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Karlsson, N. & Kilborn, W. (2023). Arithmetic and algebraic knowledge in student learning of concepts. In: Mafalda Carmo, World Institute for Advanced Research and Science (WIARS), Portugal (Ed.), Education and New Development 2023: Volume 1. Paper presented at International Conference on Education and New Developments Lisbon, Portugal, June 24-26, 2023. (pp. 3-7). Lisboa, Portugal: inSciencePress, 1
Open this publication in new window or tab >>Arithmetic and algebraic knowledge in student learning of concepts
2023 (English)In: Education and New Development 2023: Volume 1 / [ed] Mafalda Carmo, World Institute for Advanced Research and Science (WIARS), Portugal, Lisboa, Portugal: inSciencePress , 2023, Vol. 1, p. 3-7Conference paper, Published paper (Refereed)
Abstract [en]

Current research deals with students’ arithmetical and algebraical knowledge, with focus on a conceptual connection, and the relationship between two aspects of knowledge. The content in question is rational numbers, rational equations and problem-solving about proportion and ratio in grades 7, 8 and 9. The method contains three tests given to 400 students in grades 7–9. Tools for analysis were theories of generalizing arithmetic into algebra (Kieran, 2004) and the relationship between arithmetic and algebra in a conceptual context (Kaput, 2008). 

Current research shows that students’ knowledge of algebra and arithmetic often has a limited conceptual connection and a weak relationship. Their knowledge of arithmetic operations and solving rational equations use to be just procedural and rely on formulas learnt in a procedural way and often mixed up. The study also shows that students’ procedural strategies in finding formulas suitable for solving the equations, as well as carrying out the corresponding calculations, were often insufficient.  

The study pays attention to shortcomings in students' conceptualization of arithmetic operations with rational numbers and how to apply them solving rational equations. One reason for this is lack of continuity in instruction and learning.  

Place, publisher, year, edition, pages
Lisboa, Portugal: inSciencePress, 2023
Series
Education and New Developments 2023, ISSN 2184-044X, E-ISSN 2184-1489
Keywords
Rational numbers and algebra, conceptual knowledge, students’ arithmetic and algebraic knowledge
National Category
Mathematics Didactics Educational Sciences
Research subject
Studies in the Educational Sciences
Identifiers
urn:nbn:se:sh:diva-51945 (URN)10.36315/2023v1end001 (DOI)978-989-35106-3-6 (ISBN)
Conference
International Conference on Education and New Developments Lisbon, Portugal, June 24-26, 2023.
Funder
Södertörn University
Available from: 2023-07-08 Created: 2023-07-08 Last updated: 2024-05-20Bibliographically approved
Karlsson, N. & Kilborn, W. (2023). Matematikens didaktik och undervisningens innehåll i årskurs 1-6: guidat lärande med praktiska exempel (1ed.). Lund: Studentlitteratur AB
Open this publication in new window or tab >>Matematikens didaktik och undervisningens innehåll i årskurs 1-6: guidat lärande med praktiska exempel
2023 (Swedish)Book (Other academic)
Abstract [sv]

Den första delen av boken handlar om matematikens didaktik innebörd som är ett tvärvetenskapligt forskningsfält, där möts olika vetenskaper. I boken reder författarna ut  individualiserings-och konkretiseringsprocesser samt vad diagnostisering och kontinuiteten i undervisningen kan innebära. I andra delen av boken beskrivs om hur matematiskt innehåll inom olika områden i skolans matematik för årskurs 1-6 kan utformas och hur undervisningen inom dessa områden kan gå i klassrumspraktiken. 

Place, publisher, year, edition, pages
Lund: Studentlitteratur AB, 2023. p. 164 Edition: 1
Keywords
Matematikens didaktik, metodik, undervisningens innehåll, matematikundervisning
National Category
Didactics Mathematics Educational Sciences
Research subject
Studies in the Educational Sciences
Identifiers
urn:nbn:se:sh:diva-52605 (URN)978-91-44-17767-0 (ISBN)
Available from: 2023-11-04 Created: 2023-11-04 Last updated: 2023-11-06Bibliographically approved
Karlsson, N. & Kilborn, W. (2023). Pre-service teacher' knowledge of mathematics: A framework for sustainable development of student knowledge. In: Mafalda Carmo (Ed.), Education and New Developments 2023: Volume 1. Paper presented at Education and New Developments 2023, Lisboa, Portugal, June 24-26, 2023. (pp. 196-200). Lisboa, Portugal: inSciencePress, 1
Open this publication in new window or tab >>Pre-service teacher' knowledge of mathematics: A framework for sustainable development of student knowledge
2023 (English)In: Education and New Developments 2023: Volume 1 / [ed] Mafalda Carmo, Lisboa, Portugal: inSciencePress , 2023, Vol. 1, p. 196-200Conference paper, Published paper (Refereed)
Abstract [en]

