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Lösa problem: om elevers förutsättningar att lösa problem och hur lärare kan stödja processen
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. (Lärande genom imitativa och kreativa resonemang (LICR))ORCID iD: 0000-0002-9691-1995
2019 (Swedish)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Solving problems : on students’ opportunities to solve problems and how teachers can support this process (English)
Abstract [sv]

Generellt sett domineras matematikundervisning av utantillinlärning och arbete med rutinuppgifter. Om undervisning till störst del görs på detta sätt kommer elever ha svårt att att utveckla andra viktiga förmågor i matematik såsom problemlösning, resonemang och begreppsförståelse. Tidigare forskning har visat om elever får jobba med problemuppgifter (dvs. skapa egna lösningsmetoder) i större utsträckning får de en ökad matematisk förståelse, än om de enbart arbetar med rutinuppgifter.

Syftet med avhandlingen var att ge ökade insikter om varför utantillinlärning och arbete med rutinuppgifter fortsätter att vara vanligt samt undersöka och föreslå på vilket sätt elevers förutsättningar att jobba med problemuppgifter skulle kunna förbättras. Detta gjordes genom följande studier. (1) Relationen mellan vilka typer av lösningsstrategier (imitera eller skapa lösningsmetod) som krävdes och vilka som användes vid uppgiftslösning. (2) Relationen mellan elevers val av lösningsstrategi och uppfattningar om matematik. (3) Undersökning av andel problemuppgifter i läroböcker från 12 länder. (4) Karaktärisering av tidigare forskning med avseende på undervisning genom problemlösning och resonemang. (5) Interventionsstudie där ett lärarstöd, utformat för att stödja elevers problemlösning med hjälp av formativ bedömning, utvecklades, testades och utvärderades. Studierna fokuserade i första hand på skolans senare årskurser.

Elevernas förutsättningar att lösa uppgifter genom problemlösning var begränsad: av att det var mycket ovanligt med problemuppgifter bland de enklare uppgifterna i läroböckerna, av elevernas val att använda sig av imitativa lösningsstategier och av att eleverna ofta kunde lösa uppgifter genom att lotsas fram till en lösning av en annan elev eller av läraren. Elevernas förutsättningar begränsades också av elevernas uppfattningar av matematik och av elever ibland arbetade med uppgifter som inte var inom räckhåll att lösas genom problemlösning. 

För att ge elever förbättrade förutsättningar att lösa problemuppgifter bör lärare låta elever arbeta med fler problemuppgifter i en lärandemiljö som innebär att elever faktiskt skapar egna lösningsmetoder och att lärarhjälp baseras på att stödja elever utifrån elevers svårigheter och inte lotsa fram till en lösning. Resultatet ger också implikationer för hur läroböcker kan struktureras och hur det testade lärarstödet skulle kunna vara en del av en proffessionsutveckling och en del av lärarutbildningen.

Abstract [en]

In mathematics education, there is generally too much emphasis on rote learning and superficial reasoning. If learning is mostly done by rote and imitation, important mathematical competencies such as problem-solving, reasoning, and conceptual understanding are not developed. Previous research has shown that students who work with problems (i.e. constructs a new solution method to a task), to a greater extent increase their mathematical understanding than students who only solve routine tasks.

The aim of the thesis was to further understand why teaching is dominated by rote learning and imitation of procedures and investigate how opportunities for students to solve tasks through problem-solving could be improved. This was done through the following studies. (1) Investigating the relation between types of solution strategy required, used, and the rate of correct task solutions in students’ textbook task-solving. (2) Studying the relationship between students’ beliefs and choice of solution strategy when working on problems. (3) Conducting a textbook analysis of mathematics textbooks from 12 countries, to determine the proportions of tasks that could be solved by mimicking available templates and of tasks where a solution had to be constructed without guidance from the textbook. (4) Conducting a literature review in order to characterize teaching designs intended to enhance students to develop mathematical understanding through problem solving and reasoning. (5) Conducting an intervention study were a teacher guide, structured in line with central tenets of formative assessment, was developed, tested, and evaluated in real classroom settings. The teacher guide was designed to support teachers in their support of students’ in their problem-solving process. Studies I, II and V were conducted in Swedish upper secondary school settings. 

The students’ opportunities to solve tasks through problem-solving were limited: by the low proportion of problems among the easier tasks in the textbooks; by the students' choice of using imitative solution strategies; and by the guidance of solution methods that students received from other students and their teachers. The students’ opportunities were also limited by the students' beliefs of mathematics and the fact that a solution method of problem tasks was not always within reach for the students, based on the students' knowledge. In order to improve students’ opportunities, teachers should allow students to work with more problems in a learning environment that lets students engage in problem-solving and support students' work on problems by adapting their support to students' difficulties. The results also give implications for the construction and use of textbooks and how the use of the teacher guide could be part of teachers’ professional development and a tool that teacher students may meet within their education.

