As part of individual interviews incorporating whole number and rational number tasks, 323 grade 6 children in Victoria, Australia were asked to nominate the larger of two fractions for eight pairs, giving reasons for their choice. All tasks were expected to be undertaken mentally. The relative difficulty of the pairs was found to be close to that predicted, with the exception of fractions with the same numerators and different denominators, which proved surprisingly difficult. Students who demonstrated the greatest success were likely to use benchmark (transitive) and residual thinking. It is hypothesised that the methods of these successful students could form the basis of instructional approaches which may yield the kind of connected understanding promoted in various curriculum documents and required for the development of proportional reasoning in later years.

Hackenberg, A. J., & Tillema, E. S. (n.d.). Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. The Journal of Mathematical Behavior, In Press, Corrected Proof. doi: 10.1016/j.jmathb.2009.04.004.

Yanik, H. B., & Flores, A. (n.d.). Understanding rigid geometric transformations: Jeff’s learning path for translation. The Journal of Mathematical Behavior, In Press, Corrected Proof. doi: 10.1016/j.jmathb.2009.04.003.

This study investigates elementary school children’s flexible use of mental calculation strategies on additions and subtractions in the number domain 20–100. Sixty third-graders of three different mathematical achievement levels individually solved a series of 2-digit additions and subtractions in one choice and two no-choice conditions. In the choice condition, children could choose between the compensation (56 + 29 = ?; 56 + 30 = 86, 86 − 1 = 85) and jump strategy (56 + 29 = ?; 56 + 20 = 76, 76 + 9 = 85) on each item. In the two no-choice conditions, children had to solve each item with either the compensation or the jump strategy. The results demonstrated that children of all achievement levels spontaneously applied both the compensation and the jump strategy to solve the items from the choice condition. Furthermore, they all executed the compensation strategy equally accurately, but faster than the jump strategy in the no-choice conditions. Finally, children neither took into account the expected task nor individual strategy efficiency characteristics during the strategy choice process. Results are discussed in terms of recent models of adaptive strategy choices and instructional practices in the number domain 20–100.

In an effort to determine the most efficacious manner to deliver professional development training to early childhood educators, this study investigated the effect of a 2-h workshop followed by side-by-side classroom coaching. Twelve early childhood educators with 4-year degrees teaching in a university child development center participated in the study. The twice weekly classroom observations were analyzed for the use of math mediated language. Results indicate a 56% increase of math mediated language following the professional development; however, the greatest increase (39% increase over professional development condition) occurred during the side-by-side coaching phase of the treatment. These results corroborate previous findings that implementation of teaching strategies presented in professional development trainings can be enhanced by coaching teachers on the use of the strategies.