Logo: to the web site of Uppsala University

uu.sePublications from Uppsala University
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
X - men sen då? Algebrans stora idéer från första klass till högre matematik: Med fokus på tidig algebra i Sverige
Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.ORCID iD: 0000-0003-1191-5700
2021 (Swedish)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The overall aim of this thesis is to increase the knowledge of the state of algebraic thinking in the earlier years, so called early algebra, in the Swedish primary school. First, using the Big Ideas of Algebra (Blanton et al., Journal for Research in Mathematics Education, 46(1), 39–87, 2015) as a theoretical framework, the thesis investigates which types of algebraic thinking can be identified in the Swedish mathematics curriculum and in two textbooks series for Grades 1-6. Second, as students’ understanding of the equal sign as a symbol for a relation is an important factor for algebraic success, students’ knowledge of it is studied by using an assessment form based on Matthews et al. (Journal for Research in Mathematics Education, 43(3), 316-350, 2012).

The results of the empirical studies show that Equivalences, Expressions, Equations, and Inequalities (EEEI) were the most prominent Big Idea in the Swedish context. The Big Idea of Generalized Arithmetic (GA) is not represented in the Central content in Grades 1-6 and only slightly represented in the textbooks. Furthermore, there are big differences between the two textbook series, both regarding the total amount of algebraic content and regarding how well each Big Idea is represented in the textbooks. As textbooks are important artefacts in Swedish mathematics classrooms, opportunities to learn early algebra in a classroom might depend on which textbook is used. Concerning students’ understanding of the equal sign, the study shows that they, in general, are able to describe the meaning of the equal sign as relational, but they are less able to use the relational structure of an equality. This implies that “the meaning of the equal sign”, which is part of the algebraic content in the Swedish mathematics curriculum in Grades 1-3, might be learnt as a definition by word rather than by its implications in mathematics. Besides the empirical contributions, the thesis also offers a discussion whether the Big Ideas can be found in mathematics at the university level and in research in abstract algebra and it is argued that algebraic thinking is present in all levels of mathematics, from early algebra in lower primary school to research in mathematics.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2021. , p. 101
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Educational Sciences ; 23
Keywords [en]
Early algebra, curriculum, mathematics textbooks, primary school mathematics, generalized arithmetic, conceptual knowledge, knowledge levels, mathematics education
Keywords [sv]
Tidig algebra, läroplan, läromedel i matematik, matematik i tidiga skolår, generaliserad aritmetik, konceptuell kunskap, kunskapsnivåer, matematikundervisning
National Category
Didactics
Research subject
Curriculum Studies
Identifiers
URN: urn:nbn:se:uu:diva-425573ISBN: 978-91-513-1076-3 (print)OAI: oai:DiVA.org:uu-425573DiVA, id: diva2:1503354
Public defence
2021-01-22, Eva Netzelius, Blåsenhus, von Kramers allé 1, Uppsala, 13:15 (Swedish)
Opponent
Supervisors
Funder
Swedish Research Council, 2015-02043Available from: 2020-12-18 Created: 2020-11-24 Last updated: 2021-01-25
List of papers
1. Dissident maps on the seven-dimensional Euclidean space
Open this publication in new window or tab >>Dissident maps on the seven-dimensional Euclidean space
2003 (English)In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 97, p. 251-276Article in journal (Refereed) Published
Abstract [en]

Our article contributes to the classification of dissident maps on R7, which in turn contributes to the classification of 8-dimensional real division algebras. We study two large classes of dissident maps on R7. The first class is formed by all composed dissident maps, obtained from a vector product on R7 by composition with a definite endomorphism. The second class is formed by all doubled dissident maps, obtained as the purely imaginary parts of the structures of those 8-dimensional real quadratic division algebras which arise from a 4-dimensional real quadratic division algebra by doubling. For each of these two classes we exhibit a complete (but redundant) classification, given by a 49-parameter family of composed dissident maps and a 9-parameter family of doubled dissident maps respectively. The intersection of these two classes forms one isoclass of dissident maps only, namely the isoclass consisting of all vector products on R7.

