Kirsten Bjørkestøl

• Goodchild, Simon; Bjørkestøl, Kirsten; Borge, Inger Christin; Nilsen, Hans Kristian & Tonheim, Odd Helge Mjellem (2021). Educating to inspire active learning approaches in mathematics in Norwegian universities. Nordic Journal of STEM Education. ISSN 2535-4574. 5(1). doi: 10.5324/njsteme.v5i1.3972. Fulltekst i vitenarkiv
• Zakariya, Yusuf F.; Nilsen, Hans Kristian; Bjørkestøl, Kirsten & Goodchild, Simon (2021). Analysis of relationships between prior knowledge, approaches to learning, and mathematics performance among engineering students. International Journal of Mathematical Education in Science and Technology. ISSN 0020-739X. doi: 10.1080/0020739X.2021.1984596.
• Zakariya, Yusuf F.; Nilsen, Hans Kristian; Bjørkestøl, Kirsten & Goodchild, Simon (2020). Singular and combined effects of learning approaches, self-efficacy and prior knowledge on university students’ performance in mathematics. Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education (RUME). ISSN 2474-9346. s. 1003–1009.
• Zakariya, Yusuf F.; Nilsen, Hans Kristian; Bjørkestøl, Kirsten & Goodchild, Simon (2020). Impact of attitude on approaches to learning mathematics: a structural equation modelling approach. I Hausberger, T.; Bosch, M. & Chelloughi, F. (Red.), Proceedings of the Third Conference of the International Network for Didactic Research in University Mathematics. University of Carthage and INDRUM. ISSN 2496-1027. s. 268–277.
• Brodahl, Cornelia; Larson, Niclas; Wathne, Unni & Bjørkestøl, Kirsten (2020). Developing further support for in-service teachers’ implementation of a reasoning-and-proving activity and their identification of students’ level of mathematical argumentation. Journal of the International Society for Teacher Education. ISSN 1029-5968. 24(2), s. 73–87. Fulltekst i vitenarkiv Vis sammendrag

  This is the third in a series of papers focusing reasoning-and-proving. Participants were in-service teachers enrolled in a continuing university education programme in teaching mathematics for grades 5–10. Data were collected from a course assignment in 2018 and 2019, where the in-service teachers reported about their students’ work with a reasoning-and-proving task. Their reports included an identification of the levels the students’ written argumentation reached, based on Balacheff’s taxonomy of proofs. The course assignment’s instructions were expanded for the 2019-cohort. Comparing in-service teachers’ proof level identifications to the researchers’ by statistical analyses, indicated an improvement of the general quality from 2018 to 2019. A higher consensus in 2019 included identifying generic arguments and an understanding that there might be examples falling outside of the taxonomy levels. Qualitative content analysis of the two cohorts’ justifications of their identifications revealed an improved understanding of what is considered generic argumentation. The results encourage and contribute to further developments of the concept.

• Zakariya, Yusuf F.; Nilsen, Hans Kristian; Goodchild, Simon & Bjørkestøl, Kirsten (2020). Assessing first-year engineering students’ pre-university mathematics knowledge: Preliminary validity results based on an item response theory model. Journal of Technology and Science Education (JOTSE). ISSN 2014-5349. 10(2), s. 259–270. doi: 10.3926/jotse.1017. Fulltekst i vitenarkiv
• Zakariya, Yusuf F.; Nilsen, Hans Kristian; Goodchild, Simon & Bjørkestøl, Kirsten (2020). Self-efficacy and approaches to learning mathematics among engineering students: empirical evidence for potential causal relations. International Journal of Mathematical Education in Science and Technology. ISSN 0020-739X. s. 1–16. doi: 10.1080/0020739X.2020.1783006. Fulltekst i vitenarkiv
• Zakariya, Yusuf F.; Bjørkestøl, Kirsten & Nilsen, Hans Kristian (2020). Teacher job satisfaction across 38 countries and economies: An alignment optimization approach to a cross-cultural mean comparison. International Journal of Educational Research. ISSN 0883-0355. 101. doi: 10.1016/j.ijer.2020.101573. Fulltekst i vitenarkiv
• Zakariya, Yusuf F.; Bjørkestøl, Kirsten; Nilsen, Hans Kristian; Goodchild, Simon & Lorås, Madeleine (2019). University students’ learning approaches: An adaptation of the revised two-factor study process questionnaire to Norwegian. Studies in Educational Evaluation. ISSN 0191-491X. 64. doi: 10.1016/j.stueduc.2019.100816. Fulltekst i vitenarkiv
• Zakariya, Yusuf; Goodchild, Simon; Bjørkestøl, Kirsten & Nilsen, Hans Kristian (2019). Calculus Self-Efficacy Inventory: Its Development and Relationship with Approaches to Learning. Education Sciences. ISSN 2227-7102. 9(3), s. 1–14. doi: 10.3390/educsci9030170. Fulltekst i vitenarkiv
• Bjørkestøl, Kirsten; Sivertsen, Edvard & Næs, Tormod (2012). Optimization of two-step batch processes and the method of compensation for random error. Journal of Chemometrics. ISSN 0886-9383. 26(6), s. 311–321. doi: 10.1002/cem.2449.
• Kulinskaya, Elena; Dollinger, Michael B. & Bjørkestøl, Kirsten (2012). On the moments of Cochran's Q statistic under the null hypothesis, with application to the meta-analysis of risk difference. Research Synthesis Methods. ISSN 1759-2879. s. 254–270. doi: 10.1002/jrsm.54.
• Kulinskaya, Elena; Dollinger, Michael B. & Bjørkestøl, Kirsten (2011). Testing for Homogeneity in Meta-Analysis I. The One-Parameter Case: Standardized Mean Difference. Biometrics. ISSN 0006-341X. 67(1), s. 203–212. doi: 10.1111/j.1541-0420.2010.01442.x. Vis sammendrag

