Lyn D. English

Handbook of International Research in Mathematics Education

In The Age of Discontinuity: Guidelines to Our Changing Society (1992), Professor of Management Peter Drucker lays out ways in which technologies are transforming, and will continue to transform, industries throughout the world economy; for many workers, what characterizes work life now is the continual need to adapt to technological change. Such changes are not limited to the world of work: technology is transforming interactions with media, and this also relates to books. This chapter focuses on one way in which technology may transform educational processes and bring about new educational dynamics. Specifically we examine ways in which e-book technology might influence one genre of book, the (mathematics) textbook

A Modeling Perspective on Students' Mathematical Reasoning about Data.

A modeling approach to the teaching and learning of mathematics shifts the focus of the learning activity from finding a solution to a particular problem to creating a system of relationships that is generalizable and reusable. In this article, we discuss the nature of a sequence of tasks that can be used to elicit the development of such systems by middle school students. We report the results of our research with these tasks at two levels. First, we present a detailed analysis of the mathematical reasoning development of one small group of students across the sequence of tasks. Second, we provide a macrolevel analysis of the diversity of thinking patterns identified on two of the problem tasks where we incorporate data from multiple groups of students. Student reasoning about the relationships between and among quantities and their application in related situations is discussed. The results suggest that students were able to create generalizable and reusable systems or models for selecting, ranking, and weighting data. Furthermore, the extent of variations in the approaches that students took suggests that there are multiple paths for the development of ideas about ranking data for decision making.

The Development of Fifth-Grade Children's Problem-Posing Abilities.

This one-year study involved designing and implementing a problem-posing program for fifth-grade children. A framework developed for the study encompassed three main components: (a) childrens recognition and utilisation of problem structures, (b) their perceptions of, and preferences for, different problem types, and (c) their development of diverse mathematical thinking. One of the aims of the study was to investigate the extent to which childrens number sense and novel problem-solving skills govern their problem-posing abilities in routine and nonroutine situations. To this end, children who displayed different patterns of achievement in these two domains were selected to participate in the 10-week activity program. Problem-posing interviews with each child were conducted prior to, and after the program, with the progress of individual children tracked during the course of the program. Overall, the children who participated in the program appeared to show substantial developments in each of the program components, in contrast to those who did not participate.

Theories of Mathematics Education

The purpose of this Forum is to stimulate critical debate in the area of theory use and theory development, and to consider future directions for the advancement of our discipline. The Forum opens with a discussion of why theories are essential to the work of mathematics educators and addresses the possible reasons for why some researchers either ignore or misunderstand/misuse theory in their work. Other issues to be addressed include the social turn in mathematics education, an evolutionary perspective on the nature of human cognition, the use of theory to advance our understanding of student cognitive development, and models and modelling perspectives. The final paper takes a critical survey of European mathematics didactics traditions, particularly those in Germany and compares these to historical trends in other parts of the world.

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 3rd-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; (d) How the children developed important mathematical ideas; (e) Ways in which the children represented their mathematical understandings.

Children's Strategies for Solving Two- and Three-Dimensional Combinatorial Problems.

The study investigated the strategies that 7- to 12-year-old children spontaneously apply to the solution of novel combinatorial problems. The children were individually administered a set of six problems involving the dressing of toy bears in all possible combinations of tops and pants (twodimensional) or tops, pants, and tennis rackets (three-dimensional). Two sets of solution procedures were identified, each comprising a series of five increasingly complex strategies ranging from trial-and-error approaches to sophisticated odometer procedures. Results suggested that experience with the two-dimensional problems enabled children to adopt and subsequently transform their efficient 2-D odometer strategy (where one item is held constant) into the most sophisticated 3-D odometer strategy, which involved working simultaneously with two constant items. The study highlights the importance of discrete mathematics as a source of problem-solving activities in which children are motivated to create, modify, and extend their own theories.

Young children's combinatoric strategies

Children aged between 4 years 6 months and 9 years 10 months were individually administered a series of novel tasks involving the formation of different combinations of two items, selected from discrete sets of items. An analysis of the childrens performance revealed a series of six, increasingly sophisticated, solution strategies ranging from a random selection of items through to a systematic pattern in item choice (cf. Piaget and Inhelders, 1975, combinatoric operations). A significant number of children independently adopted more efficient solution procedures as they progressed on the tasks, with many displaying an algorithmic procedure reflecting the “odometer strategy” (holding one item constant while systematically varying each of the other items). Given that children as young as 7 years demonstrated this systematic strategy, it would appear that, within the appropriate learning environment, young children can discover a procedure for forming n x n combinations prior to the stage of formal operations postulated by Piaget and Inhelder. The findings support the inclusion of the combinatorial domain as a topic of investigation in the elementary school curriculum.

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.

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 (identifying complex attributes), and then progressing to organising, structuring, visualising, and representing data. Results reported here are derived from the first year of a three-year longitudinal study in which three classes of first-grade children and their teachers engaged in activities requiring 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 include children’s abilities to focus their attention on qualities of items rather than the items themselves in identifying attributes, switch their attention from one item feature to another, and create a broad range of models in organising, structuring, and representing their data. Children’s development of meta-representational knowledge facilitated their choice and nature of data representations.

Combinatorics and the Development of Children's Combinatorial Reasoning

This chapter begins by exploring some elementary ideas of combinatorics and how they support children’s development of beginning probability ideas and problem-solving skills. Consideration is then given to various types of combinatorial problems and the relevant difficulties they present children. A review of studies that have addressed childrens combinatorial reasoning is presented in the second half of the chapter. The chapter concludes by looking at ways in which we might increase childrens access to powerful ideas in combinatorics.