Flexibility in knowing school mathematics in the contexts of a Swedish and an Indian school class

Abstract: A central question in mathematics education research concerns understanding. The main objective of the present thesis has been to obtain insights into flexible modes of knowing in school mathematics in two school class contexts, and how these relate to modes of being a learner in these contexts, with specific focus on learners’ flexible ways of discerning parts and delimiting wholes, and how they understand part- and whole-relationships while doing mathematics. The theoretical exploration of knowing school mathematics was informed by perspectives from phenomenography and variation theory, as well as constructionist theoretical standpoints. Empirical material was collected from a school class in Southern Sweden and a school class in Orissa, Central-Eastern India. The meaning the learners expressed during interviews and observations, verbally or with the help of mathematics, was analysed using contextual analysis. In line with methods in phenomenographic research, the main results of the thesis are different categories of description. Three modes of knowing emerged from the empirical material. These were: associative flexible experiencing; compositional flexible experiencing and contextual flexible experiencing. These modes of knowing feature distinct differences: in the depth of understanding mathematics, in how learners use variation when dealing with an object of knowledge, and in learner identity. The associative mode of knowing involved the learner in arbitrary ways of making sense of the material s/he was working with, with a focus on arbitrarily discerned aspects in chains of associations. The compositional mode of knowing meant that the learner made an effort to understand, keeping a focus on compositions, such as number-relations or formulas. Finally, the contextual mode of knowing engaged the learners in ways of understanding the context from which critical aspects were to be discerned. The contexts gave meanings to the content. The knowledge about the context, mathematical and also reality-based, gave meaning to the theoretical constructs. The logic of the mathematical context and content was understood in a more differentiated way than within the two other modes of knowing. In all parts of empirical material, the compositional flexible mode of knowing predominated. The dominant mode of being a learner in the Swedish school class context was simultaneously independent and collaborative, as well as creative and productive. In the Indian school class context, the dominating mode of being a learner was autonomous and committed. A major finding is that in mathematics education there is a need to give pupils tasks containing possibilities both for experiencing variation and for authorship. This also demands of the teacher to observe and evaluate the individual pupil’s understanding and use of the possibilities offered.

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