Abstract:

Previous studies have revealed that students have misconceptions on numbers specifically on real numbers (Tall & Schwarzenberger, 1978; Ely, 2010). In order to eliminate the misconceptions, Voskoglou (2013) suggested that teaching should emphasize the use of multiple representations of real numbers and flexible transformations among the representations. In the current study, we conducted classroom teaching experiments (Cobb, 2000) with 19 prospective mathematics teachers in an English-medium university in İstanbul about the decimal representation of real numbers with the emphasis of quantitative reasoning (Thompson, 2011; Karagöz-Akar, 2016). The ongoing and retrospective data analysis was done through line by line analysis of the transcriptions of the video records and the written artifacts. Results showed that thinking through quantities depicted in diagrams, once prospective teachers related long division with multiple representations of rational numbers such as fractions, equivalent fractions and decimals through the mental actions of equal partitioning, grouping and counting, they were able to deduce that all these representations corresponded to the same number and squeezing the decimal representation of both rational and irrational numbers, prospective teachers were able to deduce that real numbers could be represented by the limits of rational number sequences. Results might contribute to the mathematics education field by providing task sequences showing how difficulties regarding the real numbers could be eliminated via focusing on quantities. 

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