liu.seSearch for publications in DiVA
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • 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
Elevers möjligheter till lärande av matematiska resonemang
Linköping University, Department of Social and Welfare Studies. Linköping University, Faculty of Science & Engineering.
2015 (Swedish)Licentiate thesis, comprehensive summary (Other academic)
Abstract [sv]

En av anledningarna till varför elever har svårigheter med matematik i skolan är att utantillinlärning utgör grunden för utbildningen för många av eleverna. Procedurella och konceptuella kunskaper behövs för att skapa en bred matematisk kompetens. Eleverna lär sig bara det som de får en möjlighet att lära sig, vilket innebär att de möjligheter till lärande som erbjuds eleverna i skolan måste beaktas. Ett väletablerat ramverk som gör det möjligt att analysera de resonemang som krävs för att lösa läroboksuppgifter samt de resonemang som används av eleverna vid uppgiftslösning har använts för att undersöka möjligheterna att lära sig resonera matematiskt. Genom att använda ramverket möjliggörs en mer förfinad diskussion av vilken typ av kunskap som används av eleverna. Ramverket skiljer på kreativa matematiska resonemang, där en lösning måste skapas av eleven, och imitativa resonemang som bygger på utantillinlärning eller imitering av en tillgänglig lösningsalgoritm. Möjligheterna att lära sig beror på klassrummets normer som har förhandlats fram mellan elever och lärare. Dessa normer påverkas i sin tur av flera faktorer. I denna avhandling diskuteras läroboken, både som en, av flera bilder, av undervisningen och utifrån hur den används i klassrummet, samt elevernas uppfattningar om matematik. I avhandlingen ingår tre studier. Den första studien består av en analys av uppgifterna, med avseende på kraven på resonemang, i läromedel från tolv länder, i fem världsdelar. I den andra studien har elevers resonemang då de arbetar med uppgifter från läroboken i klassrummet analyserats. I den tredje studien används en tematisk analys för att undersöka de uppfattningar som eleverna visar upp, vilka sedan kopplas till de resonemang som används.

Resultaten visar att läroböckerna från tolv olika länder har en liknande andelen uppgifter som kräver att eleverna använder kreativa matematiska resonemang. I genomsnitt krävde ungefär var tionde uppgift ett mer genomgripande kreativt matematiskt resonemang. Resultaten visar även att elever i den svenska gymnasieskolan främst löser de första, lättare uppgifterna, där andelen uppgifter som kräver ett kreativt matematiskt resonemang är lägre. Eleverna använder också i stor utsträckning imitativa resonemang. Möjligheterna för elever att träna sig på kreativa matematiska resonemang verkar utifrån mina resultat vara begränsade. Då elever guidar varandra genom uppgiftslösning verkar det som att fokus främst ligger på att komma fram till ett svar som överensstämmer med facit. Inte heller då elever får hjälp av en lärare verkar möjligheter till annat än imitativa resonemang skapas. Eleverna indikerar dessutom uppfattningar om att matematiska uppgifter i de allra flesta fall ska kunna lösas genom ett imitativt resonemang och att utantillinlärning därför bör vara en central del av undervisningen. Lärarens roll i klassrummet är viktig för att skapa och utveckla de gemensamma klassrumsnormerna. Stor vikt bör läggas vid vilka uppgifter och vilka läromedel som används i undervisningen. Även elevernas sätt att arbeta i klassrummet måste beaktas i relation till möjligheterna till lärande, och den matematiska förståelsen bör spela en större roll.

