigt vid lärande. Gester garanterar dock inte lärande utan behöver sättas i ett sammanhang där det kompletterar talet. 

  • 229.
    Knutsmark, Matilda
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Multiplikativt tänkande: Olika strategier för beräkningar av uppgifter inom multiplikation och division2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    This study focuses on multiplicative thinking among pupils in grade 3. Multiplicative thinking is abstract (Clark & Kamii, 1996), involving applying strategies to solve multiplication and division tasks. The purpose of this study is to examine how pupils use different strategies within multiplicative thinking for multiplication and division. This study was inspired by Grounded Theory. From this theory, a number of semi-structured interviews, observations and analysis of data were made. Eight pupils participated in the interviews, after an initial pilot study. The collected material was based on the pupils' solutions of tasks in multiplication and division, notes from observations of the pupils' solutions and audiotaped interviews. The results show that almost every pupil uses an additive strategy in their solutions of multiplication and division tasks. It also show that only four out of eight pupils could show understanding 0f the connection between the two basic arithmetic operations. From the results, the pupils showed different strategies and solutions within multiplicative thinking, even though they have had the same mathematic education. 

  • 230.
    Koneri, Deepak Kumar
    Jönköping University, School of Engineering, JTH, Computer and Electrical Engineering.
    Optimization of PCB Heater Heating Profile for Power-Efficient Humidity Management in Electronics Enclosures2011Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    In this master thesis, automated optimization simulation laboratory is developed and implemented for power-efficient humidity management. A concept of local heating as a humidity management measure in electronics enclosure in severe environmental conditions has been evaluated. Environmental conditions included an instant change of the temperature from 0o C to 24o C and relative humidity 30% to 95%. Evaluation process has been developed as an optimization problem solved by integrating CFD modeling and excel data sheet calculations into the automation simulation workflow environment. Heating profile vector has been defined as design variable and optimized objective was defined as the difference between surface temperature of heated IC package and local dew point temperature. The task included interfacing between floTHERM and OPTIMUS tools, sensitivity analysis and CFD model simplification in order to obtain time efficient simulation models subsequently used in optimization process. Ultimate goal in the optimization was to obtain a power efficient heating profile for the local heater that keeps local relative humidity at low level thereby minimizing moisture induced electronic reliability problems.

  • 231.
    Kullberg, Angelika
    et al.
    Department of Pedagogical, Curricular and Professional Studies, University of Gothenburg, Gothenburg, Sweden.
    Björklund, Camilla
    Department of Education, Communication and Learning, University of Gothenburg, Gothenburg, Sweden.
    Brkovic, Irma
    Department of Pedagogical, Curricular and Professional Studies, University of Gothenburg, Gothenburg, Sweden.
    Runesson Kempe, Ulla
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research. Wits School of Education, University of the Witwatersrand, Johannesburg, South Africa.
    Effects of learning addition and subtraction in preschool by making the first ten numbers and their relations visible with finger patterns2020In: Educational Studies in Mathematics, ISSN 0013-1954, E-ISSN 1573-0816, Vol. 103, no 2, p. 157-172Article in journal (Refereed)
    Abstract [en]

    In this paper, we report how 5-year-olds’ arithmetic skills developed through participation in an 8-month-long intervention. The intervention program aimed to enhance the children’s ways of experiencing numbers’ part-part-whole relations as a basis for arithmetic skills and was built on principles from the variation theory of learning. The report is based on an analysis of assessments with 103 children (intervention group n = 65 and control group n = 38) before and after the intervention and a follow-up assessment 1 year after the intervention. Our findings show that the learning outcomes of the intervention group were significantly higher compared to those of the control group after the intervention and that differences between the groups remained even 1 year after the intervention. In particular, the results show that children participating in the intervention group learned to recognize and use part-part-whole relations in novel arithmetic tasks.

