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  • 1.
    Jonsson, Bert
    et al.
    Umeå University, Faculty of Social Sciences, Department of Psychology.
    Karlsson, Linnea
    Umeå University, Faculty of Medicine, Department of Integrative Medical Biology (IMB). Umeå University, Faculty of Medicine, Umeå Centre for Functional Brain Imaging (UFBI).
    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.
    Liljekvist, Yvonne
    Karlstad University.
    Norqvist, Mathias
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Science and Mathematics Education.
    Nyberg, Lars
    Umeå University, Faculty of Medicine, Department of Integrative Medical Biology (IMB), Physiology. Umeå University, Faculty of Medicine, Umeå Centre for Functional Brain Imaging (UFBI). Umeå University, Faculty of Medicine, Department of Radiation Sciences, Diagnostic Radiology.
    Mathematical Teaching Method affects Performance and Brain Activity2012Conference paper (Refereed)
  • 2.
    Jonsson, Bert
    et al.
    Umeå University, Faculty of Social Sciences, Department of Psychology.
    Norqvist, Mathias
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Liljekvist, Yvonne
    Department of Mathematics and Computer Science, Karlstad University, Sweden ; The Centre of Science, Mathematics and Engineering Education Research, Karlstad University, 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.
    Learning mathematics through algorithmic and creative reasoning2014In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, no 36, p. 20-32Article in journal (Refereed)
    Abstract [en]

    There are extensive concerns pertaining to the idea that students do not develop sufficient mathematical competence. This problem is at least partially related to the teaching of procedure-based learning. Although better teaching methods are proposed, there are very limited research insights as to why some methods work better than others, and the conditions under which these methods are applied. The present paper evaluates a model based on students’ own creation of knowledge, denoted creative mathematically founded reasoning (CMR), and compare this to a procedure-based model of teaching that is similar to what is commonly found in schools, denoted algorithmic reasoning (AR). In the present study, CMR was found to outperform AR. It was also found cognitive proficiency was significantly associated to test task performance. However the analysis also showed that the effect was more pronounced for the AR group.

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    Learning mathematics through algorithmic and creative reasning
  • 3.
    Karlsson Wirebring, Linnea
    et al.
    Umeå University, Faculty of Social Sciences, Department of Psychology. Umeå University, Faculty of Medicine, Department of Integrative Medical Biology (IMB). Umeå University, Faculty of Medicine, Umeå Centre for Functional Brain Imaging (UFBI).
    Lithner, 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).
    Jonsson, Bert
    Umeå University, Faculty of Social Sciences, Department of Psychology.
    Liljekvist, Yvonne
    Karlstad, Sweden.
    Norqvist, Mathias
    Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC). Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyberg, Lars
    Umeå University, Faculty of Medicine, Umeå Centre for Functional Brain Imaging (UFBI). Umeå University, Faculty of Medicine, Department of Radiation Sciences, Diagnostic Radiology. Umeå University, Faculty of Medicine, Department of Integrative Medical Biology (IMB).
    Learning mathematics without a suggested solution method: durable effects on performance and brain activity2015In: Trends in Neuroscience and Education, ISSN 2211-9493, Vol. 4, no 1-2, p. 6-14Article in journal (Refereed)
    Abstract [en]

    A dominant mathematics teaching method is to present a solution method and let pupils repeatedly practice it. An alternative method is to let pupils create a solution method themselves. The current study compared these two approaches in terms of lasting effects on performance and brain activity. Seventythree participants practiced mathematics according to one of the two approaches. One week later, participants underwent fMRI while being tested on the practice tasks. Participants who had created the solution method themselves performed better at the test questions. In both conditions, participants engaged a fronto-parietal network more when solving test questions compared to a baseline task. Importantly, participants who had created the solution method themselves showed relatively lower brain activity in angular gyrus, possibly reflecting reduced demands on verbal memory. These results indicate that there might be advantages to creating the solution method oneself, and thus have implications for the design of teaching methods.

