Third grade students’ multimodal mathematical reasoning when collaboratively solving combinatorial problems in small groups

The aim of this study is to investigate Norwegian primary school students’ multimodal mathematical reasoning when solving combinatorial problems. The data collection took place in four small groups of altogether thirteen 8–9 years old third-graders. Our study shows a variety of approaches used to solve the given combinatorial problems, such as count-all and grouping. These approaches were characterized by the students’ use of inscriptions that displayed all combinations and inscriptions that did not display all combinations. Moreover, the students used gestures such as pointing and sliding. The students’ multimodal reasoning was characterized by the ways their utterances, inscriptions, and gestures emerged and supplemented each other. The students’ pointing gestures mediated the intended mathematical meaning solely when combined with inscriptions displaying all possible combinations. Sliding gestures, on the other hand, did mediate the intended mathematical meaning when their inscriptions were not displaying all possible combinations. In this latter case, the shortcomings of their inscriptions were complemented and compensated by the sliding gestures. The students’ multimodal mathematical reasoning made explicit their combinatorial thinking, mediated the intended mathematical meaning, and facilitated their solving of the given combinatorial problems.publishedVersio

In-Service Teachers’ Perceptions of the Design and Quality of Mathematics Videos in their On-Line Learning

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

Imaginary Dialogues – In-service Teachers’ Steps Towards Mathematical Argumentation in Classroom Discourse

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.publishedVersionNivå

Mathematics Video Podcasts as Integrated Elements of Online Lessons in Further University Education: In-Service Teachers’ Flow Experiences

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 theoryMathematics Video Podcasts as Integrated Elements of Online Lessons in Further University Education: In-Service Teachers’ Flow ExperiencespublishedVersionNivå

Developing further support for in-service teachers’ implementation of a reasoning-and-proving activity and their identification of students’ level of mathematical argumentation

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.publishedVersio

Engaging Mathematical Reasoning-and-Proving: A Task, a Method, and a Taxonomy

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 dialoguesNivå

Teachers' procedures when introducing algebraic expression in two Norwegian grade 8 classrooms

International audienceWe investigate similarities and differences in two teachers’ way of introducing algebraic expressions by designed examples. One teacher moves from the specific to the general, and the other moves from the general to the specific. They both mediate the passage from the students’ real world and the school mathematics they know, to algebra

Third grade students’ multimodal mathematical reasoning when collaboratively solving combinatorial problems in small groups

The aim of this study is to investigate Norwegian primary school students’ multimodal mathematical reasoning when solving combinatorial problems. The data collection took place in four small groups of altogether thirteen 8–9 years old third-graders. Our study shows a variety of approaches used to solve the given combinatorial problems, such as count-all and grouping. These approaches were characterized by the students’ use of inscriptions that displayed all combinations and inscriptions that did not display all combinations. Moreover, the students used gestures such as pointing and sliding. The students’ multimodal reasoning was characterized by the ways their utterances, inscriptions, and gestures emerged and supplemented each other. The students’ pointing gestures mediated the intended mathematical meaning solely when combined with inscriptions displaying all possible combinations. Sliding gestures, on the other hand, did mediate the intended mathematical meaning when their inscriptions were not displaying all possible combinations. In this latter case, the shortcomings of their inscriptions were complemented and compensated by the sliding gestures. The students’ multimodal mathematical reasoning made explicit their combinatorial thinking, mediated the intended mathematical meaning, and facilitated their solving of the given combinatorial problems

Engaging Mathematical Reasoning-and-Proving: A Task, a Method, and a Taxonomy

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 dialogue

Developing further support for in-service teachers’ implementation of a reasoning-and-proving activity and their identification of students’ level of mathematical argumentation

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