  • 232.
    Kullberg, Angelika
    et al.
    University of Gothenburg, Sweden.
    Ekdahl, Anna-Lena
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research. Wits School of Education, University of the Witwatersrand, Johannesburg, South Africa.
    Reis, Maria
    University of Gothenburg, Sweden.
    Björklund, Camilla
    University of Gothenburg, Sweden.
    Preschool children’s understanding of numbers shown in a partitioning task2019In: Proceedings of the 43rd Annual Meeting of the International Group for the Psychology of Mathematics Education: Volume 4: Oral Communications and Poster Presentations / [ed] Mellony Graven, Hamsa Venkat, Anthony A. Essien & Pamela Vale, Pretoria, South Africa: PME , 2019, p. 59-59Conference paper (Refereed)
    Abstract [en]

    Children’s ways of handling numbers in arithmetic tasks has been studied extensively, providing us with insights about strategies for solving tasks and development of arithmetic skills (Carpenter & Moser, 1982). Children’s ability to decompose numbers is one important part of development, since the ability allows children to apply different strategies when solving tasks (Hunting, 2003). Children’s different ways of encountering numbers in simple tasks may give a comprehensive understanding of the challenges in learning to use numbers in proficient ways. When experiencing part-part-whole relations of numbers, the child needs to consider the parts and the whole simultaneously. This paper reports on an analysis of 103 individual interviews with 5-year-old children on two occasions during their last year in preschool in Sweden. We report on the analysis of one particular task illustrating children’s experience of numbers when partitioning seven hidden marbles into two parts. The specific research question was: What different ways of experiencing numbers by 5-year-old children were exposed in a partitioning task?

    Variation theory (Marton, 2015) was used to analyse children’s ways of experiencing numbers and what aspects were critical to discern in order to solve the task. Variation theory emanates from more than thirty years of phenomenographic research, investigating different ways in which the same phenomena can be experienced. We found that children experienced numbers in six different ways: as number words, as names, as extents, as countables, as structure, or as known number facts. Our study shows that those ways of experiencing numbers that are foregrounding the cardinal, ordinal and the parts and whole simultaneously end up in plausible answers and the children initiate ways to handle the task in powerful ways. Consequently, if the children experience either the cardinal (e.g. numbers as extent) or the ordinal (e.g. numbers as names) they are not able to decompose the whole and thereby solve the task.

    References

    Carpenter, T. P., & Moser, J. M. (1982). The development of addition and subtraction problem-solving skills. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective. Hillsdale, NY: Lawrence Erlbaum.

    Hunting, R. P. (2003). Part-whole number knowledge in preschool children. Journal of Mathematical Behavior, 22(3), 217-235.

    Marton, F. (2015). Necessary conditions of learning. New York: Routledge.

  • 233.
    Kullberg, Angelika
    et al.
    Department of Pedagogical, Curricular and Professional Studies, University of Gothenburg, Gothenburg, Sweden.
    Vikström, Anna
    Department of Arts, Communication and Education, Luleå University of Technology, Luleå, Sweden.
    Runesson, Ulla
    Jönköping University, School of Education and Communication.
    Mechanisms enabling knowledge production in learning study2019In: International Journal for Lesson and Learning Studies, ISSN 2046-8253, E-ISSN 2046-8261, Vol. 9, no 1, p. 8-91Article in journal (Refereed)
    Abstract [en]

    Purpose: The purpose of this paper is to add to the discussion about practitioner research in schools – by addressing mechanisms and systematic strategies based on theory in a research model, which enables the creation of knowledge products that enhance student learning and are sharable between teachers.

    Design/methodology/approach: The research question is the following: Can a specific form of teachers’ research produce practice-based knowledge relevant beyond the borders of the local school context? This question is addressed through empirical examples from previously published papers on learning studies in natural sciences, mathematics and language.

    Findings: This paper promotes the view that teachers in learning studies can create practical public knowledge relevant beyond their local context. The authors suggest that learning studies and variation theory can offer teachers mechanisms to create such public knowledge.

    Originality/value: The paper proposes that teachers’ collaboration in professional learning communities, as in a learning study, not only has the capacity to increase students’ and teachers’ learning, but it can also be used to create practical public knowledge. 

  • 234.
    Källgren, Zandra
    Jönköping University, School of Education and Communication.
    Huvudräkning: En empirisk studie om lärares uppfattningar av huvudräkningens betydelse2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Syftet med denna studie är att få förståelse för lärares uppfattningar om vilken betydelse huvudräkning har i matematikundervisningen. Studien tar sin utgångspunkt ur fenomenografin där semistrukturerade intervjuer har använts. Fyra lärare har intervjuats på tre olika skolor.