  • 4.
    Norqvist, Mathias
    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.
    Cognitive Abilities and Mathematical Reasoning in Practice and Test Situations2018In: Proceedings of the 42ndConference of the International Group for the Psychology of Mathematics Education / [ed] Bergqvist, E., Österholm, M., Granberg, C., & Sumpter, L., Umeå, Sweden: PME , 2018, Vol. 3, p. 419-426Conference paper (Refereed)
    Abstract [en]

    Research studies have shown that to develope conceptual understanding of mathematics, practice needs to that focus this skill. In this study, the aim is to examine how different practice tasks, which promotes either imitative or creative mathematical reasoning, can influence which variables (i.e., cognitive abilities, mathematics grade, and gender) that are important for task completion. Two earlier studies show that cognitive abilities are more important in the test situation when students have practiced with imitative tasks. The result from this study indicate that although cognitive abilities are important when practicing with creative tasks, the influence of cognition is only implicit during the test. Since students often practice imitatively with given solution methods, this study suggests that a substantial part of what we test in school could be cognitive abilities rather than mathematics.

  • 5.
    Norqvist, Mathias
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Do explanations increase the efficiency of procedural tasks?Manuscript (preprint) (Other academic)
    Abstract [en]

    Studies in mathematics education often point to the necessity for students to engage in more cognitively demanding activities than just solving tasks by applying given solution methods. Lithner’s (2008) framework on mathematical reasoning address this by studying which reasoning a task promotes. Previous studies have shown that students that create their own solution methods, denoted creative reasoning (CMR), perform significantly better in follow up tests than students that are given the solution method and engage in algorithmic reasoning (AR). However, teachers and textbooks at least occasionally provide explanations together with a solution method (XAR) and this could possibly be more efficient than creative reasoning. In this study three matched groups practiced with either AR-, XAR- or CMR-tasks. The study showed that students that practiced with AR- and XAR-tasks performed similarly during both practice and test. The CMR-group did, although a worse practice score, outperform the XAR- group in the test. Additionally, there were differences in which variables predicted the test result with similar results between the XAR- and AR-group where cognitive proficiency and mathematics grade where significant. For the CMR-group the test score was predicted by the practice score alone. This would indicate that students that practice with CMR-tasks are not as dependent on their cognitive abilities for test performance as students that practice with AR- tasks. 

  • 6.
    Norqvist, Mathias
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Explanations do not improve algorithmic reasoning tasks: Volume 12016In: Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education: Volume 1 / [ed] Csíkos, C., Rausch, A., & Szitányi, J., Szeged, Hungary: International Group for the Psychology of Mathematics Education , 2016, Vol. 1, p. 110-110Conference paper (Refereed)
  • 7.
    Norqvist, Mathias
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On Mathematical Reasoning: being told or finding out2016Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    School-mathematics has been shown to mainly comprise rote-learning of procedures where the considerations of intrinsic mathematical properties are scarce. At the same time theories and syllabi emphasize competencies like problem solving and reasoning. This thesis will therefore concern how task design can influence the reasoning that students apply when solving tasks, and how the reasoning during practice is associated to students’ results, cognitive capacity, and brain activity. In studies 1-3, we examine the efficiency of different types of reasoning (i.e., algorithmic reasoning (AR) or creative mathematically founded reasoning (CMR)) in between-groups designs. We use mathematics grade, gender, and cognitive capacity as matching variables to get similar groups. We let the groups practice 14 different solution methods with tasks designed to promote either AR or CMR, and after one week the students are tested on the practiced solution methods. In study 3 the students did the test in and fMRI-scanner to study if the differing practice would yield any lasting differences in brain activation. Study 4 had a different approach and focused details in students’ reasoning when working on teacher constructed tasks in an ordinary classroom environment. Here we utilized audio-recordings of students’ solving tasks, together with interviews with teachers and students to unravel the reasoning sequences that students embark on. The turning points where the students switch subtask and the reasoning between these points were characterized and visualized. The behavioral results suggest that CMR is more efficient than AR, and also less dependent on cognitive capacity during the test. The latter is confirmed by fMRI, which showed that AR had higher activation than CMR in areas connected to memory retrieval and working memory. The behavioral result also suggested that CMR is more beneficial for cognitively less proficient students than for the high achievers. Also, task design is essential for both students’ choice of reasoning and task progression. The findings suggest that: 1) since CMR is more efficient than AR, students need to encounter more CMR, both during task solving and in teacher presentation, 2) cognitive capacity is important but depending on task design, cognitive strain will be more or less high during test situations, 3) although AR-tasks does not prohibit the use of CMR they make it less likely to occur. Since CMR-tasks can emphasize important mathematical properties, are more efficient than AR- tasks, and more beneficial for less cognitively proficient students, promoting CMR can be essential if we want students to become mathematically literate. 