National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-22144 (URN)10.4064/cm97-2-10 (DOI)
Available from: 2007-01-11 Created: 2007-01-11 Last updated: 2020-11-24Bibliographically approved
2. On the doubling of quadratic algebras
Open this publication in new window or tab >>On the doubling of quadratic algebras
2004 (English)In: Colloq. Math., no 100, p. 119-139Article in journal (Refereed) Published
Identifiers
urn:nbn:se:uu:diva-71116 (URN)
Available from: 2005-04-29 Created: 2005-04-29 Last updated: 2020-11-24
3. Doubled quadratic division algebras
Open this publication in new window or tab >>Doubled quadratic division algebras
2004 (English)Licentiate thesis, monograph (Other scientific)
Series
, ISSN 1101-3591 ; 2004:34
Identifiers
urn:nbn:se:uu:diva-86290 (URN)
Available from: 2005-04-29 Created: 2005-04-29 Last updated: 2020-11-24
4. Liftings of dissident maps
Open this publication in new window or tab >>Liftings of dissident maps
2006 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 204, no 1, p. 133-154Article in journal (Refereed) Published
Abstract [en]

We study dissident maps ηη on RmRm for m∈{3,7}m∈{3,7} by investigating liftings Φ:Rm→RmΦ:Rm→Rm of the selfbijection ηP:P(Rm)→P(Rm),ηP[v]=(η(v∧Rm))⊥ induced by ηη. Our main result (Theorem 2.4) asserts the existence and uniqueness, up to a non-zero scalar multiple, of a lifting ΦΦ whose component functions are homogeneous polynomials of degree dd, relatively prime and without non-trivial common zero. We prove that 1⩽d⩽m-21⩽d⩽m-2.

We achieve a complete description of all dissident maps of degree one and we solve their isomorphism problem (Theorems 4.8 and 4.13). As a consequence, we achieve a complete description of all real quadratic division algebras of degree one and we solve their isomorphism problem (Theorems 5.1 and 5.3). Moreover we present examples of eight-dimensional real quadratic division algebras of degree 3 and 5 (Proposition 6.3). This extends earlier results of Osborn [Trans. Amer. Math. Soc. 105 (1962) 202–221], Hefendehl [Geometriae Dedicata 9 (1980) 129–152], Hefendehl-Hebeker [Arch. Math. 40 (1983) 50–60], Cuenca Mira et al. [Lin. Alg. Appl. 290 (1999) 1–22], Dieterich [Proc. Amer. Math. Soc. 128 (2000) 3159–3166] and Dieterich and Lindberg [Colloq. Math. 97 (2003) 251–276] on the classification of real quadratic division algebras.

National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-22146 (URN)10.1016/j.jpaa.2005.04.005 (DOI)
Available from: 2007-01-11 Created: 2007-01-11 Last updated: 2020-11-24Bibliographically approved
5. Teoretiska och praktiska perspektiv på generaliserad aritmetik
Open this publication in new window or tab >>Teoretiska och praktiska perspektiv på generaliserad aritmetik
2018 (Swedish)In: Perspectives on professional development of mathematics teachers. Proceedings of MADIF11: the eleventh research seminar of the Swedish Society for Research of Mathematics Education / [ed] J. Häggström, Y. Liljekvist, J. Bergman Ärlebäck, M. Fahlgren & O. Olande, Göteborg: Svensk förening för MatematikDidaktisk Forskning - SMDF, 2018, p. 27-36Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Göteborg: Svensk förening för MatematikDidaktisk Forskning - SMDF, 2018
Series
Skrifter från Svensk Förening för MatematikDidaktisk Forskning, ISSN 1651-3274 ; 13
National Category
Didactics
Research subject
Curriculum Studies
Identifiers
urn:nbn:se:uu:diva-380059 (URN)978-91-984024-2-1 (ISBN)
Conference
MADIF11, Proceedings of MADIF 11, The eleventh research seminar of the Swedish Society for Research in Mathematics Education, Karlstad, January 23–24, 2018
Funder
Swedish Research Council, 2015–02043
Available from: 2019-03-22 Created: 2019-03-22 Last updated: 2020-11-24Bibliographically approved
6. Development of algebraic thinking: opportunities offered by the Swedish curriculum and elementary mathematics textbooks
Open this publication in new window or tab >>Development of algebraic thinking: opportunities offered by the Swedish curriculum and elementary mathematics textbooks
2019 (English)In: Nordisk matematikkdidaktikk, NOMAD: [Nordic Studies in Mathematics Education], ISSN 1104-2176, Vol. 24, no 1, p. 27-49Article in journal (Refereed) Published
Abstract [en]