     

    Resultatet visar att lärarnas uppfattningar skiljer sig åt på vilket sätt huvudräkning har betydelse i matematikundervisningen. Två av lärarna menar att huvudräkning är viktigt för utvecklande av matematikkunskaper medan de andra två lärarna ansåg att huvudräkning är ett medel för att nå ett användande av matematik i vardagen. I studien framkom det att lärarna ansåg att huvudräkningsundervisning skulle påbörjas mycket tidigare än vad den oftast gör, gärna redan i förskoleklass. De hade även olika uppfattningar om hur undervisningen skulle bedrivas. Vissa menar att man ska använda de korrekta termerna från början och undervisa om flera strategier samtidigt och plocka upp de strategier som eleverna själva använder. Andra lärare anser att det kan vara bra att utgå från elevernas ordförråd till en början, speciellt om svårigheter finns. De ansåg även att eleverna bör lotsas på de strategier som läraren undervisar om, för att få hållbara strategier. 

  • 235.
    Lantz, Björn
    et al.
    Chalmers University of Technology, Göteborg, Sweden .
    Andersson, Roy
    Jönköping University, School of Engineering, JTH, Industrial Engineering and Management. Jönköping University, School of Engineering, JTH. Research area Industrial Production.
    Manfredsson, Peter
    University of Borås, Sweden.
    Preliminary tests of normality when comparing three independent samples2016In: Journal of Modern Applied Statistical Methods, ISSN 1538-9472, Vol. 15, no 2, p. 135-148Article in journal (Refereed)
    Abstract [en]

    This paper uses simulation to explore the performance of a two-stage procedure where a preliminary Shapiro–Wilk test is used to choose between ANOVA and the Kruskal–Wallis test as three-sample location test. The results suggest that the two-stage procedure actually seems to be preferable when conducting such location tests.

  • 236.
    Lantz, Mathias
    Jönköping University, School of Education and Communication.
    Varför är matematik svårt?: En matematikdidaktisk undersökning inom ämnesområdet algebra i gymnasieskolan2016Independent thesis Basic level (university diploma), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Skolverket (2014) slår fast att svenska elever blir allt sämre i ämnet matematik. Persson (2005) menar att för att lärare ska kunna planera och strukturera undervisningen på ett effektivt sätt behöver de veta vad elever har svårt för. Syftet med denna uppsats är att identifiera några av dessa svårigheter inom en avgränsad del av gymnasieskolans algebraundervisning. Uppsatsens frågeställningar inriktar sig på elevernas uppfattningar av begreppen funktion, ekvation och graf. Frågeställningarna besvarades genom en kvalitativ undersökning i form av djupintervjuer. Intervjuobjekten bestod av sju gymnasieelever som alla undervisats av samma lärare och som läst matematikkursen 2b, vilken är obligatorisk inom ekonomi- och samhällsprogrammet. Av tidigare forskningsresultat framgår det att elever löser uppgifter inom det aktuella området i algebran på olika sätt, vilket kan tolkas som att de har förmåga att lösa uppgifter på olika förståelsenivåer. Uppsatsens resultat visar att det finns en relation mellan vilka uppgifter eleven kan lösa och den förmåga till abstraktion som eleven tycks besitta. Uppsatsens resultat pekar även på att intervjuobjekten löser andragradsekvationer mekaniskt med hjälp av en algoritm, att intervjuobjekten har svårt för det matematiska språket och att de undersökta eleverna uppfattar en relation mellan andragradspolynom och grafer till andradragradsfunktioner. 

  • 237.
    Li, Yushu
    et al.
    Department of Economic and Statistics, Center for Labor Market Policy Research (CAFO), Linaeus University, Växjö, Sweden.
    Shukur, Ghazi
    Jönköping University, Jönköping International Business School, JIBS, Economics.
    Testing for Unit Root Against LSTAR Models: Wavelet Improvement under GARCH Distortion2010In: Communications in statistics. Simulation and computation, ISSN 0361-0918, E-ISSN 1532-4141, Vol. 39, no 2, p. 277-286Article in journal (Refereed)
    Abstract [en]

    In this article, we propose a nonlinear Dickey-Fuller F test for unit root against first-order Logistic Smooth Transition Autoregressive (LSTAR) (1) model with time as the transition variable. The nonlinear Dickey-Fuller F test statistic is established under the null hypothesis of random walk without drift and the alternative model is a nonlinear LSTAR (1) model. The asymptotic distribution of the test is analytically derived while the small sample distributions are investigated by Monte Carlo experiment. The size and power properties of the test were investigated using Monte Carlo experiment. The results showed that there is a serious size distortion for the test when GARCH errors appear in the Data Generating Process (DGP), which led to an over-rejection of the unit root null hypothesis. To solve this problem, we use the Wavelet technique to count off the GARCH distortion and improve the size property of the test under GARCH error. We also discuss the asymptotic distributions of the test statistics in GARCH and wavelet environments.

