Good mathematics teaching at lower primary school level

This paper explores Norwegian lower primary teachers’ views about good mathematics teaching as revealed in a focus group interview at the end of a two-year school-based professional development program. Analyses of the empirical data indicate three main categories of findings: the teachers’ facilitation of learning, the students’ thinking in and about mathematics and the use of teaching aids in teaching. The results are discussed in relation to other Nordic studies and possible implications are also provided.publishedVersio

The 2-Hessian and sextactic points on plane algebraic curves

In an article from 1865, Arthur Cayley claims that given a plane algebraic curve there exists an associated 2-Hessian curve that intersects it in its sextactic points. In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the 2-Hessian. In addition, we present a formula for the number of sextactic points on cuspidal curves and tie this formula to the 2-Hessian. Lastly, we consider the special case of rational curves, where the sextactic points appear as zeros of the Wronski determinant of the 2nd Veronese embedding of the curve.Comment: Updated version to be published in Mathematica Scandinavica. Substantially rewritten, but with essential results unchanged. Contains results from first author's master's thesis. 21 pages, 2 figure

Introducing teacher students to digital fabrication to support children's mathematical learning

While digital fabrication has been closely linked to narratives of science, technology, engineering, and mathematics (STEM) education and design thinking, it can provide a gateway to reimagining mathematical teaching and deepening the understanding of both mathematical content and pedagogy. In this paper, we present a workshop series that engages teacher students in the creation of their own mathematical manipulatives using digital fabrication tools. Manipulatives are tangible objects that can be used to support children’s learning of mathematical concepts, such as numbers, fractions, or geometry. The workshops included activities for finding, adapting, creating, and sharing manipulatives (FACS) using 2D and 3D modeling and fabrication techniques. Our in-depth analysis of video recordings presents how teacher students successfully acquire digital fabrication skills and reflect on the use of customized manipulatives to support children’s mathematical learning. The findings suggest that introducing digital fabrication in teacher education programs can shift the focus from consuming ready-made manipulatives to creating customized materials that better suit the teaching context. The authors propose FACS as a specific approach for introducing digital fabrication to teacher students.publishedVersio

Making Manipulatives for Mathematics Education

This paper describes the efforts of an interdisciplinary team of researchers as they collaborated to create a digital fabrication curriculum module for mathematics teacher education. The initial four-day workshop design was piloted with five pre-service teachers. The design objective was to introduce digital fabrication techniques joint with mathematical concepts and the design of classroom activities to develop pre-service teachers' technological, pedagogical, and content knowledge. The workshops included activities to find, adapt, create, and share mathematical manipulatives using digital fabrication tools, techniques, and platforms. Manipulatives are tangible objects reifying mathematical concepts and one type of representation used in mathematics teaching. The paper reports on the design process and our design motivations to address contextual constraints and varying levels of exposure to digital fabrication for both pre-service teachers and teacher educators. The developed "find-adapt-create-share" framework for introducing digital fabrication was evaluated through researchers' self-reflection and pre-service teachers' feedback during concluding interviews.publishedVersio

Konkurranseutsetting av kommunal revisjon: En analyse av kostnadsutviklingen i norske kommuner 2000 - 2012

