Teaching and learning fractions has traditionally been one of the most problematic areas in primary school mathematics. Several studies have suggested that one of the main factors contributing to this complexity is that fractions comprise a multifaceted notion encompassing five interrelated subconstructs (i.e., part-whole, ratio, operator, quotient, and measure). Kieren was the first to establish that the concept of fractions is not a single construct, but consists of several interrelated subconstructs. Later on, in the early 1980s, Behr et al. built on Kieren’s conceptualization and suggested a theoretical model linking the five subconstructs of fractions to the operations of fractions, fraction equivalence, and problem solving. In the present study we used this theoretical model as a reference point to investigate students’ constructions of the different subconstructs of fractions. In particular, using structural equation modeling techniques to analyze data of 646 fifth and sixth graders’ performance on fractions, we examined the associations among the different subconstructs of fractions as well as the extent to which these subconstructs explain students’ performance on fraction operations and fraction equivalence. To a great extent, the data provided support to the associations included in the model, although, they also suggested some additional associations between the notions of the model. We discuss these findings taking into consideration the context in which the study was conducted and we provide implications for the teaching of fractions and suggestions for further research.

Drawing on a Theoretical Model to Study Students’ Understandings of Fractions

In an 8-month teaching experiment, I investigated how 4 sixth-grade students reasoned with reversible multiplicative relationships. One type of problem involved a known quantity that was a whole number multiple of an unknown quantity, and students were asked to determine the value of the unknown quantity. To solve these problems, students needed to produce a fraction of the known quantity that could be repeated some number of times to make the known, rather than repeat the known quantity to make the unknown quantity. This aspect of the problems involved reversibility because students who do not make a fraction of the known quantity tend to repeat the known quantity (Norton, 2008; Steffe, 2002). All four students constructed schemes to solve such problems and more complex versions where the relationship between known and unknown quantities was a fraction. Two students could not foresee the results of their schemes in thought—they had to carry out some activity, review its results, and then carry out more a...

/pdf/students-reasoning-with-reversible-multiplicative-4odywvsgcp.pdf

Students’ Reasoning With Reversible Multiplicative Relationships

ZDM - The International Journal on Mathematics Education

This paper reviews recent research in the area of initial fraction concepts. The common goal of the empirical studies which are represented in this analysis was to assist children develop a meaningful understanding of the rational number construct, founded on durable fraction concepts. Two interpretations of findings were derived from the research. One group of researchers identified initial fraction concepts emerging from the application of intuitive mechanisms, in particular partitioning in either continuous or discrete contexts, and leading to unit identification and iteration of the unit. The other group of researchers identified ideas of ratio and proportion present in young children's early thoughts about fractions.

A review of recent research in the area of initial fraction concepts

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.

Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction

What Might a Fraction Mean to a Child and How Would a Teacher Know

Many researchers have noted how children's whole number schemes can interfere with their efforts to learn fractions. This paper examines the persistence of whole number schemes among 14 year-old students who appear to have successfully mastered routine algorithms for working with fractions. Uncovering whole number thinking among such students is therefore difficult, and is illustrated through the use of several probing interview tasks, revealing quite different forms of whole number thinking. These forms of thinking can give correct answers also making it difficult for teachers to identify incorrect thinking about fractions. Representations of fractions using number lines can assist in identifying and correcting such thinking.

Why you have to probe to discover what Year 8 students really think about fractions

Many researchers have noted that students’ whole number knowledge can interfere with their efforts to learn fractions. In this paper we discuss a teaching experiment conducted with students in Years 5 and 6 from an eastern suburban school in Melbourne. The focus of the teaching experiment was to use number lines to highlight students’ understanding of whole numbers then fractions. This research showed that successful students had easily accessible whole number knowledge and recognised the relationship between the whole and the parts whereas the weakest students had poor number knowledge and could not see the connections.

Whole number knowledge and number lines help to develop fraction concepts

At what age do young children begin thinking mathematically? Can young children work on mathematical problems? How do early childhood educators ensure young children feel good about mathematics? Where do early childhood educators learn about suitable mathematics activities? A good early childhood start in mathematics is critical for later mathematics success. Parents, carers and early childhood educators are teaching mathematics, either consciously or unconsciously, in any social interaction with a child. Mathematical Thinking of Preschool Children in Rural and Regional Australia is an extension of a conference of Australian and New Zealand researchers that identified a number of important problems related to the mathematical learning of children prior to formal schooling. A project team of 11 researchers from top Australian universities sought to investigate how early childhood education can best have a positive influence on early mathematics learning. The investigation complements and extends the work of Project Good Start by focusing attention on critical aspects of parents, carers and early childhood educators who care for young children. Early childhood educators from regional and rural New South Wales, Queensland and Victoria were interviewed, following a set of structured questions. The questions focused on: children’s mathematics learning; support for mathematics teaching; use of technology; attitudes to mathematics; and assessment and record keeping. The researchers also reviewed research focusing on the mathematical capacities and potential foundations for further mathematical development in young children (0–5 years) published in the last decade and produced an annotated bibliography. This should provide a good basis for further research and reading. Based upon the results of this investigation, the researchers make 11 recommendations for improving the practices of early childhood education centres in relation to young children’s mathematical thinking and development. The implications for policy and decision makers are outlined for teacher education, the provision of resources and further research.

Mathematical thinking of preschool children in rural and regional Australia : research and practice

Mathematical Thinking of Preschool Children in Rural and Regional Australia : Research and Practice. Final Report to the National Centre of Science, Information Technology, and Mathematics Education in Rural and Regional Australia (SiMERR)

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