Abstract [en]

One of the main problems with learning difficulties in mathematics is that rote-learning becomes the very foundation of mathematics for many students. Procedural as well as conceptual knowledge is needed to build a broad mathematical competence. Students learn only what they get an opportunity to learn, which means that we must consider what opportunities to learn are given to school students. For the purpose of exploring what opportunities are available to learn to reason mathematically, a well established framework is used to analyze the reasoning required by textbook tasks as well as the reasoning used by students. The framework was used to refine the discussion of what type of knowledge is used by the students. Application of the framework distinguishes between creative mathematical reasoning, where a solution has to be created by the student, and imitative reasoning which is based on rote learning or following an existing template. Opportunities to learn depend on the classroom norms that have been negotiated between students and teacher. These norms are influenced by several factors. This thesis deals with the textbook, both as one of several pictures of the education, and in terms of how it is used in the classroom, as well as students’ beliefs about mathematics. There are three studies included in the thesis. In the first study, tasks in mathematics textbooks used in secondary school around the world are analyzed concerning the reasoning requirements. For the second study an analysis of students reasoning during textbook task solving in the classroom has been conducted. In the third study a thematic analysis has been used to explore students’ beliefs about mathematics and relate these beliefs to the reasoning used.

Results from analyzing textbooks from twelve different countries paint a similar picture when it comes to the proportion of tasks requiring students to use creative mathematical reasoning. On average, only every tenth task required creative mathematical reasoning to a greater extent. Furthermore, students in the Swedish upper secondary school level mainly focus on solving the easier, earlier tasks and also mainly use imitative reasoning. Opportunities for students to use creative mathematical reasoning seem limited. When students guide each other during task solving, the main focus seems to be to reach a conclusion in terms of an answer corresponding to that given in the answer-section of the book. Moreover, guidance from a teacher does not seem to lead to anything other than imitation of a procedure. Students also indicate their beliefs by expressing that most tasks should be possible to solve using imitative reasoning, and that therefore, rote learning is a central part of mathematics education. This places pressure on teachers to carefully reflect on what tasks and textbooks they use in their teaching, and also what types of classroom norms they wish to present. The manner in which students work in the classroom also needs consideration, where a greater focus should be directed toward understanding.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. , 51 p.
Series
Studies in Science and Technology Education, ISSN 1652-5051 ; 87
National Category
Pedagogy Educational Sciences
Identifiers
URN: urn:nbn:se:liu:diva-118094DOI: 10.3384/lic.diva-118094ISBN: 978-91-7519-099-0 (print)OAI: oai:DiVA.org:liu-118094DiVA: diva2:813138
Presentation
2015-03-04, K2, Kåkenhus, Campus Norrköping, Linköpings Universitet, Norrköping, 13:30 (Swedish)
Opponent
Supervisors
Available from: 2015-05-21 Created: 2015-05-21 Last updated: 2015-06-05Bibliographically approved
List of papers
1. Reasoning requirements in school mathematics textbooks: an analysis of books from 12 countries
Open this publication in new window or tab >>Reasoning requirements in school mathematics textbooks: an analysis of books from 12 countries
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A selection of secondary school textbooks from twelve countries in five continents is used as an indicator of the opportunities to learn mathematics through different forms of reasoning, in particular with respect to the distinction between rote learning and deeper learning. One aspect that is fundamental to the development of conceptual understanding as well as problem solving ability is the opportunity to learn how to construct mathematically well-founded reasoning. This study compared textbook tasks to the information provided previously in the book, determining if it is possible and reasonable to mimic available solution templates, or if a solution has to be constructed. The results show that the percentage of tasks where it is possible to mimic available templates is on average 79 %, but that this percentage varies widely within the books depending on the textbook authors own labeling of the tasks, and on the mathematical content. 13 % of the tasks can be solved mainly by mimicking provided templates but require some minor modification, and the remaining 9 % of the tasks require that the main parts of the solution are constructed without the guidance of a template. Although these distributions are relatively similar in all textbooks, the twelve countries perform differently in international tests such TIMSS and PISA.