  • 238.
    Li, Yushu
    et al.
    Department of Economic and Statistics, Center for Labor Market Policy Research (CAFO), Linnaeus University, Sweden .
    Shukur, Ghazi
    Jönköping University, Jönköping International Business School, JIBS, Economics.
    Wavelet Improvement of the Over-rejection of Unit root test under GARCH errors: An Application to Swedish Immigration Data2011In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 40, no 13, p. 2385-2396Article in journal (Refereed)
    Abstract [en]

    In this article, we use the wavelet technique to improve the over-rejection problem of the traditional Dickey-Fuller tests for unit root when the data is associated with volatility like the GARCH(1, 1) effect. The logic of this technique is based on the idea that the wavelet spectrum decomposition can separate out information of different frequencies in the data series. We prove that the asymptotic distribution of the test in the wavelet environment is still the same as the traditional Dickey-Fuller type of tests. The finite sample property is improved when the data suffers from GARCH error. The investigation of the size property and the finite sample distribution of the test is carried out by Monte Carlo experiment. An empirical example with data on the net immigration to Sweden during the period 1950-2000 is used to illustrate the performance of the wavelet improved test under GARCH errors. The results reveal that using the traditional Dickey-Fuller type of tests, the unit root hypothesis is rejected while our wavelet improved test do not reject as it is more robust to GARCH errors in finite samples.

  • 239.
    Liberg, Jesper
    et al.
    Jönköping University, School of Education and Communication.
    Stålknapp, Mathilda
    Jönköping University, School of Education and Communication.
    Matematik i det mångkulturella klassrummet: En litteraturstudie om flerspråkiga elevers problemlösningsförmåga2016Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Forskning visar att flerspråkiga elevers matematikresultat sjunker i takt med att antalet

    flerspråkiga elever ökar i klassrummen. Forskningsresultaten visar att flerspråkiga elever

    framförallt har svårt för problemlösningsuppgifter med mycket text. Frågan är om lärare

    har tillräckligt med kunskaper och resurser för att tillgodose alla elevers behov. Syftet

    med vår litteraturstudie är att undersöka vilka strategier lärare kan använda för att stödja

    flerspråkiga elever. Vidare tar litteraturstudien upp vad som är betydelsefullt för att

    flerspråkiga elever ska kunna utveckla sin problemlösningsförmåga.

    I denna litteraturstudie har vetenskapligt granskade texter sökts och upptäckts via

    användning av olika söktekniker och därefter analyserats. Sökmetoderna vi har använt är

    slumpmässig sökning, systematisk sökning och kedjesökning. I litteraturstudien

    förekommer avhandlingar, konferensbidrag, empiriska undersökningar och artiklar i

    vetenskapligt granskade tidskrifter.

    Resultatet av vår litteraturstudie visar att flerspråkiga elever påverkas av flera olika

    faktorer. Lärare har en betydande roll för att eleverna ska kunna utveckla

    problemlösningsförmågan i ämnet matematik. Vårt resultat tar upp aspekter som lärare i

    ett mångkulturellt klassrum bör ta hänsyn till i undervisningen. De aspekter som tas upp i

    litteraturstudien är huruvida lärare tar hänsyn till elevernas tidigare erfarenheter och

    deras kultur. Ett viktigt begrepp inom detta område som behandlas i litteraturstudien är

    det etnomatematiska perspektivet. Vidare visar litteraturstudiens resultat att de

    flerspråkiga elevernas tillgång till sitt modersmål är en väsentlig del för att utveckla

    matematiska färdigheter. Fortsättningsvis visar resultatet att lärares val av

    kontextanknutna uppgifter är avgörande för att flerspråkiga elever ska kunna se en helhet

    i matematiska textuppgifter. Slutligen visar resultatet att det är viktigt att flerspråkiga

    elever måste ges möjlighet att tala matematik och att lärare belyser viktiga matematiska

    begrepp i klassrummet. 