  Meta-analysis seeks to combine the results of several experiments in order to improve the accuracy of decisions. It is common to use a test for homogeneity to determine if the results of the several experiments are sufficiently similar to warrant their combination into an overall result. Cochran’s Q statistic is frequently used for this homogeneity test. It is often assumed that Q follows a chi-square distribution under the null hypothesis of homogeneity, but it has long been known that this asymptotic distribution for Q is not accurate for moderate sample sizes. Here, we present an expansion for the mean of Q under the null hypothesis that is valid when the effect and the weight for each study depend on a single parameter, but for which neither normality nor independence of the effect and weight estimators is needed. This expansion represents an order O(1/n) correction to the usual chi-square moment in the one-parameter case. We apply the result to the homogeneity test for meta-analyses in which the effects are measured by the standardized mean difference (Cohen’s d-statistic). In this situation, we recommend approximating the null distribution of Q by a chi-square distribution with fractional degrees of freedom that are estimated from the data using our expansion for the mean of Q. The resulting homogeneity test is substantially more accurate than the currently used test. We provide a program available at the Paper Information link at the Biometrics website http://www.biometrics.tibs.org for making the necessary calculations.

• Bjørkestøl, Kirsten & Næs, Tormod (2005). A Discussion of Alternative Ways of Modeling and Interpreting Mixture Data. Quality Engineering. ISSN 0898-2112. 17(4), s. 509–533. Vis sammendrag

  Artikkelen presenterer og sammenligner ulike modeller for regresjon hvor noen av variablene er blandingskomponenter (mixture components) som kan summeres til 100 %. Tolkningen av parametrene avhenger av den valgte modellen. Det vises at noen av modellene er spesielt nyttige for å få informasjon om mulige effekter. Anbefalinger og advarsler med hensyn til bruk av noe regresjonsprosedyrer in enkelte standard statistikkpakker gis. Metodene kan bli brukt også når data ikke kommer fra en ideelt statistisk design. Artikkelen diskuterer også hvordan det er mulig å finne kombinasjoner av variable som kan gi sluttprodukter med gode prediktive egenskaper. Et eksempel fra karbonindustri der en benytter flere prosess variable og to ulike sett med blandingsvariable benyttes til å illustrere ideene. The article presents and compares different models for regression where some of the variables are mixture components. The interpretations of the parameters depend on the model used. It is shown that some of the models are particularly useful for providing information about possible effects. Recommendations and cautions regarding the use of some regression procedures in some standard statistical software packages are given. The methods can be used when the data do not come from an ideal statistical design. The article also discusses how it is possible to find combinations of the variables that will provide output with good predictive properties. An example from the carbon industry with several process variables and two different mixtures is used to illustrate some of the ideas.

• Bjørkestøl, Kirsten & Dollinger, Michael B. (2001). On the moments of a Stahel-Donoho robust multivariate estimator. Journal of Statistical Computation and Simulation. ISSN 0094-9655. 70, s. 89–106. Vis sammendrag

  The Stahel-Donoho estimators (t,V) of multivariate location and scatter are defined as a weighted mean and a weighted covariance matrix with weights dependant on the "outlyingness" obtained by considering all univariate projections of the data. These estimators are one of the few known multivariate estimators which are affine equivariant and have the robustness property of having maximum possible breakdown point. However little is known about the distributions of these estimators. This article presents the results from simulations a bivariate normally distributed data which show that the moments of both t and V depend substantially on a one dimensional measure of the amount of downweighting of the data. The article investigates this measure of downweighting, and it is suggested that analysis of Stahel-Donoho estimators might be best conditioned on the amount of downweighting. Possible use of Stahel-Donoho estimators in a test is also considered.

