Problem solving for the 21st century

Mathematical problem solving has been the subject of substantial and often controversial research for several decades. We use the term, problem solving, here in a broad sense to cover a range of activities that challenge and extend one’s thinking. In this chapter, we initially present a sketch of past decades of research on mathematical problem solving and its impact on the mathematics curriculum. We then consider some of the factors that have limited previous research on problem solving. In the remainder of the chapter we address some ways in which we might advance the fields of problem-solving research and curriculum development.

Mathematical modelling in the early school years

In this article we explore young children's development of mathematical knowledge and reasoning processes as they worked two modelling problems (the Butter Beans Problem and the Airplane Problem). The problems involve authentic situations that need to be interpreted and described in mathematical ways. Both problems include tables of data, together with background information containing specific criteria to be considered in the solution process. Four classes of third-graders (8 years of age) and their teachers participated in the 6-month program, which included preparatory modelling activities along with professional development for the teachers. In discussing our findings we address: (a) Ways in which the children applied their informal, personal knowledge to the problems; (b) How the children interpreted the tables of data, including difficulties they experienced; (c) How the children operated on the data, including aggregating and comparing data, and looking for trends and patterns; (c) How the children developed important mathematical ideas; and (d) Ways in which the children represented their mathematical understandings.

Young children's early modelling with data

An educational priority of many nations is to enhance mathematical learning in early childhood. One area in need of special attention is that of statistics. This paper argues for a renewed focus on statistical reasoning in the beginning school years, with opportunities for children to engage in data modelling activities. Such modelling involves investigations of meaningful phenomena, deciding what is worthy of attention (i.e., identifying complex attributes), and then progressing to organising, structuring, visualising, and representing data. Results are reported from the first year of a three-year longitudinal study in which three classes of first-grade children and their teachers engaged in activities that required the creation of data models. The theme of “Looking after our Environment,” a component of the children’s science curriculum at the time, provided the context for the activities. Findings focus on how the children dealt with given complex attributes and how they generated their own attributes in classifying broad data sets, and the nature of the models the children created in organising, structuring, and representing their data.

Interdisciplinary modelling

This article examines one approach to promoting creative and flexible use of mathematical ideas within an interdisciplinary context in the primary curriculum, namely, through modelling. Three classes of fifth-grade children worked on a modelling problem, The First Fleet (Australia’s settlement), situated within the curriculum domains of science and studies of society and environment. Reported here are the cycles of development displayed by one group of children as they worked the problem, together with the range of models created across the classes. Children developed mathematisation processes that extended beyond their regular curriculum, including identifying and prioritising key problem elements, exploring relationships among elements, quantifying qualitative data, ranking and aggregating data, and creating and working with weighted scores. Aspects of Goldin’s (2000, 2007) affective structures also appeared to play an important role in the children's mathematical developments.

A problem-posing intervention in the development of problem-solving competence of underachieving, middle-year students

This study reported on the issues surrounding the acquisition of problem-solving competence of middle-year students who had been ascertained as above average in intelligence, but underachieving in problem-solving competence. In particular, it looked at the possible links between problem-posing skills development and improvements in problem-solving competence. A cohort of Year 7 students at a private, non-denominational, co-educational school was chosen as participants for the study, as they undertook a series of problem-posing sessions each week throughout a school term. The lessons were facilitated by the researcher in the students’ school setting. Two criteria were chosen to identify participants for this study. Firstly, each participant scored above the 60th percentile in the standardized Middle Years Ability Test (MYAT) (Australian Council for Educational Research, 2005) and secondly, the participants all scored below the cohort average for Criterion B (Problem-solving Criterion) in their school mathematics tests during the first semester of Year 7. Two mutually exclusive groups of participants were investigated with one constituting the Comparison Group and the other constituting the Intervention Group. The Comparison Group was chosen from a Year 7 cohort for whom no problem-posing intervention had occurred, while the Intervention Group was chosen from the Year 7 cohort of the following year. This second group received the problem-posing intervention in the form of a teaching experiment. That is, the Comparison Group were only pre-tested and post-tested, while the Intervention Group was involved in the teaching experiment and received the pre-testing and post-testing at the same time of the year, but in the following year, when the Comparison Group have moved on to the secondary part of the school. The groups were chosen from consecutive Year 7 cohorts to avoid cross-contamination of the data. A constructionist framework was adopted for this study that allowed the researcher to gain an “authentic understanding” of the changes that occurred in the development of problem-solving competence of the participants in the context of a classroom setting (Richardson, 1999). Qualitative and quantitative data were collected through a combination of methods including researcher observation and journal writing, video taping, student workbooks, informal student interviews, student surveys, and pre-testing and post-testing. This combination of methods was required to increase the validity of the study’s findings through triangulation of the data. The study findings showed that participation in problem-posing activities can facilitate the re-engagement of disengaged, middle-year mathematics students. In addition, participation in these activities can result in improved problem-solving competence and associated developmental learning changes. Some of the changes that were evident as a result of this study included improvements in self-regulation, increased integration of prior knowledge with new knowledge and increased and contextualised socialisation.