Keyword
Mathematics Textbooks, Mathematics Tasks, Mathematical reasoning, Opportunities to learn, Secondary school
National Category
Educational Sciences
Identifiers
urn:nbn:se:liu:diva-117560 (URN)
Available from: 2015-05-08 Created: 2015-05-04 Last updated: 2015-05-22Bibliographically approved
2. Students’ reasoning in mathematics textbook task-solving
Open this publication in new window or tab >>Students’ reasoning in mathematics textbook task-solving
2015 (English)In: International Journal of Mathematical Education in Science and Technology, ISSN 1464-5211, Vol. 46, no 4, 533-552 p.Article in journal (Refereed) Published
Abstract [en]

This study reports on an analysis of students’ textbook task-solving in Swedish upper secondary school. The relation between types of mathematical reasoning required, used, and the rate of correct task solutions were studied. Rote learning and superficial reasoning were common, and 80% of all attempted tasks were correctly solved using such imitative strategies. In the few cases where mathematically founded reasoning was used, all tasks were correctly solved. The study suggests that student collaboration and dialogue does not automatically lead to mathematically founded reasoning and deeper learning. In particular, in the often common case where the student simply copies a solution from another student without receiving or asking for mathematical justification, it may even be a disadvantage for learning to collaborate. The results also show that textbooks’ worked examples and theory sections are not used as an aid by the student in task-solving.

Place, publisher, year, edition, pages
Taylor & Francis, 2015
Keyword
Mathematical reasoning; task-solving; mathematics textbook; upper
National Category
Educational Sciences
Identifiers
urn:nbn:se:liu:diva-117559 (URN)10.1080/0020739X.2014.992986 (DOI)
Available from: 2015-05-08 Created: 2015-05-04 Last updated: 2015-05-22Bibliographically approved
3. Students’ Mathematical Reasoning and Beliefs in Non-routine Task Solving
Open this publication in new window or tab >>Students’ Mathematical Reasoning and Beliefs in Non-routine Task Solving
2017 (English)In: International Journal of Science and Mathematics Education, ISSN 1571-0068, E-ISSN 1573-1774, Vol. 15, no 4, 759-776 p.Article in journal (Refereed) Published
Abstract [en]

Students’ beliefs and problem solving are connected, and have been studied in different contexts. One of the common results of previous research is that students tend to prefer routine, algorithmic approaches to mathematical tasks. This study explores Swedish upper secondary school students’ beliefs and reasoning when solving non-routine tasks. The results regarding the beliefs indicated by the students were categorized relying on previous research and included expectations, motivational beliefs, and security. For example students expected tasks to be solvable by a known algorithm. Students also expressed an insecurity regarding their own reasoning. A variety of approaches to the tasks in terms of the reasoning used were found. Even though the tasks were designed to demand more than imitation of algorithms, students on several occasions used this method and failed to solve the tasks. Our study implies that there is more to create a problem solving learning environment than just to give students non-routine tasks.

Place, publisher, year, edition, pages
Springer Netherlands, 2017
Keyword
Beliefs, mathematical reasoning, non-routine tasks, problem solving, upper secondary school
National Category
Educational Sciences
Identifiers
urn:nbn:se:liu:diva-117561 (URN)10.1007/s10763-016-9712-3 (DOI)000395003700010 ()2-s2.0-84959365078 (Scopus ID)
Available from: 2015-05-08 Created: 2015-05-04 Last updated: 2017-04-20Bibliographically approved

Open Access in DiVA

fulltext(1557 kB)841 downloads
File information
File name FULLTEXT01.pdfFile size 1557 kBChecksum SHA-512
bb5b0e0f5f98da4929e4b37535834b5d78bab1475f7329a0a01fd634ffc60fce42268662b24d020485bb4bfc414506ebf7508db697f5e6c5e76abfff553a069c
Type fulltextMimetype application/pdf
omslag(1123 kB)45 downloads
File information
File name COVER01.pdfFile size 1123 kBChecksum SHA-512
ffa52bc1f1bbbe9ac2742a3b5f039aa3c2bdc8a6107057933926ed416350f395e8db3b028202160eff573265238336346927895f65ba533ac2e9ca22e2809c1a
Type coverMimetype application/pdf

Other links

Publisher's full text
By organisation
Department of Social and Welfare StudiesFaculty of Science & Engineering
PedagogyEducational Sciences

Search outside of DiVA

GoogleGoogle Scholar
Total: 841 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

doi
isbn
urn-nbn

Altmetric score

doi
isbn
urn-nbn
Total: 5315 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • 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