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• Borge, Inger Christin; Bjørkestøl, Kirsten; Goodchild, Simon; Nilsen, Hans Kristian & Tonheim, Odd Helge Mjellem (2022). Aktive læringsformer i matematikk - Nasjonal kartlegging av matematikkundervisning i universitets- og høgskolesektoren desember 2019 – januar 2020.
• Borge, Inger Christin; Bjørkestøl, Kirsten; Goodchild, Simon; Nilsen, Hans Kristian & Tonheim, Odd Helge Mjellem (2022). Educating to inspire active learning approaches in mathematics in Norwegian universities - Teaching approaches used in Norwegian higher education mathematics courses .
• Goodchild, Simon; Bjørkestøl, Kirsten; Nilsen, Hans Kristian; Borge, Inger Christin & Tonheim, Odd Helge Mjellem (2020). Active learning - Undersøkelse ved MatRIC i samarbeid med NMR.
• Brodahl, Cornelia; Larson, Niclas; Wathne, Unni & Bjørkestøl, Kirsten (2020). Measuring in-service teachers’ ability to analyse their students’ mathematical argumentation. Vis sammendrag

  This study draws on students’ report from a small project in an EVU-course for in-service teachers (IST) in school years 5–10. The IST should observe their students’ work with a reasoning-and-proving task and analyse their reasoning due to Balacheff’s taxonomy of mathematical proofs. The researchers scrutinised these analyses to categorise which Balacheff levels the IST concluded the students had reached. In addition, three researchers made their own Balacheff categorisation of the students’ reasoning. Thus, there are four categorisations for each student work, one derived from the IST’s analysis and three made by the researchers. Another researcher developed quantitative approaches to measure agreements between these identifications. Our framework for analysis contains nine categories, including Balacheff’s four levels, a precursor for each of these levels, as well as a “level 0” for reasoning falling outside of Balacheff’s framework. The current study draws on data from the project reports in 2018 and 2019. Based on our experiences from 2018, the project description was improved. The reports in 2019 demonstrated an increased proficiency in applying Balacheff’s taxonomy. Comparing their analyses to the researchers’ showed a significantly higher consensus in 2019. This presentation will introduce our statistical analyses of the data from 186 student works, generated by the categorisations made by IST and researchers.

• Bjørkestøl, Kirsten; Borge, Inger Christin; Goodchild, Simon; Nilsen, Hans Kristian & Tonheim, Odd Helge Mjellem (2019). Student-active learning in mathematics: Operationalisation of 'constructive alignment'. Nordic Journal of STEM Education. ISSN 2535-4574. 3, s. 220–224.
• Borge, Inger Christin; Bjørkestøl, Kirsten; Goodchild, Simon; Nilsen, Hans Kristian & Tonheim, Odd Helge Mjellem (2019). Student-active learning in mathematics: Operationalisation of ‘constructive alignment’. Vis sammendrag

  This presentation/paper describes the beginning of a developmental process leading to the creation of a survey instrument to explore the use of active learning approaches in higher education mathematics. The paper begins by arguing a relation between active learning and the conference theme “constructive alignment”. The first stages of the developmental process are based on a “Delphi-study” research design in which the meanings of “active learning” and teaching/approaches that may promote active learning are elicited from a selected group of students, mathematics instructors, mathematicians and mathematics education researchers. The Delphi-study is described and the analysis of outcomes from the first stage of the study are reported in this paper. The conference presentation will also include details from the second stage of the study that is planned at the time of writing. A working definition of “active learning” is established from the study, together with a number of teaching/learning approaches that might promote active learning in mathematics. The paper concludes with reflections on how the outcomes of the study will influence the design and content of the survey instrument. It is hoped the discussion at the conference will also inform the further development of the instrument.

