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  • 1.
    Bergqvist, Ewa
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Types of reasoning required in university exams in mathematics2007In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 26, no 4, p. 348-370Article in journal (Refereed)
    Abstract [en]

    Empirical research shows that students often use reasoning founded on copying algorithms or recalling facts (imitative reasoning) when solving mathematical tasks. Research also indicate that a focus on this type of reasoning might weaken the students' understanding of the underlying mathematical concepts. It is therefore important to study the types of reasoning students have to perform in order to solve exam tasks and pass exams. The purpose of this study is to examine what types of reasoning students taking introductory calculus courses are required to perform. Tasks from 16 exams produced at four different Swedish universities were analyzed and sorted into task classes. The analysis resulted in several examples of tasks demanding different types of mathematical reasoning. The results also show that about 70% of the tasks were solvable by imitative reasoning and that 15 of the exams could be passed using only imitative reasoning.

  • 2.
    Bergqvist, Ewa
    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).
    Theens, Frithjof
    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.
    Österholm, Magnus
    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). Department of Mathematics and Science Education, Mid Sweden University, SE-85170, Sundsvall, Sweden.
    The role of linguistic features when reading and solving mathematics tasks in different languages2018In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 51, p. 41-55Article in journal (Refereed)
    Abstract [en]

    The purpose of this study is to deepen the understanding of the relation between the language used in mathematics tasks and the difficulty in reading and solving the tasks. We examine issues of language both through linguistic features of tasks (word length, sentence length, task length, and information density) and through different natural languages used to formulate the tasks (English, German, and Swedish). Analyses of 83 PISA mathematics tasks reveal that tasks in German, when compared with English and Swedish, show stronger connections between the examined linguistic features of tasks and difficulty in reading and solving the tasks. We discuss if and how this result can be explained by general differences between the three languages.

  • 3.
    Bergqvist, Tomas
    et al.
    Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC). Umeå University, Faculty of Social Sciences, Department of applied educational science, Interactive Media and Learning (IML).
    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.
    Mathematical reasoning in teachers' presentations2012In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 31, no 2, p. 31p. 252-269Article in journal (Refereed)
    Abstract [en]

    This paper presents a study of the opportunities presented to students that allow them to learn different types of mathematical reasoning during teachers’ ordinary task solving presentations. The characteristics of algorithmic and creative reasoning that are seen in the presentations are analyzed. We find that most task solutions are based on available algorithms, often without arguments that justify the reasoning, which may lead to rote learning. The students are given some opportunities to see aspects of creative reasoning, such as reflection and arguments that are anchored in the mathematical properties of the task components, but in relatively modest ways.

  • 4.
    Boesen, Jesper
    et al.
    Göteborgs universitet.
    Helenius, Ola
    Göteborgs universitet och Örebro universitet.
    Bergqvist, Ewa
    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).
    Bergqvist, Tomas
    Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC). Umeå University, Faculty of Social Sciences, Department of applied educational science.
    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).
    Palm, Torulf
    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).
    Palmberg, Björn
    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).
    Developing mathematical competence: from the intended to the enacted curriculum2014In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 33, p. 72-87Article in journal (Refereed)
    Abstract [en]

    This study investigates the impact of a national reform in Sweden introducing mathematical competency goals. Data were gathered through interviews, classroom observations, and online surveys with nearly 200 teachers. Contrasting to most studies of this size, qualitative analyses were conducted. The results show that teachers are positive to the message, but the combination of using national curriculum documents and national tests to convey the reform message has not been sufficient for teachers to identify the meaning of the message. Thus, the teachers have not acquired the functional knowledge of the competence message required to modify their teaching in alignment with the reform. The results indicate that for complex reform messages, such as the competency message, to have intended impact on classroom practice, special attention needs to be put on the clarity of the message. To have high-stakes tests, for example, does not alone seem to be sufficient. 

  • 5.
    Granberg, Carina
    Umeå University, Faculty of Social Sciences, Department of applied educational science, Interactive Media and Learning (IML).
    Discovering and addressing errors during mathematics problem-solving — A productive struggle?2016In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 42, p. 33-48Article in journal (Refereed)
    Abstract [en]

    The present study investigates students' struggles when encountering errors in problem solving. The focus is students' problem-solving activities that lead to productive struggle and what the students might gain therefrom. Twenty-four students between the ages of 16 and 17 worked in pairs to solve a linear function problem using GeoGebra, a dynamic software application. Data in the form of recorded conversations, computer activities and post-interviews were analyzed using Hiebert and Grouws' (2007. Second handbook of research on mathematics teaching and learning (Vol. 1). 404) concept of productive struggles and Schoenfeld's (1985. Mathematical problem solving: ERIC) framework for problem-solving. The study showed that all students made errors concerning incorrect prior knowledge and erroneously constructed new knowledge. All participants engaged in superficial, unproductive struggles moving between a couple of Schoenfeld's episodes. However, a majority of the students managed to transform their efforts into productive struggle. They engaged in several of Schoenfeld's episodes and succeeded in reconstructing useful prior knowledge and constructing correct new knowledge i.e., solving the problem.

  • 6.
    Granberg, Carina
    et al.
    Umeå University, Faculty of Social Sciences, Department of applied educational science, Interactive Media and Learning (IML).
    Olsson, Jan
    Umeå University, Faculty of Science and Technology, Department of Science and Mathematics Education.
    ICT-supported problem solving and collaborative creative reasoning: Exploring linear functions using dynamic mathematics software2015In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 37, p. 48-62Article in journal (Refereed)
    Abstract [en]

    The present study investigates how a dynamic software program, GeoGebra, may support students' collaboration and creative reasoning during mathematical problem solving. Thirty-six students between the ages of 16 and 17 worked in pairs to solve a linear function using GeoGebra. Data in the form of recorded conversations, and computer activities were analyzed using Lithner's (2008) framework of imitative and creative reasoning in conjunction with the collaborative model of joint problem space (Roschelle & Teasley, 1994). The results indicated that GeoGebra supported collaboration and creative reasoning by providing students with a shared working space and feedback that became the subject for students' creative reasoning. Furthermore, the students' collaborative activities aimed toward sharing their reasoning with one another enhanced their creative reasoning. There were also examples of students using GeoGebra for trial-and-error strategies and students engaging in superficial argumentation.

  • 7.
    Johansson, Helena
    et al.
    Department of Mathematics and Science Education, Mid Sweden University, Sundsvall, Sweden.
    Österholm, Magnus
    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). Department of Mathematics and Science Education, Mid Sweden University, Sundsvall, Sweden.
    Objectification of upper-secondary teachers’ verbal discourse in relation to symbolic expressions2019In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 56, article id 100722Article in journal (Refereed)
    Abstract [en]

    Research literature points to the importance of objectification when learning mathematics, and thereby in the discourse of mathematics. To increase the field’s understanding of aspects and degrees of objectification in various mathematical discourses, our study uses the combination of two sub-processes of objectification in order to analyse upper-secondary teachers’ word use in relation to any type of mathematical symbols. Our results show that the verbal discourse around symbols is very objectified. This can put high demands on students understanding of their teacher, since it might be needed that the students have reached a certain degree of objectification in their own thinking in order to be able to participate in a more objectified discourse. The results also show that there exist patterns in the variation of the degree of objectification, in particular that the discourse tends to be more objectified when more familiar symbols are used. This exploratory study also reveals several phenomena that could be the focus of more in-depth analyses in future studies.

  • 8.
    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.

  • 9.
    Lithner, Johan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Mathematical reasoning in calculus textbook exercises2004In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 23, no 4, p. 405-427Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is to study some of the strategies that are possible to use in order to solve the exercises in undergraduate calculus textbooks. It is described in detail how most exercises may be solved by mathematically superficial strategies, often without actually considering the core mathematics of the book section in question.

  • 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.

  • 11.
    Sundström (Raman), Manya
    Rutgers University, USA.
    Coordinating informal and formal aspects of mathematics: student behavior and textbook messages2002In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 21, no 2, p. 135-150Article in journal (Refereed)
    Abstract [en]

    In this paper I illustrate difficulties students have coordinating informal and formal aspects of mathematics. I also discuss two ways in which precalculus and calculus textbooks treat mathematics that may make this coordination difficult: emphasizing the informal at the expense of the formal and emphasizing the formal at the expense of the informal. By looking at student difficulties in light of textbook treatments, we see evidence that student difficulties are not merely developmental. Students are not given many opportunities to make the kinds of connections which, while difficult, are an essential component of mathematical thinking.

  • 12.
    Sundström (Raman), Manya
    Rutgers University, New Brunswick, USA.
    Epistemological messages conveyed by three high school and college mathematics textbooks2004In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 23, no 4, p. 389-404Article in journal (Refereed)
    Abstract [en]

    Mathematics textbooks embody a particular set of assumptions about mathematics or the mathematics intended for students at a particular level. Thus, an epistemological analysis of textbooks can provide some context for understanding, for example, the difficulties many students encounter when moving from high-school to collegiate mathematics. In this study, we consider how typical pre-calculus, calculus, and analysis texts treat the topic of continuity. We find that these texts send conflicting messages regarding the status and purpose of mathematical definitions.

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