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.

Aligned with the enhanced international commitment to early childhood education, recognition of the importance of providing young children with opportunities to develop mathematical understandings and skills is increasing. While there is much research about effective mathematics pedagogy in the school sector, less research activity is evident within the early childhood sector. Focused on people, relationships and the learning environment, this article draws on a synthesis of research on effective pedagogical practices to describe effective learning communities that can enhance the development of young children’s mathematical identities and competencies. Concerned that the wider synthesis noted limited cross-sector collaboration within the mathematics education community, this article aims to act as a bridge for researchers currently working within the preschool and school sectors. The authors argue that understandings of effective pedagogies that enhance young children’s mathematics learning will benefit from more cross-sector research studies.

• Towards new documentation systems for mathematics teachers? by Ghislaine Gueudet and Luc Trouche
• Experiencing equivalence but organizing order, by Amir H. Asghari
• A categorization of the “whys” and “hows” of using history in mathematics education, by Uffe Thomas Jankvist
• Intuitive vs analytical thinking: four perspectives, by Uri Leron and Orit Hazzan
• Using graphing software to teach about algebraic forms: a study of technology-supported practice in secondary-school mathematics, by Kenneth Ruthven, Rosemary Deaney and Sara Hennessy
• Perceptions that may affect teachers’ intention to use technology in secondary mathematics classes, by Robyn Pierce and Lynda Ball

• Learning opportunities from group discussions: Warrants become the objects of debate
  K. Weber, C. Maher, A. Powell, & H. Lee
• Transitions among different symbolic generalizations by algebra beginners in a computer intensive environment
  M. Tabach, A. Arcavi, & R. Hershkowitz
• Abstraction and consolidation of the limit precept by means of instrumented schemes: The complementary role of three different frameworks
  I. Kidron
• Signifying “students”, “teachers” and “mathematics”: A reading of a special issue
  T. Brown
• On semiotics and subjectivity: A response to Tony Brown’s “signifying ‘students’, ‘teachers’ and ‘mathematics’: a reading of a special issue”.
• Cognitive styles, dynamic geometry and measurement performance.
  D. Pitta-Pantazi & C. Christou
• Embodied design: Constructing means for constructing meaning
  D. Abrahamson
• Constructing competence: An analysis of student participation in the activity system of mathematics classrooms
  M. Gresalfi, T. Martin, V. Hand, & J. Greeno
• Teachers’ perspectives on “authentic mathematics” and the two-column proof form
  M. Weiss, P. Herbst, & C. Chen
• From arithmetical thought to algebraic thought: The role of the “variable”
  E. Malisani & F. Spagnolo