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  • 8.
    Norqvist, Mathias
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    The effect of explanations on mathematical reasoning tasks2018In: International journal of mathematical education in science and technology, ISSN 0020-739X, E-ISSN 1464-5211, Vol. 49, no 1, p. 15-30Article in journal (Refereed)
    Abstract [en]

    Studies in mathematics education often point to the necessity for students to engage in more cognitively demanding activities than just solving tasks by applying given solution methods. Previous studies have shown that students that engage in creative mathematically founded reasoning to construct a solution method, perform significantly better in follow up tests than students that are given a solution method and engage in algorithmic reasoning. However, teachers and textbooks, at least occasionally, provide explanations together with an algorithmic method, and this could possibly be more efficient than creative reasoning. In this study, three matched groups practiced with either creative, algorithmic, or explained algorithmic tasks. The main finding was that students that practiced with creative tasks did, outperform the students that practiced with explained algorithmic tasks in a post-test, despite a much lower practice score. The two groups that got a solution method presented, performed similarly in both practice and post-test, even though one group got an explanation to the given solution method. Additionally, there were some differences between the groups in which variables predicted the post-test score.

  • 9.
    Norqvist, Mathias
    et al.
    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.
    Jonsson, Bert
    Umeå University, Faculty of Social Sciences, Department of Psychology.
    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.
    Eye-tracking data and mathematical tasks with focus on mathematical reasoning2019In: Data in Brief, E-ISSN 2352-3409, Vol. 25, article id 104216Article in journal (Refereed)
    Abstract [en]

    This data article contains eye-tracking data (i.e., dwell time and fixations), Z-transformed cognitive data (i.e., Raven's Advanced Progressive Matrices and Operation span), and practice and test scores from a study in mathematics education. This data is provided in a supplementary file. The method section describes the mathematics tasks used in the study. These mathematics tasks are of two kinds, with and without solution templates, to induce different types of mathematical reasoning.

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  • 10.
    Norqvist, Mathias
    et al.
    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.
    Jonsson, Bert
    Umeå University, Faculty of Social Sciences, Department of Psychology.
    Lithner, 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).
    Qwillbard, Tony
    Umeå University, Faculty of Social Sciences, Department of Psychology.
    Holm, Linus
    Umeå University, Faculty of Social Sciences, Department of Psychology.
    Investigating algorithmic and creative reasoning strategies by eye tracking2019In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 55, article id 100701Article in journal (Refereed)
    Abstract [en]

    Imitative teaching and learning approaches have been dominating in mathematics education. Although more creative approaches (e.g. problem-based learning) have been proposed and implemented, a main challenge of mathematics education research is to document robust links between teaching, tasks, student activities and learning. This study investigates one aspect of such links, by contrasting tasks providing algorithmic solution templates with tasks requiring students’ constructions of solutions and relating this to students’ learning processes and outcomes. Information about students’ task solving strategies are gathered by corneal eye-tracking, which is related to subsequent post-test performances and individual variation in cognitive proficiency. Results show that students practicing by creative tasks outperform students practicing by imitative algorithmic tasks in the post-test, but also that students that perform less well on creative tasks tend to try ineffective imitative strategies.