Denne masteroppgaven studerer hvordan organisering av kommunal revisjon påvirker revisjonskostnadene i norske kommuner. Norsk offentlig sektor har de siste tiår vært preget av debatten om ny offentlig styring. Kommuneloven fra 1992 åpnet for større frihet for kommunene til selv å organisere og administrere virksomheten. Staten definerte allikevel i stor utstrekning hva kommunene skal gjøre. Fra 1. juli 2004 ble kommune gitt mulighet til å konkurranseutsette kommunal revisjon etter eget tiltak. Den frihet kommunene har fått til selv å organisere sin virksomhet kombinert med et økende antall lovpålagte plikter innenfor kommunenes ansvarsområde har gitt et generelt større fokus på kontrollarbeid. Oppgavens teoretiske rammeverk er transaksjonskostnadsanalyse. Oppgaven er bygget opp av innsamlet datamaterialet fra 224 norske kommuner, i tillegg benyttes data for undersøkelsen til Åge Johnsen mfl. fra 2000, samt data fra Statistisk sentralbyrå. Resultatene fra undersøkelsen viser at majoriteten av norske kommuner fortsatt velger ulike samarbeidsløsninger når valget på organisasjonsform for den kommunale revisjonen skal tas. I dag har 53 norske kommuner helt eller delvis konkurranseutsatt sin kommunale revisjon. Kommuner som velger å konkurranseutsette sin kommunale revisjon har lavere kostnad per innbygger når kostnadene kontrolleres for kommunestørrelse, organisasjonsform og politisk regime. Kostnader knyttet til kommunal revisjon har gått ned i perioden 2000 til 2012.This master thesis evaluates how deregulation of municipal auditing gives different organizational opportunities which influence the audit costs. Norwegian public sector has in recent decades been characterized by a debate about new public management. The local Government Act was changed in 1992. The reform allowed the local authorities greater freedom to organize and manage the public services. National government still defines the main tasks for the municipalities. In 2004 the local authorities were allowed to reorganize and contract out municipal auditing on their own motion. The freedom local authorities achieved combined with increasing statutory duties has given a generally greater focus on control and audit. The theoretical framework is transaction cost analysis and agent theory. This thesis was put together by collecting data from 224 Norwegian municipalities. In addition to data form the survey conducted by Åge Johnsen et al in 2000, and data form Statistisk sentral byrå. The results from the survey shows that the majority of Norwegian municipalities still prefers to use cooperative organizational structures to perform the local audit. Today 53 Norwegian municipalities have chosen to use the method of contracting out their local audit. The municipalities that have chosen the contracting out method, either hole og partly, have lover cost per capita than the other municipalities, controlled for municipality size, organizational form og political regime. Cost per capita has decreased in the period 2000 to 2012.Master i styring og ledels

Muntlig eksamen - får studentene bedre vist hva de kan? En analyse av åtte studenters forståelse på muntlig eksamen i matematikk: rapport - NFR-prosjektet

Denne rapporten beskriver en kvalitativ studie av åtte informanters muntlige eksamensbesvarelser i det obligatoriske matematikkurset i allmennlærerutdanningen. Forskningsarbeidet er en del av et større prosjekt med midler fra Norges forskningsråd. Forståelse har stått sentralt i studien der problemstillingen har vært: Hva kan muntlig eksamen fortelle oss om studentenes forståelse i den obligatoriske delen av matematikkfaget i allmennlærerutdanningen? I materialet har jeg fokusert på to ulike kjennetegn på forståelse som vises tydelig: 1. studentenes evne til å begrunne og resonnere og 2. å kunne og å anvende definisjoner av begreper i ulike kontekster. Jeg har prøvd å finne noen kjennetegn ved de som forstår godt og de som sliter med forståelse. Tre slike ytterpunkter har jeg med: 1. studenter som tar initiativ contra de som må spørres ut. 2. studenter som har presise begreper contra de som har upresise begreper. 3. studenter som har gode regneferdigheter contra de som har dårlige regneferdigheter. Konklusjonen er at flere studenter har mangler i den grunnleggende forståelsen, samtidig som graden av forståelse varierer fra svært god forståelse og evne til resonnement til dårligere forståelse med fokus på mekanisk læring av formler og framgangsmåter. Muligheten for å stille oppfølgingsspørsmål er et godt argument for å benytte muntlig eksamen som evalueringsform i dette kurse

The professional identities of prospective mathematics teachers who teach through programming

As has been the case in many countries around the world, the new Norwegian curriculum from 2020 included programming as part of mathematics education. However, little is known about how prospective teachers perceive this addition in regard to their developing professional identities. When the results from an electronic survey of 394 prospective teachers showed unexpected findings, five of the subjects were asked to participate in a focus group interview in order to explore some of these results. The focus group interview was conducted to understand how prospective teachers considered the past, present and future aspects of their professional identities as teachers of mathematics through programming. The results reveal that, although the prospective teachers had little experience of programming, they were positive regarding its implementation in mathematics lessons because they identified themselves as digital natives; they therefore believed that learning to program would be easy. They aligned themselves with their students, as masters of technology, in contrast to their future colleagues, whom they implicitly described as digital immigrants. The findings of this study have implications for teacher education. Even if the prospective teachers have a positive attitude toward programming and consider themselves digitally competent, a limited understanding of how programming can be integrated into their mathematics teaching will affect the identities that they see for themselves as teachers who teach mathematics through programming.publishedVersio

