Unni Wathne

• Wathne, Unni & Carlsen, Martin (2022). Third grade students’ multimodal mathematical reasoning when collaboratively solving combinatorial problems in small groups. Mathematical Thinking and Learning. ISSN 1098-6065. doi: 10.1080/10986065.2022.2099611. Full text in Research Archive
• 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), p. 73–87. Full text in Research Archive Show summary

  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.

• Wathne, Unni; Reinhardtsen, Jorunn; Borgersen, Hans Erik & Cestari, Maria Luiza (2019). Designed examples as mediating tools: Introductory algebra in two Norwegian grade 8 classrooms. In Kilhamn, Cecilia & Säljö, Roger (Ed.), Encountering algebra. A comparative study of classrooms in Finland, Norway, Sweden, and the USA. Springer Nature. ISSN 978-3-030-17576-4. p. 71–109. doi: 10.1007/978-3-030-17577-1_5.
• Wathne, Unni & Brodahl, Cornelia (2019). Engaging Mathematical Reasoning-and-Proving: A Task, a Method, and a Taxonomy. Journal of the International Society for Teacher Education. ISSN 1029-5968. 23(1), p. 6–17. Full text in Research Archive Show summary

  This article is the second paper in a series of papers on studies focusing on teaching mathematical reasoning-and-proving in elementary mathematics classroom. Participants are in-service teachers enrolled in a continuing university education program in mathematics. Results from the first paper suggested the method of imaginary dialogues to have the potential to support in-service teachers in engaging their students in mathematical reasoning-and-proving, and Balacheff’s taxonomy of proofs to support in-service teachers in identifying students’ argumentation. This study is on the following years’ in-service teachers in the program. It examines their perceptions of the usefulness of two constituent parts of this approach, and what insights students’ written dialogues might provide. The study draws on G. J. Stylianides’ analytic framework for reasoning-and-proving. Main data were obtained from a questionnaire taken by 32 in-service teachers and follow-up interviews with four of them. The study reveals engaging students to reason, argue, and prove, while making students’ argumentation visible for teachers was perceived the most useful with imaginary dialogues. The teachers’ increasing awareness of levels of argumentation, was perceived to be the most useful with getting exposed to Balacheff’s distinctions. Keywords: Balacheff’s four levels of proofs, mathematical reasoning-and-proving, written imaginary dialogues

• Brodahl, Cornelia & Wathne, Unni (2018). Imaginary Dialogues – In-service Teachers’ Steps Towards Mathematical Argumentation in Classroom Discourse. Journal of the International Society for Teacher Education. ISSN 1029-5968. 22(1), p. 30–42. Full text in Research Archive Show summary

  The purpose of this qualitative study was to explore in-service teachers’ first experiences with imaginary dialogues – a form of mathematical writing where students are introduced to a written and unfinished dialogue between two imaginary persons discussing a mathematical problem. Students are supposed to continue working with the problem and to complete the initial dialogue between these persons. In-service teachers were enrolled in a continuing university education mathematics course. They were given the task to try out imaginary dialogues in their classes from grades 4 to 10. Based on in-service teachers’ responses in open-ended self-evaluation forms, the study examined how the in-service teachers perceived imaginary dialogues as a tool to approach students’ mathematical argumentation. The study also sought to investigate how they identified levels of argumentation in their students’ written dialogues based on the background of Balacheff’s levels of proofs in school mathematics practices.

• Wathne, Unni & Brodahl, Cornelia (2017). Mathematics Video Podcasts as Integrated Elements of Online Lessons in Further University Education: In-Service Teachers’ Flow Experiences. Journal of the International Society for Teacher Education. ISSN 1029-5968. 21(1), p. 33–44. Full text in Research Archive Show summary

  This case-study examined in-service teachers’ perceptions of learning by means of online mathematic lessons consisting of a mix of text and video podcasts. The investigation is part of further university education directed at practicing teachers in lower secondary schools. The course was a distant education course, with in-service teachers learning online only. The research, based on a series of questionnaires and follow-up interviews, examined whether in-service teachers perceived that video podcasts embedded in online lessons fostered their learning compared to reading similar material. The study focused on efficiency, enjoyment, and concentration as perceived conditions for learning in conjunction with flow theory