The purpose of this article is to draw attention to, analyze and discuss the following issues: (1) What mathematics should teacher education include, in order for student teachers to gain knowledge of a teaching practice that ensures the progression in students' mathematics development, and (2) How can the subject-specific content in an algebra course for student teachers be designed through an interaction between formal concepts in mathematics and the content of practical mathematics teaching with focus on algebra. An analysis of these issues is carried out within a theoretical framework of didactics of mathematics, related to a research context.

This article is based on two research projects, MIL (Mathematics in teacher education) and SKUM (Student teachers’ knowledge and perceptions of mathematics) as well as ongoing research work with a focus on the quality of student teacher education in mathematics and the didactics of mathematics in the K–3 and 4–6 programs at Södertörn University. 

Place, publisher, year, edition, pages
Lisboa, Portugal: inSciencePress, 2023
Series
Education and New Developments 2023, ISSN 2184-044X, E-ISSN 2184-1489
Keywords
SMK model and pre-service teacher knowledge in mathematics, student teacher learning for teaching, algebra in teacher education, Abelian groups and teaching, and rational numbers and teaching
National Category
Mathematics Didactics Educational Sciences
Research subject
Studies in the Educational Sciences
Identifiers
urn:nbn:se:sh:diva-51947 (URN)978-989-35106-3-6 (ISBN)
Conference
Education and New Developments 2023, Lisboa, Portugal, June 24-26, 2023.
Funder
Södertörn University
Available from: 2023-07-08 Created: 2023-07-08 Last updated: 2023-12-05Bibliographically approved
Karlsson, N. & Kilborn, W. (2023). The impact of teachers' matter knowledge on students' learning of rational numbers and proportion. In: Mafalda Carmo (Ed.), Education Applications & Developments VIII: Advances in Education and Educational Trends (pp. 13-24). Lisboa, Portugal: inSciencePress
Open this publication in new window or tab >>The impact of teachers' matter knowledge on students' learning of rational numbers and proportion
2023 (English)In: Education Applications & Developments VIII: Advances in Education and Educational Trends / [ed] Mafalda Carmo, Lisboa, Portugal: inSciencePress , 2023, p. 13-24Chapter in book (Refereed)
Abstract [en]

This study examines the impact of teachers’ subject matter knowledge on students’learning. The mathematical content deals with rational number as fractions andproportion. The study includes pre- and post-tests from 99 students, classroomobservations, students’ written solutions and interviews with 48 selected studentsafter the post-tests. Findings from this study show that the impact of teachers’ subjectmatter knowledge and ability to identify the objects of learning, and apply this inteaching, strongly influenced the development of students’ conceptual learning aboutfractions and proportion.

Place, publisher, year, edition, pages
Lisboa, Portugal: inSciencePress, 2023
Series
Education Applications & Developments, ISSN 2183-2978, E-ISSN 2184-0210 ; VIII
Keywords
teachers’ subject matter knowledge, fraction and proportion, students’ learning
National Category
Mathematics Didactics Educational Sciences
Research subject
Studies in the Educational Sciences
Identifiers
urn:nbn:se:sh:diva-51942 (URN)10.36315/2023eadVIII02 (DOI)978-989-53614-6-5 (ISBN)
Funder
Södertörn University
Available from: 2023-07-08 Created: 2023-07-08 Last updated: 2024-05-28Bibliographically approved
Karlsson, N. & Kilborn, W. (2023). Towards a generalization: what students learn about multiplication. In: Mafalda Carmo (Ed.), Education Applications & Developments VIII: Advances in Education and Educational Trends (pp. 51-62). Lisboa, Portugal: inSciencePress
Open this publication in new window or tab >>Towards a generalization: what students learn about multiplication
2023 (English)In: Education Applications & Developments VIII: Advances in Education and Educational Trends / [ed] Mafalda Carmo, Lisboa, Portugal: inSciencePress , 2023, p. 51-62Chapter in book (Refereed)
Abstract [en]

