Understandhrg of rat io an d proport ion is critical t o t he development of higher level mathematical skills. Followhrg Carpenter, Gomez, et al.'s (1999) proposal of a four-level t rajectory hr t he development o f proportional reasonhrg, a 12-week hr vestigation was undertaken of t he developmental trajectory of pro portional reas onhrg of girls hr two 5t h-grade cl asses i n Iceland. Students hr these classes were accustomed to hrstructional practices that en couraged t hem to de vise an d explahr their own solution s to mathematical problems. Results of the study cmffirm the learnhrg trajectory with the addition of a further disthrct level of development between Levol 2 and Level 3. R esults sh owed that girls moved easily, with mhr imum scaffoldhrg, from Level i to 2 and from Level 2 to 3. The transition to Level 4, wh ich hr volves explicit awareness of 'wit hhr' an d' between' multiplicative r elationships, t ook great er time an d effort.Teacher awareness of the four-level learnhrg strategy, with the new emerghrg Level 3, assists hr th e design o f appropriate pro blems, cla ss structure, and teachhrg strategies. Building on Lamon's notions of un itizhrg and normhrg as by Carpenter et al.'s (1999) developmental model, this study contributes to our com prehension of s tudents' unders tandhrg of proportionality an d how it develops.
Students' mathematical achievement in Iceland, as reported in PISA 2003, showed significant and (by comparison) unusual gender differences in mathematics: Iceland was the only country in which the mathematics gender gap favored girls. When data were broken down and analyzed, the Icelandic gender gap appeared statistically significant only in the rural areas of Iceland, suggesting a question about differences in rural and urban educational communities. In the 2007 qualitative research study reported in this paper, the authors interviewed 19 students from rural and urban Iceland who participated in PISA 2003 in order to investigate these differences and to identify factors that contributed to gender differences in mathematics learning. Students were asked to talk about their mathematical experiences, their thoughts about the PISA results, and their ideas about the reasons behind the PISA 2003 results. The data were transcribed, coded, and analyzed using techniques from analytic induction in order to build themes and to present both male and female student perspectives on the Icelandic anomaly. Strikingly, youth in the interviews focused on social and societal factors concerning education in general rather then on their mathematics education.
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