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 2019. , p. 74
Series
Umeå Studies in the Educational Sciences ; 33
Keywords [en]
problem solving, reasoning, beliefs, textbook, teaching, formative assessment, design research, secondary school
Keywords [sv]
problemlösning, resonemang, uppfattningar, lärobok, lärare, undervisning, formativ bedömning, designforskning, gymnasieskola
National Category
Didactics
Identifiers
URN: urn:nbn:se:umu:diva-157557ISBN: 978-91-7855-049-4 (print)OAI: oai:DiVA.org:umu-157557DiVA, id: diva2:1303310
Public defence
2019-05-17, N420, Johan Bures väg 16, Umeå, 12:30 (Swedish)
Opponent
Supervisors
Projects
Lärande genom imitativa och kreativa resonemang (LICR)Available from: 2019-04-16 Created: 2019-04-09 Last updated: 2019-05-14Bibliographically 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 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.

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: 2019-04-09Bibliographically 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, 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.

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)
Available from: 2016-11-16 Created: 2016-11-03 Last updated: 2019-04-12Bibliographically approved
3. Mathematical problem solving in textbooks from twelve countries
Open this publication in new window or tab >>Mathematical problem solving in textbooks from twelve countries
(English)Manuscript (preprint) (Other academic)
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.

Keywords
mathematics textbooks; mathematics tasks; mathematical problem solving; secondary school
National Category
Didactics
Identifiers
urn:nbn:se:umu:diva-157562 (URN)
Available from: 2019-03-26 Created: 2019-03-26 Last updated: 2019-04-09
4. Literature review of mathematics teaching design for problem solving and reasoning
Open this publication in new window or tab >>Literature review of mathematics teaching design for problem solving and reasoning
2019 (English)In: Nordisk matematikkdidaktikk, NOMAD: [Nordic Studies in Mathematics Education], ISSN 1104-2176, Vol. 24, no 1, p. 51-74Article in journal (Refereed) Published
Abstract [en]

To characterize teaching designs intended to enhance students’ problem solving and reasoning skills or to develop other mathematical competencies via problem solving and reasoning, a literature review was conducted of 26 articles published in seven top-ranked journals on mathematics education from 2000 to 2016. Teaching designs were characterized by a) the educational goals of the designs, b) the claims about how to reach these goals, and c) the empirical and theoretical arguments underlying these claims. Thematic analysis was used to analyze the retrieved articles. All but two studies had goals concerned with developing students’ mathematical competencies. The overarching ideas of the identified emergent claims regarding the achievement of stipulated goals, concerned scaffolding students’ learning and letting students construct their own mathematics. Four recurring theoretical arguments were found to support emergent claims: hypothetical learning trajectories, realistic mathematics education, theory of didactical situations and zone of proximal development.

Place, publisher, year, edition, pages
Göteborgs universitet, 2019
National Category
Educational Sciences
Identifiers
urn:nbn:se:umu:diva-156533 (URN)
Projects
Learning by Imitative and Creative Reasoning
Available from: 2019-02-19 Created: 2019-02-19 Last updated: 2019-04-09Bibliographically approved
5. Supporting Teachers to Support Students’ Problem-solving
Open this publication in new window or tab >>Supporting Teachers to Support Students’ Problem-solving
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The purpose of this intervention study was to develop, test, and evaluate a teacher guide structured in line with central tenets of formative assessment in a real classroom setting. The teacher-guide was designed to support teachers’ diagnosis of student difficulties and their choice of feedback to help students to continue the construction of solution methods during problem- solving if they become stuck. By using an approach inspired by design research, five teachers used the teacher guide for two plus two weeks in 12 mathematics courses in upper secondary school with revisions of the teacher guide in between the iterations. Ninety-six teacher-student interactions were observed, and teacher interviews were conducted. The results showed that the teacher guide supported the teachers in providing less algorithmic information and instead focusing on the problem-solving process, and by that helping the students to themselves construct solutions during their problem-solving activity. The use of the teacher guide was sometimes constrained by the type of tasks the students were working on, by difficulties in making reasonable diagnoses of students’ difficulties, and by students’ insufficient ability and/or willingness to communicate.

Keywords
teacher-guide, problem-solving, formative assessment, design research
National Category
Didactics
Identifiers
urn:nbn:se:umu:diva-157561 (URN)
Available from: 2019-03-26 Created: 2019-03-26 Last updated: 2019-04-09

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