  • 240.
    Lidberg, Lovisa
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research.
    Lärarkunskaper inom programmering i matematikundervisningen.: En kvalitativ intervjustudie som synliggör lärares kunskaper inom programmering i årskurs 1-3.2019Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Programming is a relatively new element in the school, which teachers should work with. My perception is that teachers have mixed reactions about programming and its content. Therefore, this study was conducted with the aim of creating an understanding of teachers’ knowledge in programming in mathematics teaching in grade 1-3. Three questions were used to achieve the aim of this study, they are: what knowledge and competence do teachers in mathematics have in programming, what knowledge do teachers have of the subject matter in the curriculum in relation to programming, and how teachers work with the content programming in mathematics teaching.

    The study is an interview study that has been used to gather information about the teachers' knowledge in programming. In total, five teachers were interviewed, of whom 4 worked in grades 1-3 and one in grade 4. The theoretical framework MKT were used to collect the data, interview and for the analysis of the data. The result has shown that it is a benefit if the teachers know words and concepts that are used in programming and how different programming tools works. Furthermore, the results shows how the teachers' worked with programming and how it can be used in teaching mathematics.

  • 241.
    Lidberg, Lovisa
    et al.
    Jönköping University, School of Education and Communication.
    Myrta, Zelfi
    Jönköping University, School of Education and Communication.
    IKT i matematikundervisningen: En litteraturstudie om elevers matematiklärande i skolår F-3.2018Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Informations- och kommunikationsteknik (IKT) har successivt ökat under de senaste åren och blivit en större del i skolans matematikundervisning. Kombinationen IKT och matematik är inte speciellt utforskat, vilket vi ställer oss frågade till eftersom digitala verktyg upptar en större del i läroplanen. Därför är syftet med litteraturstudien att belysa hur didaktisk forskning beskriver elevers matematiklärande, genom användning av IKT i matematikundervisningen i skolår F-3. Frågeställningar såsom vilka möjligheter och hinder som finns inom IKT samt vilka kunskaper elever kan utveckla med hjälp av IKT i matematikundervisningen, har skapats för att besvara syftet med studien. De frågeställningarna har använts för att lyfta fram olika perspektiv inom området. 

    De internationella vetenskapliga texterna som utgör grunden för resultatdelen fick vi fram genom att använda olika databaser och sökord på engelska. Litteratursökningen visade på begränsad forskning inom skolåren F-3, därför inkluderades även förskolan samt årskurs 4-9. När materialanalysen genomfördes av de vetenskapliga texterna användes dels forskningsfrågorna, dels en översiktstabell. Resultatet av litteraturstudien visar att IKT skapar möjligheter för elever att bli mer motiverade och engagerade i matematiklärandet, även olika hinder såsom tekniska problem och bristande kunskaper inom IKT kan påverka elevers lärande. Dessutom visar resultatet att IKT ger möjlighet för elever att utveckla matematikkunskaper inom algebra, geometri, problemlösning samt taluppfattning och tals användning. 

  • 242.
    Lindeberg, Ludvig
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research.
    Växande geometriska mönster i skolan: En studie om undervisningsmetoder, svårigheter och utveckling.2019Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    This study investigates how teachers work with growing geometric patterns, with a focus on students aged 10-12 in the west of Sweden. Specifically, it uncovers what methods teachers use and the difficulties they experience when teaching growing geometrical patterns, as well as outlining how they believe teaching in this subject can be developed. 

    Teaching growing geometrical patterns is part of the Swedish curriculum, because it is seen as a pathway to algebra. Current research says that there are two main theories on teaching growing geometrical pattern, hierarchically or dynamically. This study takes a qualitative approach to investigate these theories further. It uses phenomenology as a theoretical framework, as a way to understand the complex phenomenon of teaching. Semi-structured interviews were conducted with four primary school teachers in Sweden. The data gained from these interviews was analysed multiple times using the hermeneutical spiral, to ensure scientific rigour. The findings demonstrate that teachers prefer using hands on methods when teaching growing geometrical patterns to students, such as a method called the “math machine” and using real objects to create patterns. However, they experience difficulties when instructing students how to turn the pattern into a formula, as students often find this progression difficult to grasp. Additionally, the teachers explained that study material has a big impact on how much time teaching growing geometrical patterns is allocated in the classroom, as most of them will only teach this topic if it is covered in the textbook. For the teaching to improve, teachers need more resources regarding how to teach growing geometrical patterns, particularly when it comes to developing formula, and the teaching of the topic needs to be further promoted by the Swedish National Agency for Education.

