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Med uppgift att lära: om matematikuppgifter som en resurs för lärande
Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC). Umeå University, Faculty of Science and Technology, Department of Science and Mathematics Education.
2019 (Swedish)Doctoral thesis, comprehensive summary (Other academic)Alternative title
A task to teach : on mathematical tasks as a resource for learning (English)
Abstract [sv]

Elevers möjligheter att utveckla sin kunskap i matematik påverkas av de uppgifter de arbetar med. Det är möjligt att göra en distinktion mellan rutinuppgifter och matematiska problem. En rutinuppgift är en uppgift som en elev kan lösa genom att använda en välbekant metod, eller genom att imitera en förlaga. För att lösa ett matematiskt problem behöver däremot eleven konstruera en för henne ny lösningsmetod. För att utveckla sin matematiska kunskap behöver elever möta såväl rutinuppgifter som matematiska problem. Problemlösning kan skapa förutsättningar för en elev att utveckla såväl en kreativ problemlösningsförmåga, som en konceptuell, matematisk förståelse.

Avhandlingen består av fem studier med ett fokus på matematikuppgifter, där studie 1-3 syftade till att undersöka vilka möjligheter att arbeta med matematisk problemlösning som elever i gymnasieskolan erbjuds. Detta undersöktes genom läroboksanalyser, studier av elevers arbete med uppgifter och av elevers uppfattningar om matematik. Uppgifter i läroböcker från 12 länder analyserades (studie 1) och ungefär 10 procent av dessa var matematiska problem. Eleverna arbetade (studie 2) nästan uteslutande med de uppgifter som av läroboksförfattarna kategoriserats som enkla och utan att arbeta problemlösande. Bland dessa uppgifter var andelen matematiska problem 4 procent. Inte heller bland uppgifter som kategoriserats som till exempel ’problemlösning’ eller ’utforska’ var matematiska problem i övervikt. Resultaten var relativt lika för de tolv ländernas läroböcker. Elevers uppfattningar om att rutinarbete är säkrare och något som är rimligt att förvänta sig i matematik (studie 3) kan ha en ytterligare påverkan på deras möjligheter att arbeta problemlösande. Med tanke på de positiva effekter som påvisats för elever som arbetar med problemlösning verkar elevers möjligheter att arbeta med problemlösning begränsade. Det finns potential i att såväl utveckla innehållet i läroböckerna för att öka andelen matematiska problem, som i ett medvetet uppgiftsurval från dessa läroböcker.

Syftet med studie 4 och 5 var att fördjupa förståelsen för problemlösning. Ett analytiskt ramverk har utvecklats för att identifiera kreativa, konceptuella och andra utmaningar i elevers problemlösning. Respektive utmaning karaktäriserades för att ytterligare fördjupa förståelsen för dessa och för problemlösning. Elevers arbete med matematiska problem (studie 4) och lärares förväntningar på de utmaningar elever möter vid problemlösning (studie 5) studerades. Konceptuella och kreativa utmaningar visade sig vara de mest centrala vid elevers problemlösning. Genom den karaktäristik som knöts till respektive utmaning kan svårigheter med att identifiera, framför allt kreativa utmaningar, och relationen mellan uppgift och utmaning diskuteras.

Abstract [en]

Students' ability to develop their mathematical competency is influenced by the tasks they work with. A routine task is a task that a student can solve by using a familiar method, or by imitating a template. In order to solve a mathematical problem however, the student needs to construct a to her new solution method. To develop their mathematical competency, students need to work with routine tasks as well as mathematical problems. A creative problem-solving skill, as well as a conceptual understanding may be developed through problem solving.

The thesis consists of five studies, of which the purpose of studies 1-3 was to explore the opportunities to work with mathematical problem solving offered to students in secondary school. Tasks in textbooks from 12 countries were analyzed (study 1), and approximately 10 percent of these were mathematical problems. The students worked (study 2) almost exclusively with tasks categorized by the textbook authors as easy. Among these tasks, the proportion of mathematical problems was 4 percent. Nor among tasks categorized as 'problem solving' or 'exploring', mathematical problems were predominant. The proportions were relatively similar in textbooks from the twelve countries. Students' beliefs that routine work is more secure and something that is reasonable to expect in mathematics (study 3) can have an additional impact on their opportunites to mathematical problem solving. Given the positive effects of problem solving, students' opportunities to work with problem solving seem limited. There is potential in an increased proportion of mathematical problems in textbooks, as well as in a more deliberate task selection from these textbooks.

