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Att lära sig resonera: Om elevers möjligheter att lära sig matematiska resonemang
Linköping University, Department of Social and Welfare Studies. Linköping University, Faculty of Educational Sciences.ORCID iD: 0000-0002-3691-1995
2015 (Swedish)Licentiate thesis, comprehensive summary (Other academic)Alternative title
Learning to Reason : On students' opportunities to learn mathematical reasoning (English)
Abstract [en]

Students only learn what they get the opportunity to learn. This means, for example, that students do not develop their reasoning- and problem solving competence unless teaching especially focuses on developing these competencies. Despite the fact that it has for the last 20 years been pointed out the need for a reform-oriented mathematics education, research still shows that in Sweden, as well as internationally, an over-emphasis are placed on rote learning and procedures, at the cost of promoting conceptual understanding. Mathematical understanding can be separated into procedural and conceptual understanding, where conceptual understanding can be connected to a reform oriented mathematics education. By developing a reasoning competence conceptual understanding can also be developed. This thesis, which deals with students’ opportunities to learn to reason mathematically, includes three studies (with data from Swedish upper secondary school, year ten and mathematics textbooks from twelve countries). These opportunities have been studied based on a textbook analysis and by studying students' work with textbook tasks during normal classroom work. Students’ opportunities to learn to reason mathematically have also been studied by examining the relationship between students' reasoning and their beliefs. An analytical framework (Lithner, 2008) has been used to categorise and analyse reasoning used in solving tasks and required to solve tasks.

Results support previous research in that teaching and mathematics textbooks are not necessarily in harmony with reform-oriented mathematics teaching. And that students indicated beliefs of insecurity, personal- and subject expectations as well as intrinsic- and extrinsic motivation connects to not using mathematical reasoning when solving non-routine tasks. Most commonly students used other strategies than mathematical reasoning when solving textbook tasks. One common way to solve tasks was to be guided, in particular by another student. The results also showed that the students primarily worked with the simpler tasks in the textbook. These simpler tasks required mathematical reasoning more rarely than the more difficult tasks. The results also showed a negative relationship between a belief of insecurity and the use of mathematical reasoning. Furthermore, the results show that the distributions of tasks that require mathematical reasoning are relatively similar in the examined textbooks across five continents.

Based on the results it is argued for a teaching based on sociomathematical norms that leads to an inquiry based teaching and textbooks that are more in harmony with a reform-oriented mathematics education.

Abstract [sv]

Elever kan bara lära sig de det de får möjlighet att lära sig. Detta innebär till exempel att elever inte utvecklar sin resonemangs- och problemlösningsförmåga i någon större utsträckning om inte deras undervisning fokuserar på just dessa förmågor. Forskning, nationellt och internationellt visar att det finns en överbetoning på utantillinlärning och på procedurer. Detta verkar ske på bekostnad av en konceptuell förståelse, trots att det under 20 års tid pekats på behovet av en reforminriktad matematikundervisning. Matematisk förståelse kan delas in i procedurell- och konceptuell förståelse där en konceptuell förståelse kan kopplas till en reforminriktad matematikundervisning. Genom att utveckla förmågan att resonera matematiskt utvecklas också den konceptuella förståelsen. Denna avhandling, som inbegriper tre studier (med empiri från gymnasiet år ett och matematikläroböcker från tolv länder) behandlar elevers möjlighet att lära sig att resonera matematiskt. Dessa möjligheter har studerats utifrån att undersöka vilka möjligheter läroboken ger att lära sig matematiska resonemang, dels via en läroboksanalys och dels genom att studera elevers arbete med läroboksuppgifter i klassrumsmiljö. Elevers möjligheter att lära sig att resonera matematiskt har också studerats genom att undersöka relationen mellan elevers matematiska resonemang och deras uppfattningar om matematik. Ett analytiskt ramverk (Lithner, 2008) har används för att kategorisera och analysera resonemang som använts för att lösa uppgifter och som behövs för att lösa en uppgift.

Resultaten från studierna har givit stöd åt tidigare forskning vad gäller att undervisning och läroböckerna inte nödvändigtvis harmonierar med en reforminriktad matematikundervisning. Och att elever har uppfattningar om matematik som bygger på osäkerhet, förväntan på ämnet och sin egen förmåga samt motivation och att dessa uppfattningar delvis kan kopplas till att eleverna inte använder matematiska resonemang för att försöka lösa icke-rutinuppgifter. Det vanligaste sättet att lösa läroboksuppgifter var att välja andra strategier än att använda sig av matematiska resonemang. Ett vanligt sätt att lösa uppgifter var att låta sig guidas, av främst en annan elev. Eleverna arbetade framförallt med de enklare uppgifterna i läroböckerna. Bland dessa enklare uppgifter var det mer sällsynt med uppgifter som krävde matematiska resonemang för att lösas relativt de svårare uppgifterna. Resultaten visade även att det fanns en negativ relation mellan en uppfattning av osäkerhet hos elever och ett användande av matematiska resonemang. Resultaten visade vidare att fördelningen av uppgifter som krävde matematiska resonemang var relativt lika i alla undersökta läroböcker från fem världsdelar.

Utifrån resultaten argumenteras för en förändrad undervisning mot en undersökande undervisning och läroböcker som är mer i harmoni med en reforminriktad matematikundervisning.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. , 59 p.
Series
Studies in Science and Technology Education, ISSN 1652-5051 ; 86
Keyword [en]
Mathematical reasoning, problem solving, mathematics textbook, beliefs, mathematics tasks, opportunities to learn, upper secondary school
Keyword [sv]
Mathematical reasoning, problem solving, mathematics textbook, beliefs, mathematics tasks, opportunities to learn, upper secondary school
National Category
Educational Sciences
Identifiers
URN: urn:nbn:se:liu:diva-117759DOI: 10.3384/lic.diva-117759ISBN: 978-91-7519-100-3 (print)OAI: oai:DiVA.org:liu-117759DiVA: diva2:810757
Presentation
2015-03-18, K2, Kåkenhus, Campus Norrköping, Linköpings universitet, Norrköping, 10:15 (Swedish)
Opponent
Supervisors
Available from: 2015-05-22 Created: 2015-05-08 Last updated: 2016-05-04Bibliographically approved
List of papers
1. 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
2. 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
3. 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

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