This chapter examines and analyzes students’ learning about aspects of the concept of multiplication with a focus on perceptions and representations, and how they apply this to handling multiplicative situations and patterns in the multiplication tables. The analysis has been performed in the context of the generalization process related to teaching activities, with a focus on students’ perception of multiplication. The theoretical approach is based on Davydov’s (1990) view of theoretical generalization as a perception-conception-elementary concept (PCE model). The current mathematical content was classified according to: (1) multiplicative structures (Vergnaud, 1983);and (2) basic laws of algebra (van der Waerden, 1971). The relationship between students’ learning and the teaching process was studied in order to identify students’ learning in action. The study comprises two teachers and 40 students in two classes in grade 3 and was followed up two years later in grade 5 with one teacher and 25 students. The findings of this study can provide knowledge about students’ learning about multiplication using structures and multiplication tables in a conceptual context.

Place, publisher, year, edition, pages
Lisboa, Portugal: inSciencePress, 2023
Series
Education Applications & Developments, ISSN 2183-2978, E-ISSN 2184-0210
Keywords
multiplication, multiplicative structures, basic laws of algebra, teaching and learning multiplication
National Category
Mathematics Didactics Educational Sciences
Research subject
Studies in the Educational Sciences
Identifiers
urn:nbn:se:sh:diva-51943 (URN)978-989-53614-6-5 (ISBN)
Funder
Södertörn University
Available from: 2023-07-08 Created: 2023-07-08 Last updated: 2023-07-10Bibliographically approved
Karlsson, N. & Kilborn, W. (2022). Teachers' mathematical content knowledge and students' progression in learning of fraction and proportion. In: Mafalda Carmo (Ed.), Education and New Developments 2022: Volyme 1. Paper presented at Education and New Developments 2022, Madeira island, June 18-20, 2022. (pp. 7-11). Lisboa: inScience Press, 1
Open this publication in new window or tab >>Teachers' mathematical content knowledge and students' progression in learning of fraction and proportion
2022 (English)In: Education and New Developments 2022: Volyme 1 / [ed] Mafalda Carmo, Lisboa: inScience Press, 2022, Vol. 1, p. 7-11Conference paper, Published paper (Refereed)
Abstract [en]

Proportional reasoning causes considerable difficulty for students. One reason for this is, that a lack of basic understanding of fractions in earlier years, causes difficulties in the middle years. Moreover, learning of fractions and proportion is a long-term process and students encounter it continuously from grade 1 to 9. It is also difficult to teach, and teachers’ mathematical content knowledge plays a crucial role in students’ learning of mathematics. The purpose of this study is to analyse the effects of teaching, and the influence of teachers’ mathematical content knowledge about fractions and proportion on students learning in progression from grade 4 and 5 to grade 8. The method contains pre- and post-tests for 86 students, classroom observations, checking students’ written solutions and interviews of 35 selected students after the post-test. The intervention includes construction of educational materials (EM), teachers participating in seminars-training related to EM and teacher’s implementation of EM. The EM contains tasks for students and a teachers’ guide with aims and goals for teaching and a theoretical background. Tools for analysis were a methodological design, Variation theory, and a theoretical approach, Mathematical Content Knowledge for Teaching. In focus of the analysis was students learning in progression, related to variation and crucial aspects of learning. Findings from this study shows that teachers’ mathematical content knowledge and their ability to identify the objects of learning and apply this in teaching is very important for students’ learning and progression in their learning. Most students showed an ability to learn, but their performance was intimately linked to teachers’ perception of the crucial aspects in teaching, and variation. Moreover, anomalies in students’ perceptions of basic concepts caused obstacles in their learning. Some anomalies seem to have followed students from middle School to grade 8. Finally, the study illustrated how anomalies arise if misconceptions are not noticed by teachers. The outcome of the study can explain more about crucial steps in teaching and learning of fraction and proportion. The study pay attention to challenges in mathematics teaching.

Place, publisher, year, edition, pages
Lisboa: inScience Press, 2022
Series
Education and new developments, ISSN 2184-044X, E-ISSN 2184-1489
Keywords
Mathematical Content Knowledge, fraction and proportion, variation, progression and learning
National Category
Mathematics Didactics Pedagogy
Research subject
Studies in the Educational Sciences
Identifiers
urn:nbn:se:sh:diva-49208 (URN)978-989-53614-3-4 (ISBN)
Conference
Education and New Developments 2022, Madeira island, June 18-20, 2022.
Note

The project has been funded by School of Natural Sciences, Technology and Environmental Studies, Södertörn university 