Teaching middle years : rethinking curriculum, pedagogy and assessment [2nd. ed.]

The middle years of schooling are increasingly recognised as a crucial stage in students' lives, one that has significant consequences for ongoing educational success. International research indicates that young adolescents benefit from programs designed especially for their needs. Teaching Middle Years offers a systematic overview of the philosophy, principles and issues in middle schooling. It includes contributions from academics and school-based practitioners on intellectual and emotional development in early adolescence, pedagogy, curriculum and assessment of middle years students. This second edition is fully revised to reflect the latest research findings. It includes new chapters on students with diverse needs, school partnerships with families and community, and effective team teaching. Also new to this edition is a chapter that brings middle schooling concepts to life by providing real examples of reform in action.

Teaching middle years : rethinking curriculum, pedagogy and assessment [1st ed.]

The middle years of schooling are increasingly recognised as a crucial stage in students' lives, one that has significant consequences for ongoing educational success. International research indicates that young adolescents benefit from programs designed especially for their needs, and the middle years have become an important reform issue for education systems. Teaching Middle Years offers a systematic overview of the philosophy, principles and issues in middle schooling. It includes contributions from academics and school-based practitioners on intellectual and emotional development in early adolescence, pedagogy, curriculum and assessment of middle years students. Written for teachers, student teachers, education leaders and policy makers, Teaching Middle Years is an essential resource for anyone involved in educating young adolescents. Teaching Middle Years is the first comprehensive Australian book to match and surpass the quality of many overseas publications.'

Trailblazers

A recent study in the United Kingdom (Ofsted Report 2008) provides strong evidence that well-organized activities outside the classroom contribute significantly to the quality and depth of children's learning, including their personal, social, and emotional development. Outdoor math trails supply further evidence of such enhanced learning: They are meaningful, stimulating, challenging, and exciting for children. Most important, these trails invite all students, irrespective of their classroom achievement level, to participate successfully in the problem activities and gain a sense of pride in the mathematics they create. Additionally, Math trails empower lifelong learning. Integrating "outside" mathematics with "inside" classroom mathematics can sow the seeds to develop flexible, creative, future-oriented mathematical thinkers and problem solvers. Here, English et al discuss how to design and implement math trails to promote active, meaningful, real-world mathematical learning beyond the classroom walls.

Adventures in pedagogy : the trials and tribulations of teaching common law in China

China is now seen as arguably, the next economic giant of the 21st century. From a country closed in the past to the external world, the Chinese market now presents as one of the most lucrative in the world economy. One area that has drawn increasing international interest is education - it has been estimated that by 2020 there will be 25 million excess demands for higher education places that the Chinese tertiary educational system cannot meet. Many overseas institutions have developed programs to cater for this immense potential market. In 2000 the Law Faculty of the University of Technology, Sydney (UTS)introduced a new postgraduate program specifically targeting the Chinese market. This paper is a brief assessment of the program - it examines general issues in the pedagogical delivery of programs in LOTE (Language Other Than English) and the use of 'proxies' in the delivery of LOTE programs. The paper concludes that while the UTS program demonstrates that it is feasible to use proxy lecturers or interpreters in the delivery of programs in LOTE, the exercise entails significant problems that can undermine the integrity of such programs.

Transition pedagogy : a third generation approach to FYE : a case study of policy and practice for the higher education sector

Current research and practice related to the first year experience (FYE) of commencing higher education students are still mainly piecemeal rather than institution-wide with institutions struggling to achieve cross-institutional integration, coordination and coherence of FYE policy and practice. Drawing on a decade of FYE-related research including an ALTC Senior Fellowship and evidence at a large Australian metropolitan university, this paper explores how one institution has addressed that issue by tracing the evolution and maturation of strategies that ultimately conceptualize FYE as “everybody's business.” It is argued that, when first generation co-curricular and second generation curricular approaches are integrated and implemented through an intentionally designed curriculum by seamless partnerships of academic and professional staff in a whole-of-institution transformation, we have a third generation approach labelled here as transition pedagogy. It is suggested that transition pedagogy provides the optimal vehicle for dealing with the increasingly diverse commencing student cohorts by facilitating a sense of engagement, support and belonging. What is presented here is an example of transition pedagogy in action.

Solving graphics tasks : gender differences in middle-school students

The capacity to solve tasks that contain high concentrations of visual-spatial information, including graphs, maps and diagrams, is becoming increasingly important in educational contexts as well as everyday life. This research examined gender differences in the performance of students solving graphics tasks from the Graphical Languages in Mathematics (GLIM) instrument that included number lines, graphs, maps and diagrams. The participants were 317 Australian students (169 males and 148 females) aged 9 to 12 years. Boys outperformed girls on graphical languages that required the interpretation of information represented on an axis and graphical languages that required movement between two- and three-dimensional representations (generally Map language).