Gloria Stillman

A framework for identifying student blockages during transitions in the modelling process

In this article we present, illustrate, test and refine a framework developed by Galbraith, Stillman, Brown and Edwards (2006) for identifying student blockages whilst undertaking modelling tasks during transitions in the modelling process. The framework was developed with 14~15 year old students who were engaging in their first experiences of modelling at the secondary level.

Applying Metacognitive Knowledge and Strategies in Applications and Modelling Tasks at Secondary School

The importance of reflective metacognitive activity during mathematical modelling activity has been recognised by scholars and researchers over the years. The metacognitive activity (or lack of it) of secondary students associated with transitions between stages in the modelling process – especially in relation to the identification and release of blockages to progress – is considered. Productive metacognitive acts are identified as occurring at three levels. Routine metacognition together with metacognitive responses to Goos’ red flag situations are elaborated together with the notion of meta-metacognition being engaged in by teachers trying to foster students’ development of independent modelling competencies especially their metacognitive competencies.

Identifying Challenges within Transition Phases of Mathematical Modeling Activities at Year 9

The Galbraith, Stillman, Brown, and Edwards Framework (2007) for identifying blockages hindering progress in transitions in the modeling process is applied to a modeling task undertaken by 21 Year 9 students. The Framework identified where challenges occurred; but, because some blockages proved to be more robust than others, another construct “level of intensity” was added. The blockages described here occurred during the formulation phase of the modeling cycle. We infer that blockages induced by lack of reflection, or by incorrect or incomplete knowledge, are different in nature and cognitive demand from those involving the revision of mental schemas (i.e., cognitive dissonance). The nature and intensity of the blockage have consequences for teacher intervention and task implementation.

Trends in Teaching and Learning of Mathematical Modelling: ICTMA14

This book contains suggestions for and reflections on the teaching, learning and assessing of mathematical modelling and applications in a rapidly changing world, including teaching and learning environments. It addresses all levels of education from universities and technical colleges to secondary and primary schools. Sponsored by the International Community of Teachers of Mathematical Modelling and Applications (ICTMA), it reflects recent ideas and methods contributed by specialists from 30 countries in Africa, the Americas, Asia, Australia and Europe. Inspired by contributions to the Fourteenth Conference on the Teaching of Mathematical Modelling and Applications (ICTMA14) in Hamburg, 2009, the book describes the latest trends in the teaching and learning of mathematical modelling at school and university including teacher education. The broad and versatile range of topics will stress the international state-of-the-art on the following issues:Theoretical reflections on the teaching and learning of modellingModelling competenciesCognitive perspectives on modellingModelling examples for all educational levelsPractice of modelling in school and at university levelPractices in Engineering and Applications

Turning Ideas into Modeling Problems

We show how the nucleus of an idea can be developed into modeling problems for secondary school using principles for problem design enunciated in Galbraith (2007). Once the germ has been developed for a task, the idea can be extended as necessary into related problems closer to the personal experience of the adolescents in secondary school. The issues and contexts secondary students choose to investigate, and the questions that they pose when given free reign or minimal constraints, are illustrated from an Australian modeling challenge. Finally, using these contexts as starting points, it is suggested such situations can be developed to engage students in important teaching issues involving necessary constituents of the modeling process.

Strategies Employed by Upper Secondary Students for Overcoming or Exploiting Conditions Affecting Accessibility of Applications Tasks.

A cognitive/metacognitive framework is presented for analysing applications tasks and responses to these. Conditions facilitating or impeding access to such tasks at the upper secondary level were identified using qualitative data analysis techniques within this framework. Strategies employed in exploiting, or overcoming these conditions were identified. A well-developed repertoire of cognitive and metacognitive strategies together with a rich store of mathematical knowledge, real-world knowledge and experiences, and comprehension skills facilitated access. This was enhanced by metacognitive knowledge encouraging student engagement with the task and by students imagining they were in the task situation. Once moderate skill had been achieved in accessing these applications, coordination and integration of multiple representations, further cues, and mathematical processes and procedures became critical.

Implementation Case Study: Sustaining Curriculum Change

An Australian study is being undertaken into influences on the sustaining of curriculum change designed to place real-world mathematical applications and modelling centrally within mainstream senior secondary mathematics curricula. Conditions that impeded or sustained this curriculum change in two Australian are currently being investigated. Findings from the implementation in one of the states are presented.

Evolution of Applications and Modelling in a Senior Secondary Curriculum

In Queensland Australia, mathematical modelling and applications have featured in senior secondary mathematics curricula for two decades. Part of a longitudinal study of the implementation of this initiative, as seen through the eyes of selected teachers and administrators who have been centrally involved in its development and on-going practice, is reported. The data consist of responses to structured and open interview questions, syllabus documents, and application and modelling tasks designed and implemented by teachers. Perceptions of why modelling and applications are valuable at this level of schooling, the distinction between applications and modelling, how established applications and modelling are in the curriculum, the sources of such tasks, and the sufficiency of support for the development of these tasks by teachers are presented.

Challenging Mathematics: Classroom Practices

In this chapter we examine classroom practice issues related to teachers providing mathematical challenges in their everyday classrooms. We examine how challenging mathematics can become the essence of mathematics classrooms, how challenging mathematics can be designed for the everyday classroom and how classroom artifacts and practices can be designed for mathematical challenges. Finally, the question of suitable research designs for research into classroom practices associated with the use of challenging mathematics in everyday classrooms is addressed and illustrated.

Challenges in modelling challenges: Intents and purposes

Extra curricular modelling events for school students are increasing in popularity. The experience from the perspective of participating Australian and Singaporean Year 10 and 11 students in one such event, the A. B. Paterson College Mathematical Modelling Challenge, is investigated. The aim is to gauge whether their experience meets the intended purposes of mentors that the whole event be considered an inherently valuable learning experience about modelling and application of mathematics to real situations. Findings from a questionnaire and other data included motivation for choice of situation to model, decision making about approach, and reporting of findings were mainly goal oriented rather than performance oriented, despite the Challenge ostensibly being a competition.