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  • 1.
    Jäder, Jonas
    et al.
    Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC). School of Education, Health and Social Studies, Dalarna University, Falun, Sweden.
    Lithner, Johan
    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.
    Sidenvall, Johan
    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.
    Mathematical problem solving in textbooks from twelve countries2019In: International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, E-ISSN 1464-5211Article in journal (Refereed)
    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.

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  • 2. Jäder, Jonas
    et al.
    Sidenvall, Johan
    Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Mathematical Reasoning and Beliefs2014In: Proceedings of the Joint Meeting of PME 38 and PME-NA 36 / [ed] Oesterle, S., Nicol, C., Liljedahl, P., & Allan, D., Vancouver, 2014, Vol. 6, p. 114-Conference paper (Refereed)
    Abstract [en]

    We present a research project on students’ mathematical reasoning and how beliefs are indicated in their arguments. Preliminary results show that students express beliefs that task solving does not include reflection or much struggle. The results underpin earlier studies stressing expectations as a theme of belief.

  • 3.
    Jäder, Jonas
    et al.
    School of Education, Health and Social Studies, Dalarna University, Sweden; Department of Social and Welfare Studies, Linköping University, Norrköping, Sweden.
    Sidenvall, Johan
    Department of Social and Welfare Studies, Linköping University, Norrköping, Sweden; School Administration, Municipality of Hudiksvall, Hudiksvall, Sweden.
    Sumpter, Lovisa
    Students' mathematical reasoning and beliefs in non-routine task solving2017In: International Journal of Science and Mathematics Education, ISSN 1571-0068, E-ISSN 1573-1774, Vol. 15, no 4, p. 759-776Article in journal (Refereed)
    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.

  • 4.
    Sidenvall, Johan
    Linköpings universitet, Institutionen för samhälls- och välfärdsstudier.
    Att lära sig resonera: om elevers möjligheter att lära sig matematiska resonemang2015Licentiate thesis, comprehensive summary (Other academic)
    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. 

  • 5.
    Sidenvall, Johan
    Institutionen för samhälls- och välfärdsstudier, Linköpings universitet.
    Att lära sig resonera: Om elevers möjligheter att lära sig matematiska resonemang2015Licentiate thesis, comprehensive summary (Other academic)
    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. 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.

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  • 6.
    Sidenvall, Johan
    Umeå University, Faculty of Science and Technology, Department of Science and Mathematics Education. Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Literature review of mathematics teaching design for problem solving and reasoning2019In: Nordisk matematikkdidaktikk, NOMAD: [Nordic Studies in Mathematics Education], ISSN 1104-2176, Vol. 24, no 1, p. 51-74Article in journal (Refereed)
    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.

  • 7.
    Sidenvall, Johan
    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.
    Lösa problem: om elevers förutsättningar att lösa problem och hur lärare kan stödja processen2019Doctoral thesis, comprehensive summary (Other academic)
    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.

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  • 8.
    Sidenvall, Johan
    Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Students’ Reasoning in Mathematics Textbook Task Solving2013In: FontD 10-year anniversary meeting with the Scientific Committee / [ed] K. Schönborn & L. Tibell, Norrköping: Linköping University Electronic Press, 2013Conference paper (Refereed)
    Abstract [en]

    To a large extent school mathematics consists of procedures to be memorized without understanding. Using procedures without understanding is one of the main causes behind the difficulties when learning mathematics. By developing mathematically founded reasoning, other mathematical competences are also developed; problem solving and conceptual understanding. The textbook is important in mathematical education. This study reports an analysis of students’ textbook task solving in Swedish upper secondary school were the relation between types of mathematical reasoning used and the success and failure to complete tasks was studied. Results show that using rote learning and superficial reasoning is common, although only about 50 % of the tasks where successfully solved using such strategies. In the case that mathematically founded reasoning was used, the tasks were successfully solved. Conclusions from the study are that students’ work with textbook tasks mainly let the students develop rote learning and superficial reasoning and students are not enhanced to develop mathematically founded reasoning in their work with textbook tasks. Students have to work with textbook tasks in other ways or be given other additional opportunities in order to develop mathematically founded reasoning.

  • 9.
    Sidenvall, Johan
    et al.
    Umeå University, Faculty of Science and Technology, Department of Science and Mathematics Education. Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Granberg, Carina
    Umeå University, Faculty of Social Sciences, Department of applied educational science, Interactive Media and Learning (IML). Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Lithner, Johan
    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.
    Palmberg, Björn
    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.
    Supporting Teachers to Support Students’ Problem-solvingManuscript (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.

  • 10.
    Sidenvall, Johan
    et al.
    Linköping University.
    Jäder, Jonas
    Sumpter, Lovisa
    Mathematical reasoning and beliefs in non-routine task solving2015In: Current State of Research on Mathematical Beliefs XX: Proceedings of the MAVI-20 / [ed] Lovisa Sumpter, Falun: Högskolan Dalarna, 2015Conference paper (Refereed)
    Abstract [en]

    This paper explores low performing upper secondary school students’ mathematical reasoning when solving non-routine tasks in pairs. Their solutions were analysed using a theoretical framework about mathematical reasoning and a model to study beliefs as arguments for choices. The results confirm previous research and three themes of beliefs are used by the student. These themes are safety, expectations, and motivation. The results also show a connection between beliefs and imitative reasoning as a way to solve non-routine tasks.

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  • 11.
    Sidenvall, Johan
    et al.
    Department of Social and Welfare Studies, Linköping University, Sweden.
    Lithner, Johan
    Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Jäder, Jonas
    Department of Social and Welfare Studies, Linköping University, Sweden.
    Students' reasoning in mathematics textbook task-solving2015In: 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)
    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.

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