The purpose of studies 4 and 5 was to contribute towards a better understanding of mathematical problems and mathematical problem solving. An analytical framework was developed to identify creative, conceptual and other challenges in students' problem solving. Each challenge was characterized to be able to understand and describe these components of problem solving. Students' work with mathematical problems (study 4) and the, by teachers anticipated challenges students face in problem solving (study 5) were studied. Conceptual and creative challenges proved to be the most central to students' problem solving. Through the characteristics of each of the challenges, the relation between task and challenge, and difficulties in identifying, especially the creative challenge, was discussed. 

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 2019. , p. 82
Keywords [en]
task, mathematical problem, textbook, problem solving, reasoning, beliefs, mathematical challenge, upper secondary school
Keywords [sv]
Matematikuppgift, Matematiskt problem, Problemlösning, Lärobok, Matematiskt resonemang, Gymnasieskolan, Utmaningar
National Category
Didactics
Research subject
didactics of mathematics
Identifiers
URN: urn:nbn:se:umu:diva-166139ISBN: 978-91-7855-179-8 (electronic)ISBN: 978-91-7855-155-2 (print)OAI: oai:DiVA.org:umu-166139DiVA, id: diva2:1378919
Public defence
2020-01-10, N360, Naturvetarhuset, Umeå, 13:00 (Swedish)
Opponent
Supervisors
Available from: 2019-12-20 Created: 2019-12-16 Last updated: 2019-12-16Bibliographically approved
List of papers
1. Mathematical problem solving in textbooks from twelve countries
Open this publication in new window or tab >>Mathematical problem solving in textbooks from twelve countries
2019 (English)In: International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, E-ISSN 1464-5211Article in journal (Refereed) Epub ahead of print
Abstract [en]

A selection of secondary school mathematics textbooks from twelve countries on five continents was analysed to better understand the support they might be in teaching and learning mathematical problem solving. Over 5700 tasks were compared to the information provided earlier in each textbook to determine whether each task could be solved by mimicking available templates or whether a solution had to be constructed without guidance from the textbook. There were similarities between the twelve textbooks in the sense that most tasks could be solved using a template as guidance. A significantly lower proportion of the tasks required a solution to be constructed. This was especially striking in the initial sets of tasks. Textbook descriptions indicating problem solving did not guarantee that a task solution had to be constructed without the support of an available template.

Place, publisher, year, edition, pages
Taylor & Francis, 2019
Keywords
mathematics textbooks, mathematics tasks, mathematical problem solving, secondary school
National Category
Didactics
Identifiers
urn:nbn:se:umu:diva-157562 (URN)10.1080/0020739X.2019.1656826 (DOI)000484213400001 ()
Note

Originally included in thesis 1 in manuscript form 

Available from: 2019-03-26 Created: 2019-03-26 Last updated: 2019-12-16
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 0020-739X, E-ISSN 1464-5211, Vol. 46, no 4, p. 533-552Article 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
Keywords
mathematics textbook, task-solving, mathematical reasoning, upper secondary school
National Category
Pedagogy
Identifiers
urn:nbn:se:umu:diva-112771 (URN)10.1080/0020739X.2014.992986 (DOI)000354280800005 ()
Available from: 2015-12-15 Created: 2015-12-14 Last updated: 2019-12-16Bibliographically 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, p. 759-776Article in journal (Refereed) Published
Abstract [en]

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 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 found deductively and include expectations, motivational beliefs and security. When it comes to reasoning, a variety of approaches were found. Even though the tasks were designed to demand more than imitation of algorithms, students used this method and failed to solve the task.

Place, publisher, year, edition, pages
Springer, 2017
Keywords
Beliefs, Mathematical reasoning, Non-routine tasks, Problem solving, Upper secondary school
National Category
Learning
Identifiers
urn:nbn:se:umu:diva-127216 (URN)10.1007/s10763-016-9712-3 (DOI)000395003700010 ()
Available from: 2016-11-16 Created: 2016-11-03 Last updated: 2019-12-16Bibliographically approved
4. The challenges of mathematical problem solving: the conceptual and the creative challenge
Open this publication in new window or tab >>The challenges of mathematical problem solving: the conceptual and the creative challenge
(English)Manuscript (preprint) (Other academic)
National Category
Didactics
Identifiers
urn:nbn:se:umu:diva-166137 (URN)
Available from: 2019-12-12 Created: 2019-12-12 Last updated: 2019-12-16
5. The anticipated challenges of students’ problem solving: teachers’ perception of conceptual and creative challenges
Open this publication in new window or tab >>The anticipated challenges of students’ problem solving: teachers’ perception of conceptual and creative challenges
(English)Manuscript (preprint) (Other academic)
National Category
Didactics
Identifiers
urn:nbn:se:umu:diva-166138 (URN)
Available from: 2019-12-12 Created: 2019-12-12 Last updated: 2019-12-16

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