Anne Birgitte Fyhn

The fanasváhter is a special tool used by boatbuilders to determine how steeply the boatboards should be positioned. Sámi mathematics teacher educators, together with Sea Sámi boatbuilders and a pre-service mathematics teacher, present a descriptive case study of a Sea Sámi boatbuilder’s use of this tool. The aim is to reveal mathematical understanding that might be relevant for vocational school teaching. Firstly, we analyze a semi-structured interview with a skilled boatbuilder with respect to knowledge and values embedded in the use of the fanasváhter. Because the Sea Sámi boatbuilding tradition was almost extinct after the Nazis’ devastation in World War II, there is a need for some creativity in the boatbuilders’ regeneration of Sea Sámi boatbuilding. An analysis of the use of the fanasváhter with respect to creativity reveals how creativity is important in Sea Sámi boatbuilding. The analysis further reveals that Sea Sámi boatbuilders compare angles, but they do not refer to any angles measured in degrees. This contrasts with traditional school mathematics. Thus, the Sea Sámi boatbuilders’ mathematics is less abstract and more intuitive than traditional school mathematics.

The use of Norwegian numerals in the Sámi language is widespread among Sámi native speakers. Like the Sámi languages and the minority language Kven, Welsh is an endangered minority language in a Western European country with one school system. A study from Wales revealed that children who either spoke Welsh only at home or both at home and at school read and compared two-digit numbers more accurately than monolingual English children. Unlike the Norwegian and English languages, the Sámi, Kven, and Welsh languages have strictly regular counting systems. Analyses of the counting systems for the numerals 11–20 in eight Sámi languages and Kven and comparisons with the counting system in Standard Welsh have resulted in a categorization of the counting systems into three groups regarding transparency and possible support for children’s grouping of ten ‘ones’ into one ‘ten’. The analysis gives reason to believe that reversing the increased use of Sámi and Kven numerals may contribute to Sámi and Kven children’s grasping of the base-10 system because of the counting systems’ transparency. Understanding the base-10 system is fundamental for further learning in school mathematics. Based on the findings, we recommend that Sámi and Kven numerals be included in the mathematics curriculum.

According to Norway’s Core Curriculum all students are to learn about Sámi issues, and the subject of mathematics is not exempt from it. Central to the Sámi tradition is non-standardized individual body-based measuring. Two Sámi pre-service teachers taught Sámi traditional measuring in an urban school where most students are non-Sámi. Their idea was for students to experience the usefulness of using their own bodies as measuring tools. Afterwards, a teacher educator interviewed the pre-service teachers about their experiences. The teaching unit was planned as if it were to be taught in a Sámi school, except that the language of instruction was Norwegian. Analysis shows how the teaching unit aligns with the suggestions and recommendations of Norway’s Truth and Reconciliation Commission. Analysis with respect to the Theory of Didactical Situations framework, TDS, which has not been developed for Indigenous education, show that TDS contributes to improving the teaching unit’s plan. We recommend that Sámi traditional measuring should be taught by Sámi teachers to more non-Sámi students, and we present a revised teaching plan. However, because there is a significant lack of Sámi teachers, we recommend that non-Sámi teachers use our plan as a model for teaching Sámi traditional measuring.

Northern Norway’s population comprises many different cultural groups. According to the Norwegian Education Act, education must give students insight into and a firm foundation in history and culture. This paper aims to present a proposal for how to start working with the creation of local rural mathematics curricula for which Sámi culture in particular, and Northern Norwegian culture in general, is the basis and foundation. It examines three activities that are examples of intangible cultural heritage from different non-urban Northern Norwegian cultures: i) Sámi traditional measuring, ii) fishermen’s traditional navigation at sea and iii) ruvden (a Sámi way of braiding). The activities are analysed with respect to the framework cultural symmetry, which was developed in research in Māori mathematics education. The analysis shows that the three activities are of great significance to local cultural reasoning to such an extent that they should be included in local rural mathematics education. Each of the three activities provides opportunities for developing a culture-based mathematics teaching that values the language and culture in which the activities are embedded. We conclude that cultural symmetry seems to function as a tool for developing a Sámi mathematics curriculum.

