Developing mathematical understanding

The notion of scaffolding is used to introduce research on differentiated learning trajectories that make use of activities termed 'prompts'. The prompts are used to enable children to develop the necessary mathematical understanding and skills to keep up with the rest of the class.  They focus on aspects of teaching that teachers identified as important aspects of classroom interactivity that contribute to the understanding: the use of physical representation of concepts: actions aimed at building conceptual links; and the use of language based activity.

Developing mathematical concepts in Australian pre-school settings : the background

The SIMERR project Mathematical Thinking of Pre-school Children in Rural and Regional Australia: Research and Practice included a review of relevant research literature with the aim of making this accessible to researchers as well as early childhood teachers and educators. This paper introduces the methods used in the project and provides a brief summary of the literature pertaining to the development of mathematical concepts.

Assessment of mathematical development in early childhood : some views of preschool practitioners in regional and rural Australia

As part of the project Mathematical Thinking of Preschool Children in Rural and Regional Australia: Research and Practice directors, teachers, and assistants in prior-to-school settings from regional and rural eastern Australia were interviewed to ascertain their beliefs and practices concerning early childhood mathematics. This paper reports the responses to  uestions about their assessment of children’s mathematical activity and development. The practitioners provided examples of both incidental and planned assessment activities, the different forms these took, methods of recording, and how the results were used.

Tasks and pedagogies that facilitate mathematical problem solving

This is a report from one aspect of a project seeking to identify teacher actions that support mathematical problem solving. The project developed a planning and teaching model that describes the type of classroom tasks that can facilitate mathematical problem solving, the sequencing of the tasks, the nature of teaching heterogeneous groups, ways of dfferentiating tasks, and particular pedagogies. We report here one teacher's implementation of the
model using a unit of work that he planned and taught. The report provides important insights into the implementation of the theoretically founded model and the responses of students. We found that the model can be used for planning and teaching and for encouraging problem solving. The model has a positive effect on the learning of most students. Specific teachers actions were identified in order to address the needs of the students we are most
keen to support, those experiencing dfficulties.

How early childhood practitioners view young children's mathematical thinking

Interviews were held with 64 professionals in childcare centres. This paper reports on their responses to 3 questions seeking their perceptions of mathematical thinking in very young children. Generally, the interviewees were found to have a good sense of mathematical concepts relevant to babies and toddlers, and they cited evidence of young children’s mathematical development. It is concluded that this practical knowledge would provide a strong foundation for further professional development.

Using web-based mathematical interactive exercises and exploratory investigations : the possibilities and pitfalls

This case study looks at how a teacher used interactive exercises and exploratory investigations to reinforce concepts that have been learnt in a Year 8 Mathematics class. It highlights the practical problems faced by the teacher in terms of technical issues such as slow download times and logging on problems as well as class management issues. The findings are discussed in the light of the teacher’s mediation with the affordances of the Internet.

A primer on mathematical modelling in economics

The Commission on Graduate Education in Economics had raised several concerns regarding the role of mathematics in graduate training in economics (Krueger, 1991; Colander, 1998, 2005). This paper undertakes a detailed scrutiny of the notion of a utility function to motivate and describe the common patterns across mathematical concepts and results that are used by economists. In the process one arrives at a classification of mathematical terms which is used to state mathematical results in economics. The usefulness of the classification scheme is illustrated with the help of a discussion of Arrow's impossibility theorem. Common knowledge of the patterns in mathematical concepts and results could be effective in enhancing communication between students, teachers and researchers specializing in different sub-fields of economics.

Building the (mathematical) future.

Here are some questions to ponder during the school year, and other moments. Personal reflection and professional critical analysis are crucial components of an effective teacher's life.

Mathematical modelling of deep bed filtration

Numerous mathematical models have been developed to evaluate both initial and transient stage removal efficiency of deep bed filters. Microscopic models either using trajectory analysis or convective-diffusion equations were used to compute the initial removal efficiency. These models predicted the removal efficiency under favorable filtration conditions quantitatively, but failed to predict the removal efficiency under unfavorable conditions. They underestimated the removal efficiency under unfavorable conditions. Thus, semi-empirical formulations were developed to compute initial removal efficiencies under unfavorable conditions. Also, correction for the adhesion of particles onto filter grains improved the results obtained for removal efficiency from the trajectory analysis. Macroscopic models were used to predict the transient stage removal efficiency of deep bed filters. O’Melia and Ali’s model assumed that the particle removal is due to filter grains as well as the particles that are already deposited onto the filter grain. Thus, semi-empirical models were used to predict the ripening of filtration. Several modifications were made to the model developed by O’Melia and Ali to predict the deterioration of particle removal during the transient stages of filtration. Models considering the removal of particles under favorable conditions and the accumulation of charges on the filter grains during the transient stages were also developed. This paper evaluates those models and their applicability under different operating conditions of filtration.