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Mathematical problem solving in textbooks from twelve countries
Dalarna University, School of Education, Health and Social Studies, Mathematics Education. Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).ORCID iD: 0000-0002-8498-8014
Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).
Umeå universitet, Umeå forskningscentrum för matematikdidaktik (UFM).ORCID iD: 0000-0002-9691-1995
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 [en]
mathematics textbooks, mathematics tasks, mathematical problem solving, secondary school
National Category
Didactics
Research subject
Education and Learning
Identifiers
URN: urn:nbn:se:du-31727DOI: 10.1080/0020739X.2019.1656826ISI: 000484213400001OAI: oai:DiVA.org:du-31727DiVA, id: diva2:1388733
Note

Originally included in thesis 1 in manuscript form 

Available from: 2020-01-27 Created: 2020-01-27 Last updated: 2020-01-27
In thesis
1. Med uppgift att lära: om matematikuppgifter som en resurs för lärande
Open this publication in new window or tab >>Med uppgift att lära: om matematikuppgifter som en resurs för lärande
2019 (Swedish)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[en]
A task to teach : on mathematical tasks as a resource for learning
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
task, mathematical problem, textbook, problem solving, reasoning, beliefs, mathematical challenge, upper secondary school, Matematikuppgift, Matematiskt problem, Problemlösning, Lärobok, Matematiskt resonemang, Gymnasieskolan, Utmaningar
National Category
Didactics
Research subject
Education and Learning
Identifiers
urn:nbn:se:du-31724 (URN)978-91-7855-179-8 (ISBN)978-91-7855-155-2 (ISBN)
Public defence
2020-01-10, N360, Naturvetarhuset, Umeå, 13:00 (Swedish)
Opponent
Supervisors
Available from: 2020-01-27 Created: 2020-01-27 Last updated: 2020-01-27Bibliographically approved

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Jäder, JonasLithner, JohanSidenvall, Johan

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