Cecilia Kilhamn Phd thesis, University of Gothenburg

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Making Sense of Negative Numbers

Cecilia Kilhamn

PhD Thesis, University of Gothenburg

Supervisors: Prof Berner Lindström and Dr Ola Helenius

The general aim of this research project is to investigate how students make sense of negative numbers, and more specifically what role models and metaphorical reasoning play in that process. The topic of integers is chosen because it is a topic where metaphors are useful at the start but where at some point they break down. It is a topic where the transition from intuitive mathematics to formal mathematics becomes a necessity. The study is based on assumptions about mathematics as both a social and an abstract science and of metaphors as an important link between the social and the cognitive. Largely the study can be characterized as explorative, illuminating the complexity of mathematical thinking and the richness of the concept of negative numbers.
The empirical data was collected over a period of three years, following one Swedish school class being taught by the same teacher, using recurrent interviews, observation protocols and video recordings. As the data collection developed, a number of key issues emerged as central to the aims, namely students’ development of number sense, the use of metaphors in the mathematics classroom discourse and the influence of sociomathematical norms.
Conceptual metaphor theory was found to be useful to analyse teaching and learning about negative numbers, and showed that the extension of metaphors to incorporate integers is a sophisticated process that needs to be further investigated. In addition to the four grounding metaphors for arithmetic described in the theory a metaphor of Number as Relation is suggested as essential for the extension of the number domain. Different metaphors give very different meanings to statements such as finding the difference between two numbers; with different and incoherent mappings onto mathematical symbols. The analyses show affordances but also many constraints of the metaphors in their role as tools for sense making. Extending a metaphor will in fact change the metaphor, with unfamiliarity, inconsistency and limited applicability as a result. The results highlight the importance of understanding conditions of use for different metaphors, something that is not explicitly brought up during the lessons or in the textbook in the study. Findings also indicate that students are less apt to make explicit use of metaphorical reasoning than the teacher. Although metaphors help students initially to make sense of negative numbers, extended and inconsistent metaphors create discomfort and confusion. This suggests that the goal to give metaphorical meaning to specific tasks with negative numbers can be counteractive to the transition from intuitive to formal mathematics. Comparing and contrasting different metaphors could give more insight to the meaning embodied in mathematical structures than trying to fit the mathematical structure into any particular embodied metaphor.
Participants in the study showed quite different learning trajectories concerning their development of number sense when negative numbers were introduced. Problems that students had were often related to similar problems in the historical evolution of negative numbers, suggesting that teachers and students could benefit from deeper knowledge of the history of mathematics. Students with a highly developed number sense for positive numbers seemed to incorporate negatives more easily than students with a poorly developed numbers sense, implying that more time should be spent on number sense issues in the earlier years, particularly with respect to subtraction and to the number zero. The close study of the classroom mathematical discourse highlighted a few weaknesses such as; the lack of a useful Swedish word for integers or signed numbers; the many meanings of the Swedish word for number; the lack of clarifying language to distinguish between subtraction and negative numbers; and the implicit double interpretation of size of number as both value and magnitude.

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