In search of the reasons for Swedish students’ low achievement in algebra in inter-national and national evaluations, we investigate how the development of algebraic thinking is addressed in the Swedish national mathematics curriculum and two widely used mathematics textbook series for grades 1–6 in Sweden. The analytical tool used is based on the classification of ”big ideas” which research has shown as important for developing pupils’ algebraic understanding in early school grades. The results show that functional thinking, expressions, and equations are well represented topics both in the curriculum and the textbooks; however generalized arithmetic is a topic that is poorly developed in both the curriculum and the textbooks.

National Category
Didactics
Identifiers
urn:nbn:se:uu:diva-377441 (URN)
Funder
Swedish Research Council, 2015-02043
Available from: 2019-02-19 Created: 2019-02-19 Last updated: 2020-11-24Bibliographically approved
7. Primary School Students’ Knowledge of the Equal Sign – the Swedish Case
Open this publication in new window or tab >>Primary School Students’ Knowledge of the Equal Sign – the Swedish Case
2021 (English)In: International Journal of Science and Mathematics Education, ISSN 1571-0068, E-ISSN 1573-1774, Vol. 20, p. 321-343Article in journal (Refereed) Published
Abstract [en]

This case study investigates, at three Swedish schools, primary school students’ knowledge of the equal sign. The schools were chosen as representatives of schools whose students have different socioeconomic backgrounds. The data consist of Grade 3 and 6 students’ responses to an assessment form based on Matthews et al. (Journal for Research in Mathematics Education, 43(3), 316-350, 2012). A comparison between the schools indicates that there are great differences in the students’ knowledge of the equal sign depending on which school they attend. The Swedish students perform better than South Korean students responding to the same assessment form when it comes to knowing that the equal sign does not mean an operation (“the answer to a problem”). We find this interesting to discuss, as South Korea is one of the countries that performs the best on the TIMSS and PISA.

Place, publisher, year, edition, pages
Springer Nature, 2021
Keywords
Algebraic learning, Early algebra, Equal sign, Primary school
National Category
Didactics
Identifiers
urn:nbn:se:uu:diva-425571 (URN)10.1007/s10763-020-10144-z (DOI)000607508200002 ()
Funder
Swedish Research Council, 2015-02043
Available from: 2020-11-17 Created: 2020-11-17 Last updated: 2022-04-19Bibliographically approved

Open Access in DiVA

fulltext(1335 kB)1503 downloads
File information
File name FULLTEXT01.pdfFile size 1335 kBChecksum SHA-512
942b849aa785fabb0516ac8c0119c97bf6e433f52fa3ab846b03d74fd23ced3f78305762d6857472770afcdb8b9cef5e66d3a7ce115ffdf65750dc2bc830dae4
Type fulltextMimetype application/pdf

Authority records

Madej, Lars

Search in DiVA

By author/editor
Madej, Lars
By organisation
Department of Education
Didactics

Search outside of DiVA

GoogleGoogle Scholar
Total: 1503 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 2950 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf