Begrunnelse og resonnement i divisjon av brøk : En case studie av hvordan åttende trinns elever begrunner og resonnerer når de løser divisjon av brøk oppgaver
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Fractionsarethe most complex numbers students meet in Elementaryschool and a complex concept (Bulgar, 2002).In addition, division of fraction is considered themost mechanical andleast understood algorithm(Tirosh, 2000).In this studyI have thereforeconsidered how students justify and reason in tasks with division of fractions. The study was based on these research questions: 1)How do eight-degreestudents justify and reason in tasks with division of fractions?a)Which representations areused and how aretheyused in the students ́justification and reasoning? b)Which errors and misconceptionsoccur in students ́justification and reasoning? To answer these research questions,eight-degree students have solvedone task that involvedmeasurement division and one task about partitive division. Each student ́sassignment hasbeen considered relative to which justification and reasoning that’s used: Contradiction, recognition of patterns, natural numbers, fractionsand measurement. In addition, representations, errors and misconceptions have been considered. To give a deeper explanation of the student’sassignments and their thinking processwhen solvingthe tasks, six students have been chosen for aninterview. To provide a more comprehensive representation of division of fraction, the fraction conceptand other arithmetic operations are described in the theoretical framework. This part also consistsof earlier research performedon justification and reasoning duringdivision of fractions. Relevant theory about representations and errors are also presented in thispart. In this way,the analysis of the students ́assignments can be justified with relevant theory. Measurement division contributed revealmany different justifications andreasonings, where different representations were used to explain the justifications. In det partitive division task,the use ofdifferentjustification and reasoning, and different systems of representations were not clear as it was in themeasurement division task. The main findings in the study arestill observable in both of thetasks.Students ́justification and reasoning can be categorizedintothree main categories: Fraction, natural numbersand measurement. Students ́use of representations can be categorizedintofour main categories: Symbols as numbers, symbols as mathematical signs, drawing as justification and reasoning and language as justification and reasoning. In addition, the errors students have madecan be categorized intothree different types of errors: Algorithmically based errors, intuitive errors, and errors based on formal knowledge. It turned out that thereis a big variation withineach category.In several assignments it occurs that one task has been solved with a combinationof justifications and reasonings. Different semiotic representations can be used in the same justification and reasoning.Lack of different types of knowledge can result in errors being categorized into different categories.
Masteroppgave matematikkdidaktikk MA502 – Universitetet i Agder 2019