  • 243.
    Lindeberg, Ludvig
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Johansson, Tom
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Undervisning med växande geometriska mönster: En litteraturstudie om arbetssätt och metoder.2018Independent thesis Basic level (university diploma), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Växande geometriska mönster ses som ett redskap för att få elever från ett aritmetiskt till ett algebraiskt tänk om mönster successivt ersätts med tal och bokstavsbeteckningar. TIMSS- rapporten från 2015 redovisar att svenska elever i årskurs 8 har dåliga resultat inom algebra och geometri men samtidigt är aritmetik ett av de starkaste områdena. Syftet är därför att undersöka vilka teorier om lärande av växande geometriska mönster som finns samt vilka arbetssätt och metoder som används i undervisning om växande geometriska mönster. Studien är en litteraturstudie, där vetenskapligt material analyseras utifrån forskningsfrågor. I denna studie består materialet av tidskriftsartiklar, konferensbidrag och en avhandling.

     

    Resultatet visar på två skilda teorier angående hur undervisning om växande geometriska mönster ska bedrivas, hierarkisk och dynamisk undervisning. Den hierarkiska teorins ståndpunkt är att kunskap byggs från grunden och att gå från det enkla till det komplexa för att elever ska lära sig generalisera. Tvärtemot anser förespråkare för en dynamisk undervisning att elever bör tvingas generalisera redan från början. Resultatet visar också att elever har lättare att generalisera och se samband mellan formel och mönster om de får lära sig analysera det växande geometriska mönstrets delar och samband mellan figurer redan från början.

  • 244.
    Linder, Sanna
    Jönköping University, School of Education and Communication.
    Elevers användning av räknelagar, räkneregler och räknestrategier2019Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    In mathematics, there are laws, rules and strategies of arithmetic. That is why it is important that young students are given the opportunity to develop knowledge about them. The purpose of this study it to investigate whether students in the early school years are aware of the laws, rules and strategies that build the structure of arithmetic. The data for this study is 16 semi-structured interviews with Swedish 1:st grade students.

    The students in this study have, during four sub-tasks, shown how they add three or four numbers. The study shows that students choose numbers according to different principles, place numbers according to different principles and change the placement of the numbers in calculations. Students can also give reasons for why the order of number can be changed. Students used the associative law even though they lack formal education about it. For the various sub-tasks, it is obvious that arithmetic laws, rules and strategies have been used. Particularly several students have shown that they can make generalizations of the commutative and associative law. The conclusion of the study shows that students are aware of the laws, rules and strategies that build the structure of arithmetic expressions. 

  • 245.
    Lindgren, Mats
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Belov, Ilja
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Experimental Evaluation of Glob-top Materials for use in Harsh Environments2005In: Journal of Microelectronics and Electronic Packaging, ISSN 1551-4897, E-ISSN 1555-8037, Vol. 2, no 4, p. 253-268Article in journal (Refereed)
    Abstract [en]

    This article presents results of experimental evaluation of glob-top materials for multi-chip-modules (MCM) in harsh environments. Material and process tests have been performed with the purpose to find a material which would fulfill the reliability requirements for use e.g. in military or automotive applications. Seven polymer materials, i.e. four epoxies, two silicones and one polyurethane material have been selected and evaluated in the experiments. The most critical material and process parameters for glob-top have been identified and measured. Based on the experimental results, application-based scoring of studied epoxy materials has been performed. Material evaluation results have been summarized in conclusions about the most suitable glob-top material for use in harsh environments.

  • 246.
    Lindgren, Mats
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Belov, Ilja
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Thermal and thermo-mechanical analysis for design evaluation of an automotive radar module2004In: Quality and Reliability Engineering International, ISSN 0748-8017, E-ISSN 1099-1638, Vol. 20, no 7, p. 709-726Article in journal (Refereed)
    Abstract [en]

    The influence of the substrate technology, assembly method, and housing material on the thermal, thermo-mechanical and cost performance of a radar module for automotive applications has been studied to address the product reliability aspects during the design phase. Flip chip and wire bonding have been evaluated for Multi-Chip Module—Laminate/Deposition (MCM-L/D) and Multi-Chip Module—Deposition (MCM-D) substrate technologies used for electronic packaging solutions in a harsh environment. Solder ball and direct attachment have been investigated as second-level assembly. As a result of thermal and thermo-mechanical simulations and cost analysis, radar module designs combining MCM-D and MCM-L/D with wire bonding have been revealed, which are preferable for use in different temperature environments with respect to two performance criteria, the maximum junction temperature and the manufacturing cost. Simulation-based guidelines have been developed for designing radar modules used in automotive applications while satisfying temperature and stress constraints provided for the module.