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  • 11.
    Norqvist, Mathias
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Liljekvist, Yvonne
    Karlstads universitet.
    Jonsson, Bert
    Umeå University, Faculty of Social Sciences, Department of Psychology.
    Lithner, 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).
    Increased efficiency when engaging in creative mathematical founded reasoning2013In: Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Vol. 5. / [ed] Lindmeier, A. M. & Heinze, A., Kiel, Germany: PME , 2013, p. 136-136Conference paper (Refereed)
  • 12.
    Norqvist, Mathias
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Lithner, 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).
    Jonsson, Bert
    Umeå University, Faculty of Social Sciences, Department of Psychology.
    Liljekvist, Yvonne
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap..
    Creative Reasoning More Beneficial For Cognitively Weaker Students2015In: Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education, Prague: Charles University , 2015, p. 502-503Conference paper (Refereed)
    Abstract [en]

    In a study with 91 upper-secondary students the efficiency of two different types of mathematical practice tasks, procedural based algorithmic tasks and creative reasoning tasks, were studied. It was found that although the algorithmic group outperformed the creative group during practice the latter performed significantly better on a follow-up test. Closer inspection revealed that the difference in test performance was, contrary to common beliefs, driven by the cognitively weaker students.

  • 13.
    Van Steenbrugge, Hendrik
    et al.
    School of Education, Culture and Communication , Mälardalen U niversity, Sweden.
    Norqvist, Mathias
    Umeå University, Faculty of Science and Technology, Department of Science and Mathematics Education.
    Students' Reasoning in the Classroom: An Approach for Analysis2018In: Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education / [ed] Bergqvist, E., Österholm, M., Granberg, C., & Sumpter, L., Umeå: PME , 2018, Vol. 5, p. 305-305Conference paper (Refereed)
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  • 14.
    Van Steenbrugge, Hendrik
    et al.
    Mälardalens Högskola.
    Norqvist, Mathias
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Unraveling students’ reasoning: analyzing small-group discussions during task solvingManuscript (preprint) (Other academic)
    Abstract [en]

    The aim of this study is to examine students’ mathematical reasoning, suggested by Lithner (2008), to see how reasoning sequences will unfold in actual classroom situations. We visited two classrooms in an upper secondary school and observed two student groups in each classroom for the time it took them to complete a task, constructed and presented to them by their teacher. Initial analysis showed that there were two interesting dimensions to regard, group characteristics (i.e., motivation and persistence) and task design (i.e., reasoning promoted by the task). We recorded conversations between the students and after transcribing we utilized Lithner’s (2008) framework of mathematical reasoning to analyze students’ reasoning. We classified the moments (vertices) when the students’ reasoning took a new trajectory and characterized the segment (edge) between two such vertices according to the students’ reasoning (i.e., either creative mathematically founded reasoning or algorithmic reasoning). We then visualized the students’ reasoning in graphs (see Figure 10) and analyzed the patterns, the progress and types of reasoning, as well as how the group characteristics and task design would influence reasoning and progress. The result showed that task design is important for which reasoning the students will use. Although an algorithmic-task does not exclude creative reasoning, it only occurs in our data if the students have difficulties and strive to handle them by themselves. We also observed that group characteristics were important for the chosen reasoning type. 

  • 15.
    Vennberg, Helena
    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).
    Norqvist, Mathias
    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.
    Counting on: Long Term Effects of an Early Intervention Programme2018In: Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education / [ed] Bergqvist, E., Österholm, M., Granberg, C., & Sumpter, L., Umeå: PME , 2018, Vol. 4, p. 355-362Conference paper (Refereed)
    Abstract [en]

    This paper reports the long-term results of an intervention study with 134 six-year-old students from seven preschool-classes in northern Sweden to evaluate whether the Think, Reason and Count in Preschool-class programme (TRC) could prevent at-risk students from becoming low-performing students in mathematics. Whereas the pre-test score revealed that the intervention and the control group preformed equally, scores on the delayed follow-up-test in Grade 3 showed that the intervention group performed better than the control group and that at-risk students had closed the performance gap between themselves and their not-at-risk peers.

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