Available from: 2022-06-13 Created: 2022-06-13 Last updated: 2022-07-07Bibliographically approved
Karlsson, N. & Kilborn, W. (2022). Teaching and learning the multiplication table by using multiplicative structures: Variation and crutial patterns. In: Mafalda Carmo (Ed.), Education and New Developments 2022: Volyme 1. Paper presented at Education and New Developments 2022, Madeira island, June 18-20, 2022. (pp. 45-49). Lisboa: inScience Press, 1
Open this publication in new window or tab >>Teaching and learning the multiplication table by using multiplicative structures: Variation and crutial patterns
2022 (English)In: Education and New Developments 2022: Volyme 1 / [ed] Mafalda Carmo, Lisboa: inScience Press, 2022, Vol. 1, p. 45-49Conference paper, Published paper (Refereed)
Abstract [en]

This paper examines and analyzes how students learn multiplication tables, specifically the role of multiplicative structures and how these are used as students learn to master the tables. The analysis is performed in the context of the generalization process related to the teaching activity focusing students’ perception of concepts. The theoretical approach applies Davydov’s concept of theoretical generalization as perception-conception-elementary concept (PCE model) and Vergnaud’s theory of multiplicative structures in three classes: mapping rule (MR), multiplicative comparison (MC), and Cartesian product (CP). For the methodological design, Marton’s variation theory has been chosen.

This study includes two teachers and 40 students in two Year 3 classes, followed two years later by one teacher and 25 students in one Year 5 class. The analysis of the outcome is based on documented classroom observations, one-on-one interviews with students and teachers’ reflections on students’ learning outcomes. The conclusion of the study is that the generalization of multiplication is a difficult process for students, especially in the classes MC and PC, and one that sometimes results in challenges to identifying multiplicative situations and relating these to the multiplication tables. This illustrates that teaching activities and teachers’ support are necessary conditions for students’ learning. The study also shows that multiplicative structures can help students to find and systematize crucial patterns in the multiplication table, allowing them to learn the multiplication table in a more efficient and structured manner. During the one-on-one interviews, students actively searched for and found structures and solutions that did not come up during lessons. This shows that multiplicative structures are a suitable didactic tool for identifying patterns in multiplication tables, thereby facilitating learning other than by rote.

Place, publisher, year, edition, pages
Lisboa: inScience Press, 2022
Series
Education and new developments, ISSN 2184-044X, E-ISSN 2184-1489
Keywords
Multiplicative structures, multiplication table, generalization, student’s perception, elementary basic concepts.
National Category
Mathematics Didactics Pedagogy
Identifiers
urn:nbn:se:sh:diva-49209 (URN)10.36315/2022v1end010 (DOI)978-989-53614-3-4 (ISBN)
Conference
Education and New Developments 2022, Madeira island, June 18-20, 2022.
Note

The project has been funded by School of Natural Sciences, Technology and Environmental Studies, Södertörn university 

Available from: 2022-06-13 Created: 2022-06-13 Last updated: 2024-05-21Bibliographically approved
Karlsson, N. & Kilborn, W. (2021). Development and progression in students' experience of fractions and proportion. In: Maitree Inprasitha; Narumon Changsri; Nisakorn Boonsena (Ed.), Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1): . Paper presented at 44th Conference of The International Group for The Psychology of Mathematics Education, Khon Kaen, Thailand [DIGITAL], July 19-22, 2021. (pp. 150). Khon Kaen, Thailand: PME, 1
Open this publication in new window or tab >>Development and progression in students' experience of fractions and proportion
2021 (English)In: Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1) / [ed] Maitree Inprasitha; Narumon Changsri; Nisakorn Boonsena, Khon Kaen, Thailand: PME , 2021, Vol. 1, p. 150-Conference paper, Oral presentation with published abstract (Refereed)
Abstract [en]

In a recent study we followed the teaching and learning of fractions and proportion with focus on how students develops their knowledge from grade 4 to grade 8. Another focus was on teachers’ ability to identify the object of learning and its crucial aspects.

A qualitative analysis of the date shows that the students often failed in understanding crucial aspects of fractions and proportion and for that reason they made systematic mistakes later on. One reason for this appeared to be teachers’ lack of subject matter knowledge and an insufficient focus on the object of teaching. This became most evident in grade 8, where students’ misconceptions of fractions and extending of fractions, forced the teachers to introduce procedural methods like cross-multiplication. However, such teaching methods caused still more confusion and new misconceptions.