Mapping the Kitchen Microbiota in Five European Countries Reveals a Set of Core Bacteria across Countries, Kitchen Surfaces, and Cleaning Utensils

The residential kitchen is often heavily colonized by microbes originating from different sources, including food and human contact. Although a few studies have reported the bacterial composition in cleaning utensils and surface samples there is limited knowledge of the bacterial diversity across different sample types, households, and countries. As part of a large European study, we have identified the microbiota of 302 samples from cleaning utensils (sponges and cloths), kitchen surfaces (sinks, cutting boards, countertops, tap handles, and a pooled sample of other handles) in 74 households across 5 countries (France, Hungary, Norway, Portugal, and Romania). In total, 31 bacterial phyla were identified, with Proteobacteria, Firmicutes, Bacteroidota, and Actinobacteria being the most abundant. Despite large variations in households with respect to kitchen standards, kitchen practices, cleaning regimes, and diet and considerable differences in bacterial diversity between samples, eight bacterial genera/families commonly associated with environmental sources were identified in most samples and defined as a core microbiota: Acinetobacter, Pseudomonas, Enhydrobacter, Enterobacteriaceae, Psychrobacter, Chryseobacterium, Bacillus, and Staphylococcus. These genera/families were also among the bacteria with the highest relative abundance across all samples, in addition to Yersiniaceae, Kocuria, Pantoea, and Streptococcus. Taxa associated with potential pathogens and fecal indicators were low in abundance but broadly distributed throughout the households. The microbial composition of surface samples indicated that the microbial composition on kitchen surfaces is more characteristic for the particular country than the object type, while the microbiota of cleaning utensils was similar across countries but differed between types (sponge or cloth).publishedVersio

Hvem skrives individuell plan for? : En kvalitativ analyse av deltakelse og medvirkning i arbeidet med individuelle planer for døve med ulike funksjonshemninger og/eller psykiske lidelser

Bakgrunn: Norsk politikk for mennesker med funksjonshemming har hatt ”full deltakelse og likestilling” som overordnet mål siden FNs år for funksjonshemmede i 1981. Individuell plan har vært en rettighet for personer med behov for langvarige og koordinerte tjenester siden 2001. Medvirkning og deltakelse er viktige elementer i den individuelle planen. Studien tar for seg hvilke erfaringer medarbeidere i en privat virksomhet har med å realisere medvirkning og deltakelse i arbeidet med individuelle planer. Problemstilling: Hvilke erfaringer har ansatte i en privat virksomhet med realisering av medvirkning og deltakelse i arbeidet med utvikling av individuell plan for døve med psykiske lidelser og/eller ulike funksjonshemminger? Teoretisk referanseramme: Funnene diskuteres i forhold til empowermentteori. Empowermentprosessene retter seg både mot individ- og systemnivå samt relasjonen mellom disse. Metode: Det er valgt en kvalitativ tilnærming til problemstillingen. Det ble gjennomført to fokusgruppeintervjuer med medarbeidere ved virksomheten. Materialet er analysert ved meningsfortetting. Funn og resultater: Den individuelle planen er tjenestemottakerens helhetlige og koordinerte plan. Virksomhetene inngår avtaler med tjenestemottakernes hjemkommuner og den individuelle planen skal sendes hjemkommunen. Medarbeiderne deler erfaringer om hvordan den individuelle planen også får en funksjon som kontraktunderlag. Dette medfører at enkelte tjenestemottakere ikke opplever den grad av medvirkning og deltakelse som medarbeiderne mener er mulig. Dette skyldes to forhold. Språket i planen kan virke fremmedgjørende, slik at planen i mindre grad enn ønskelig blir tilgjengelig for vedkommende som planen gjelder for. Det forekom også at de målene som presenteres i planen ikke har kommet til i samarbeid med brukerne, men etter krav fra en hjemkommune. Konklusjon: Tjenestemottakerens eierforhold til den individuelle planen har betydning for hvordan målet om medvirkning og deltakelse realiseres i arbeidet med planen. Hvis planen benyttes som kontraktunderlag for tildeling av tjenester kan dette også påvirke planens innhold