Respect and listening are two issues that are complicated to research. This paper presents how respectful listening may constitute one aspect of a teacher's role in child-centered learning. The analysis focus on a teacher's reflections about events that took place after she and a colleague carried out a mathematics teaching unit on culturally diverse children's understanding of 'pattern'. The teacher observed situations that she found interesting and relevant for the children's learning. She communicated with the researcher about this for some months. A closer look at the teacher's reflections caused the research focus to change from what the teacher observed, to how she carried out the observations. So, the research focus is the teacher's application of respectful listening skills when these observations were made. Our analysis reveals the outcomes of two situations. Situation 1 is about communication between the teacher and a child's mother, while situation 2 is about communication between the teacher and a child.

Artikkelen redegjør først for induktive arbeidsformer i de to fagfeltene matematikk og fremmedspråk. Deretter presenterer forfatterne lektorstudenters oppfatning av induktive arbeidsformer i sitt fagfelt. Dagens norske læreplaner vektlegger elevenes evne til å stille spørsmål, utforske og eksperimentere. Utforsking er en form for induktivt arbeid, og forfatterne ser induktivt arbeid som svært relevant for læreplanen. Fjerdeårs lektorstudenter fra matematikk og fremmedspråk deltok i hver sitt fokusgruppeintervju, der tema var studentenes oppfatninger av induktive arbeidsformer i sitt fag. Studentene tilhørte lektorutdanning for 8.–13. skoleår. De hadde avlagt eksamener tilsvarende minst 60 studiepoeng i hvert av to realfag eller i hvert av to fag fra humaniora/samfunnsfag. I forkant av intervjuene hadde studentene en økt med felles undervisning om induktivt arbeid. Artikkelen presenterer en komparativ casestudie av de to intervjuene. Vi gjør rede for svar som gikk igjen i begge gruppene, knyttet til fordeler ved induktive arbeidsformer. Svarene fra begge grupper viser at studentene mener induktive arbeidsformer fører til aktive elever, dybdelæring og forståelse. Én student var kritisk til selve spørsmålet, og dette løfter vi også frem. Studentene var positive til å prøve slikt tverrfaglig arbeid i sin fremtidige jobb, men flere av dem peker på utfordringer ved dette. En oppfølgingsstudie etter noen år kan gi informasjon om hvorvidt studentene har lyktes med å prøve ut induktive arbeidsformer i klasserommet.

Students in Norway and other countries experience vectors as a difficult topic. Four young skilled climbers, who all did well in mathematics at school, participated in the Vector Study (VS). They participated for free and each lesson lasted until the students decided it was over. The idea was to investigate how climbing may function as a basis for students’ development of a vector concept. The teaching goal was the parallelogram law of vector addition. The students investigated what happens when a climber falls. They discussed the situation, and they tested it out in practice. They also performed a supporting activity, a rope-pulling situation, which provides insight into what happens in the climbing situation. The students’ development is analysed by identifying a) Bishop’s six basic activities, b) the role of angles, and c) manipulation of mental objects. The analysis reveals that relations between a vector’s magnitude and direction was central in the students’ investigations. It is important that students develop two aspects of vectors before the parallelogram law of addition is introduced. These are a) relations between angles and vectors, and b) the zero vector.

This paper describes two Sámi mathematics teachers’ development of an innovative instructional practice. Having recognized that Norway's national written exam disadvantaged their students, the teachers developed and established a culturally responsive local oral mathematics exam as part of a five-year research project that took place between 2010 and 2015 in Guovdageaidnu, Norway. The aim of the paper is to illuminate the role of teachers’ autonomy in the process towards Indigenous educational self-determination. We analyse the teachers’ development from a state of recovery to a state of self-determination with respect to a framework consisting of (a) the four states in Smith's (2012) Indigenous research agenda: survival, recovery, development and self-determination; and (b) Deci and Ryan's (2012) distinction between supporting autonomy and controlling behaviour. The teachers’ development of a culturally responsive mathematics exam reflects their development towards self-determination. We draw on data consisting of audio recordings and handwritten notes from meetings between the teachers and researchers.