• Brodahl, Cornelia & Wathne, Unni (2016). In-Service Teachers’ Perceptions of the Design and Quality of Mathematics Videos in their On-Line Learning. Journal of the International Society for Teacher Education. ISSN 1029-5968. 20(2), p. 67–78. Full text in Research Archive Show summary

  This investigation is part of a continuing education program in mathematics, directed at in-service teachers in lower secondary schools holding teacher certificates. Online mathematics lessons, offered through a distant education course, consisted of a combination of text and video podcasts. University educators’ podcast development was guided by research-based design principles related to e-learning and multimedia instruction. The question arose as to whether in-service teachers enrolled in the course would perceive the podcast design as supportive for their learning. Using questionnaires, this study monitored how in-service teachers perceived podcast quality based on design dimensions. It sought to identify participants’ preferences and their recommendations for video development improvement. Key factors for quality included podcast length and the speaker’s narration. Inservice teachers perceived the podcasts as being useful for their learning processes and indicated efficiency, enjoyment, and concentration as critical learning conditions.

• Wathne, Unni; Reinhardtsen, Jorunn & Cestari, Maria Luiza (2013). Introductory lessons on algebra: A Video Study (Videomat). In Ubuz, Behiye; Haser, Çiğdem & Mariotti, Maria Alessandra (Ed.), Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education. European Society for Research in Mathematics Education. ISSN 978-975-429-315-9. p. 576–577.
• Wathne, Unni (2013). Inquiry in the classroom: Teaching and learning mathematics in alternative training. In Fuglestad, Anne Berit (Eds.), Special Needs Education in Mathematics New Trends, Problems and Possibilities. Portal forlag. ISSN 978 82 92712 98 6. p. 125–130.

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• Carlsen, Martin; Wathne, Unni & Blomgren, Gro (2017). Matematikk for barnehagelærere (3. utgave). Cappelen Damm Akademisk. ISBN 978820248267-1. 325 p.
• Carlsen, Martin; Wathne, Unni & Blomgren, Gro (2012). Matematikk for barnehagelærere, 2. utgave. Cappelen Damm Akademisk. ISBN 978-82-7634-978-8. 288 p.
• Carlsen, Martin; Wathne, Unni & Blomgren, Gro (2011). Matematikk for førskolelærere. Cappelen Damm Høyskoleforlaget. ISBN 9788276348613. 288 p.

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• Brodahl, Cornelia; Larson, Niclas; Wathne, Unni & Bjørkestøl, Kirsten (2020). Measuring in-service teachers’ ability to analyse their students’ mathematical argumentation. Show summary

  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.

• Brodahl, Cornelia & Wathne, Unni (2018). In-service teachers' implementing of imaginary dialogues for mathematical reasoning-and-proving in the classroom. Show summary

  In the context of teacher education, the method of imaginary dialogues, a form of mathematical writing where students continue a written and unfinished dialogue on a mathematical problem, may have the potential to support in-service teachers in engaging their students in mathematical reasoning, as well as Balacheff’s levels of proofs may support in-service teachers in identifying students’ argumentation. Subjects in this exploratory case study are in-service teachers, enrolled in a continuing university education program in mathematics. The study investigates in-service teachers’ implementing imaginary dialogues and adopting Balacheffs’ levels in their identification. It examines their perceptions of the usefulness of this approach, and what insights it might provide. Both quantitative and qualitative methods are applied to collect data. Main data were obtained from a questionnaire taken by 32 in-service teachers and follow-up interviews with four of them. The study reveals engaging students to reason, argument and prove, while getting students’ argumentation visible for teachers, was perceived the most useful with imaginary dialogues in class. Further, the teachers’ increasing awareness of levels of argumentation, was perceived to be the most useful with getting exposed to Balacheff’s distinctions. According to the implementation of imaginary dialogues in the mathematical classroom, the study also indicated challenges to be overcome in order to further develop and implement this approach in teacher education. Teacher educator-directed researcher recommendations for preparing teachers for designing a teaching segment, are included, with focus on mathematical reasoning-and-proving and with imaginary dialogues involved.