Place, publisher, year, edition, pages
Khon Kaen, Thailand: PME, 2021
Keywords
Mathematics education, Rational numbers and Proportion in teaching and learning, Variation and Learning, Teachers Content Knowledge
National Category
Mathematics Didactics Educational Sciences
Research subject
Other research area; Studies in the Educational Sciences
Identifiers
urn:nbn:se:sh:diva-46203 (URN)978-616-93830-0-0 (ISBN)
Conference
44th Conference of The International Group for The Psychology of Mathematics Education, Khon Kaen, Thailand [DIGITAL], July 19-22, 2021.
Available from: 2021-08-17 Created: 2021-08-17 Last updated: 2021-10-01Bibliographically approved
Klang, N., Karlsson, N., Kilborn, W., Eriksson, P. & Karlberg, M. (2021). Mathematical Problem-Solving Through Cooperative Learning - The Importance of Peer Acceptance and Friendships. Frontiers in Education, 6, Article ID 710296.
Open this publication in new window or tab >>Mathematical Problem-Solving Through Cooperative Learning - The Importance of Peer Acceptance and Friendships
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2021 (English)In: Frontiers in Education, E-ISSN 2504-284X, Vol. 6, article id 710296Article in journal (Refereed) Published
Abstract [en]

Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional approaches supporting student learning in this area. This study aims to contribute to previous research by studying the effects of an instructional approach of cooperative learning on students’ mathematical problem-solving in heterogeneous classrooms in grade five, in which students with special needs are educated alongside with their peers. The intervention combined a cooperative learning approach with instruction in problem-solving strategies including mathematical models of multiplication/division, proportionality, and geometry. The teachers in the experimental group received training in cooperative learning and mathematical problem-solving and implemented the intervention for 15 weeks. The teachers in the control group received training in mathematical problem-solving and provided instruction as they would usually. Students (269 in the intervention and 312 in the control group) participated in tests of mathematical problem-solving in the areas of multiplication/division, proportionality, and geometry before and after the intervention. The results revealed significant effects of the intervention on student performance in overall problem-solving and problem-solving in geometry. The students who received higher scores on social acceptance and friendships for the pre-test also received higher scores on the selected tests of mathematical problem-solving. Thus, the cooperative learning approach may lead to gains in mathematical problem-solving in heterogeneous classrooms, but social acceptance and friendships may also greatly impact students’ results. 

Place, publisher, year, edition, pages
Frontiers Media S.A., 2021
Keywords
Cooperative learning, Mathematical problem-solving, Intervention, Heterogeneous classrooms, Hierarchical linear regression analysis
National Category
Pedagogy Didactics Mathematics
Research subject
Studies in the Educational Sciences
Identifiers
urn:nbn:se:sh:diva-46269 (URN)10.3389/feduc.2021.710296 (DOI)000728833400001 ()2-s2.0-85114439569 (Scopus ID)
Funder
Swedish Research Council, 2016-04,679
Available from: 2021-08-24 Created: 2021-08-24 Last updated: 2022-09-15Bibliographically approved
Karlsson, N. & Kilborn, W. (2021). Teachers' and students' perception of rational numbers. In: Maitree Inprasitha; Narumon Changsri; Nisakorn Boonsena (Ed.), Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3): . Paper presented at 44th Conference of the International Group for the Psychology of Mathematics Education, Khon Kaen,Thailand [DIGITAL], July 19-22, 2021. (pp. 120-127). Khon Kaen, Thailand: PME, 3
Open this publication in new window or tab >>Teachers' and students' perception of rational numbers
2021 (English)In: Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3) / [ed] Maitree Inprasitha; Narumon Changsri; Nisakorn Boonsena, Khon Kaen, Thailand: PME , 2021, Vol. 3, p. 120-127Conference paper, Published paper (Refereed)
Abstract [en]

It is known from several studies, that students have problems when treating rational numbers. In this paper, there is a focus on rational numbers as equivalent classes, in teaching and learning. 

Place, publisher, year, edition, pages
Khon Kaen, Thailand: PME, 2021
Keywords
Mathematics education, Rational numbers in teaching and learning, Variation and Learning, Teachers Content Knowledge
National Category
Mathematics Didactics Educational Sciences
Research subject
Other research area; Studies in the Educational Sciences
Identifiers
urn:nbn:se:sh:diva-46204 (URN)978-616-93830-2-4 (ISBN)
Conference
44th Conference of the International Group for the Psychology of Mathematics Education, Khon Kaen,Thailand [DIGITAL], July 19-22, 2021.
Available from: 2021-08-17 Created: 2021-08-17 Last updated: 2021-08-18Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-0417-2574

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