• Bjørkestøl, Kirsten (2018). Matte-2_2017-2018-Prelim-analyses. About studies and podcast.
• Bjørkestøl, Kirsten (2017). Analyser av karaktersetting i MA-115/138, høsten 2016.
• Bjørkestøl, Kirsten (2017). Statistics requested by crime investigators.
• Bjørkestøl, Kirsten (2017). Statistikk etterspurt av etterforskere.
• Bjørkestøl, Kirsten (2016). Om sikkerhet knyttet til andel i en stor gruppe basert på andel i et utvalg.
• Bjørkestøl, Kirsten (2015). Om bruk av matematikk i tolkninger av statistiske modeller.
• Bjørkestøl, Kirsten (2014). Om betingelser og noen paradokser knyttet til sannsynlighet og statistikk.
• Bjørkestøl, Kirsten (2012). Hvor mange observasjoner? - En balanse mellom nivå og styrke for sammenligning av to grupper.
• Bjørkestøl, Kirsten; Borge, Inger Christin; Goodchild, Simon; Nilsen, Hans Kristian & Tonheim, Odd Helge Mjellem (2020). Nasjonal kartlegging av matematikkundervisning i universitets- og høgskolesektoren desember 2019 – januar 2020 - Foreløpig rapport: September 2020. Norsk matematikkråd og MatRIC.
• Bjørkestøl, Kirsten (2005). Modeling and optimization of processes; with main emphasis on two-step strategies Modellering og optimalisering av prosesser; med hovedvekt på to-trinns strategier. Norwegian University of Life Sciences. ISSN 8257506311. Vis sammendrag

  Denne avhandlingen tar for seg ulike metoder som kan benyttes for å få en dypere innsikt og forståelse av faktorer som påvirker prosesser. Avhandlingen består av fem artikler. Den første artikkelen er en teoretisk studie av hvordan det er mulig å lage en robust estimator for senteret og spredningen i et multivariabelt datasett, som for eksempel kan bestå av punkt i et plan istedenfor punkt på ei linje. Estimatorene er robuste i den forstand at de er mindre sårbare for ekstreme observasjoner. Alle observasjoner blir vektet etter hvor ekstreme de er, og disse vektene utnyttes i et vektet gjennomsnitt og i et mål for spredningen i datasettet. Egenskaper for disse estimatorene studeres. Den neste artikkelen ser på ulike måter å modellere sammenhenger på når noen av variablene er blandingskomponenter som kan summeres til 100%. Noen av modellene er egnet til å tolke spesielle typer effekter. De tre siste artiklene tar for seg optimalisering i to-trinns prosesser. Råvarene kommer inn til behandling i det første trinnet, og deretter går produktet videre til en ny behandling i det neste trinnet. Kvaliteten på sluttproduktet blir målt. Denne kvaliteten avhenger både av innstillingene på kontrollvariablene knyttet til behandlingene i to trinnene og av støy (tilfeldig variasjon) knyttet til de to trinnene. Kontrollvariablene skal innstilles slik at sluttproduktet tilfredsstiller et gitt kvalitetskrav, samtidig som kostnaden knyttet til innstillingene blir minimalisert. En spesiell metode blir utviklet for prosesser der det er mulig å foreta en mellommåling mellom de to trinnene. Verdien av denne mellommålingen legges til grunn for hvordan kontrollvariablene i det siste trinnet innstilles. Ved denne strategien blir det mulig å senke kostnadene, samtidigsom det kompenseres for støy slik at sluttproduktet kan bli mer stabilt. The thesis consists of five papers. Two of the papers focus on modeling, and three of the papers focus on optimization. One of the papers on modeling studies a robust multivariate Stahel-Donoho estimator of the location and of the scatter. The paper describes properties of the moments of these estimators, and it is shown that the moments depend on a measure of the amount of downweighting of the data. The other modeling paper compares different models, where some of the variables are mixture components, and some are process variables. Some of the models are particularly useful for providing information about possible effects. The paper discusses how it is possible to find combinations of the variables that will provide good predictive properties. Anexample from the carbon industry with several process variables and two different mixtures is used to illustrate some of the ideas. The three last papers present methods for optimizations in two-step processes. The processes can be described by stochastic models, where some of the variables are control variables that can be used to adjust the output of the processes. Restrictions are given for the final product of the process. The cost of the process is related to the use of the control variables. The optimization problem is to find a strategy for how to the control variables should be set to minimize the cost with the given restrictions of the final product. One type of strategy, here called the combined strategy, treats the process as a whole, and the optimization problem searches for settings of the control variables that can be fixed before the process starts. This strategy can often be treated by standard linear or nonlinear optimization methods. The solution can be found numerically or numerically by software procedures. Different aspects of such processes are compared to another type of strategy, called the adjustable strategy. This strategy utilizes the possibilities of t

• Bjørkestøl, Kirsten & Næs, Tormod (2001). A discussion of alternative ways of modelling and interpreting mixture data. Universitetet i Agder. ISSN 8271174401. Vis sammendrag

  This paper compares different models for regression where some of the variables are mixture components. The focus is on ideas about how to set up a model that is useful for analysing data which do not necessarily come from an ideal statistical design. Recommendations and cautions about the use of som regression procedures in some statistical software are given.

• Bjørkestøl, Kirsten & Dollinger, Michael B. (1999). On the moments of a Stahel-Donoho robust multivariate estimator. Universitetet i Agder. ISSN 8271174126.

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