  • 247.
    Lindgren, Mats
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Belov, Ilja
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Törnvall, Magnus
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Application of simulation-based decision making in product development of an RF module2007In: Microelectronics and reliability, ISSN 0026-2714, E-ISSN 1872-941X, Vol. 47, no 2-3, p. 302-309Article in journal (Refereed)
    Abstract [en]

    This paper presents results of simulation-based design evaluation for thermal and thermo-mechanical performance and cost of packaging technology of a RF module for automotive application. Combination of thermal, thermo-mechanical and cost analysis within the multi-attribute decision making enabled design ranking and revealed two MCM-L/D and MCM-D designs with wire bonding assembly preferred for use in automotive applications for different temperature environments. Simulation-based design guidelines were developed for designing electronic modules exhibiting good thermal and thermo-mechanical performance. By application-based partitioning of the importance weights assigned to the reliability and cost criteria, the guidelines were extended to cover other application areas.

  • 248.
    Lindgren, Mats
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Belov, Ilja
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Törnvall, Magnus
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Application of simulation-based decision making in product development of an RF module2004In: 5th IEEE conference in Thermal and Mechanical Simulation and Experiments in Micro-Electronics and Micro-Systems: EuroSimE 2004, 2004, p. 233-240Conference paper (Refereed)
    Abstract [en]

    This paper presents results of simulation-based design evaluation for thermal and thermo-mechanical performance and cost of packaging technology of an RF module for automotive applications. A combination of thermal, thermo-mechanical and cost analysis within the multi-attribute decision making framework enabled design ranking and revealed two MCM-L/D (multi-chip module-laminate/deposition) and MCM-D (deposition) designs with the wire bonding assembly preferred for use in automotive applications for different temperature environments. Simulation-based design guidelines were developed for designing electronic modules exhibiting good thermal and thermo-mechanical performance. By application-based partitioning of the importance weights assigned to the reliability and cost criteria, the guidelines were extended to cover other application areas.

  • 249. Lindqvist, Maria
    et al.
    Myrta, Arbnora
    Elevers strategier för att fortsätta avtagande kvadratiska talföljder2015Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The aim of this paper is to explore the strategies students use when they continue decreasing, quadratic number sequences. The investigated grades are 3 and 6. The study was quantitative and a questionnaire survey was used. The questionnaire consisted of four different tasks. We found that students in both grades used the same strategies, but the distribution of the strategies were different. Our study shows that one strategy was more common than others in grade 3, and this strategy was looking for differences between the numbers. In grade 6 two strategies were more common than the others. These strategies were looking for differences and looking for differences between differences. We also found differences between the pupils way to express the different strategies.

  • 250.
    Lindqvist, Maria
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Myrta, Arbnora
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Elevstrategier i arbetet med mönster och talföljder2014Independent thesis Basic level (university diploma), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Syftet med vår undersökning var att undersöka elevers strategier och hur dessa kan utvecklas i arbetet med mönster och talföljder. Detta undersöktes genom en litteratursökning i olika databaser för att få fram nationella och internationella studier. Den första tilltänkta åldersgruppen var 6-8 år, men den fick utökas till åldrarna 3-15 år, på grund av begränsad forskning i den tidigare tänkta åldersgruppen. Materialet som användes var vetenskapliga publikationer. Resultatet av undersökningen visade att de vanligaste strategierna eleverna använde var: visualisering, hypotes, räknestrategier samt att försöka se samband. Valet av strategier grundar sig på elevernas förkunskaper om mönster och talföljder. Resultatet visade bland annat att även läraren har en stor roll i elevers utveckling av strategier vad gäller arbetet med mönster och talföljder. Detta är viktigt eftersom utvecklingen av strategier i arbetet med mönster och talföljder är en grund för det fortsatta algebraiska tänkandet.

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