• Carlsen, Martin & Wathne, Unni (2017). Primary students’ reasoning when collaboratively solving combinatorial problems in small groups.
• Brodahl, Cornelia & Wathne, Unni (2017). Imaginary dialogues – In-service teachers’ steps towards mathematical argumentation in classroom discourse. Show summary

  The purpose of the study is to explore in-service teachers’ first experiences with imaginary dialogues, a form of mathematical writing where students are introduced to a written and unfinished dialogue between two thought persons discussing a mathematical problem. Students are supposed to continue working with the problem and to complete the initial dialogue between these persons. The dialogue is to be written while the students themselves are exploring and arguing the problem. In-service teachers were enrolled in a continuing university education mathematics online course. They were given the task to try out imaginary dialogue in their classes, from grade 4 to 11 respectively, with their students working in pairs. On the base of in-service teachers’ responses in self-evaluations forms with open-ended questions, the study examines how the in-service teachers perceived the benefits of introducing imaginary dialogues as a tool to approach students’ mathematical argumentation. The study also sought to investigate how they identified types and levels of argumentation in their students’ written dialogues, on the background of Balacheff’s framework of proofs in school mathematics practices. Students’ writing on the arithmetical “handshake problem” served to exemplify teachers’ reasoning on students’ line of argumentation.

• Birkeland, Per Arne; Carlsen, Martin; Erfjord, Ingvald; Hundeland, Per Sigurd & Wathne, Unni (2016). Matematikklærere må kunne mer enn faget. Utdanning. ISSN 1502-9778. p. 38–38.
• Birkeland, Per Arne; Carlsen, Martin; Erfjord, Ingvald; Hundeland, Per Sigurd & Wathne, Unni (2016). Mattelærere må kunne mer enn matte. Aftenposten (morgenutg. : trykt utg.). ISSN 0804-3116. p. 11–11.
• Hundeland, Per Sigurd; Wathne, Unni; Carlsen, Martin; Birkeland, Per Arne & Erfjord, Ingvald (2016). Mattelærere må kunne mer enn matte. Aftenposten (morgenutg. : trykt utg.). ISSN 0804-3116.
• Birkeland, Per Arne; Carlsen, Martin; Erfjord, Ingvald; Hundeland, Per Sigurd & Wathne, Unni (2016). Må kunne mer enn matte. Fædrelandsvennen. ISSN 0805-3790. 142(151), p. 20–20.
• Brodahl, Cornelia & Wathne, Unni (2016). Math video podcasts as integrated parts of online lessons: Trained teacher students' perceptions of effective use and enjoyment in learning. Show summary

  This work examines students’ perceptions of learning by means of online Math lessons that consist of a mix of text and video podcasts. The investigation is part of further university education, directed at working teachers. The course was a distant education course, and the students were only learning online. The exploratory part of the study sought, by using questionnaires, to identify potentials for improvement for the further development of online lessons and video podcast resources. The research part, also based on questionnaires and follow-up interviews with five teacher students, focused on learning with video podcasts as integrated parts of online lessons. The paper discusses concentration, enjoyment and efficiency as perceived conditions for their learning.

• Wathne, Unni (2015). Teachers’ procedures when introducing algebraic expression in two Norwegian grade 8 classrooms. In Krainer, Konrad & Vondrova, Nada (Ed.), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education. European Society for Research in Mathematics Education. ISSN 978-80-7290-844-8. p. 508–509.
• Wathne, Unni (2015). Introducing algebraic expression in two Norwegian Grade 8 classroom.
• Wathne, Unni; Reinhardtsen, Jorunn & Cestari, Maria Luiza (2013). Videomat – Introduction of algebra in Norwegian classrooms.
• Wathne, Unni (2013). Introductory lessons on algebra: A Video Study (Videomat).
• Carlsen, Martin & Wathne, Unni (2011). Matematikk i barnehagen - en undrende og utforskende tilnærming.
• Wathne, Unni (2011). Inquiry in the classroom - Teaching and learning mathematics in alternative training.
• Wathne, Unni (2008). Barns tilnærming til analogiske og kombinatoriske resonnement En longitudinell studie av samspill i smågrupper. Universitetet i Agder. ISSN 978-82-7117-650-1.
• Wathne, Unni (2002). Geometri i byggekunst: fasadeproporsjoner ved bygninger på Sanden i Mandal. Universitetet i Agder.

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