The University of Montana
Department of Mathematical Sciences

Technical report #1/2007

Icelandic 5th grade girls developmental trajectories in proportional reasoning

Olof Steinthorsdottir
University of North Carolina - Chapel Hill, USA

Bharath Sriraman
The University of Montana, USA


Understanding of ratio and proportion are critical to the development of higher level mathematical skills. Following the proposal by Carpenter et al (1999) of a four-level trajectory in the development of proportional reasoning, a twelve week investigation was undertaken of the developmental trajectory of proportional reasoning of girls in two fifth-grade classes in Iceland. Students in these classes were used to instructional practices that encouraged them to devise and explain their own solutions to mathematical problems. Results of the study confirm the learning trajectory with the addition of a further distinct level of development between Level 2 and Level 3. Results showed that girls moved easily, with minimum scaffolding, from Level 1 to 2 and from Level 2 to 3. The transition to Level 4, which involves explicit awareness of ‘within’ and ‘between’ multiplicative relationships, took greater time and effort. Teacher awareness of the four-level learning strategy, with the new emerging Level 3, will assist in the design of appropriate problems, class structure and teaching strategies.

Keywords: Iceland, Proportion, ratio, proportional reasoning, teaching proportion, learning trajectory, multiplicative thinking, mathematical learning, mathematical understanding.

AMS Subject Classification: 97

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