Robert P. Hunting

A Review of Recent Research in the Area of Initial Fraction Concepts.

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 childrens early thoughts about fractions.By generating links between studies, integrated research is created and consensus regarding critical problems and future directions is reached. Concluding remarks pose questions for further investigation.

Rachel's Schemes for Constructing Fraction Knowledge.

This article focuses on the behaviour of nine year old Rachel as she constructed knowledge about fractions over a five month period. In particular, it is concerned to explain behavioural regularities for which certain mental processes, called ‘schemes’, are deemed responsible. Particular patterns of behaviour are accounted for by different kinds of scheme. Schemes that Rachel used to solve fractional number problems will first be distinguished. The roles played by Rachels schemes are examined in the dynamic setting of a teaching experiment. An analysis of interactions between certain schemes is provided in the context of Rachels efforts to learn about fifths.

Spontaneous partitioning: Pre-schoolers and discrete items

This study addresses the question of whether a dealing strategy that is widely used by young children in clinical interviews occurs in less structured situations. Our findings are that it did not in the setting we examined, namely the performance of a routine counting task, by pre-schoolers with the opportunity of sharing sweets when the task was completed. We discuss reasons for the apparent discrepancies between the results for clinical interviews and less structured situations.

Dimensions of Young Children’s Conceptions of the Fraction One Half

One-half is a fundamental building block in elementary mathematics. As a rational number one-half may be understood at the level of an equivalence class of ordered pairs of integers. However, very few outside the discipline of mathematics understand one-half at this level of sophistication, and very few mathematicians function with one half at this level of abstraction. We propose that knowledge of the fraction one-half develops through three general levels: as a qualitative unit, as a quantitative unit, and as an abstract unit. In particular, we will attempt to explain how one-half is conceived of by children at the onset of formal schooling. We will propose several categories of meaning for one-half that typifies the behavior of young children who are passing from qualitative to quantitative understandings of this number.

Pre-fraction Concepts of Preschoolers

Much school mathematics is devoted to teaching concepts and procedures based on those units that form the core of whole number arithmetic, such as ones, tens, and hundreds. Other topics such as fractions and decimals demand new and extended understanding of units and their relationships. Researchers have noted how children’s whole number ideas interfere with their efforts to learn fractions (Behr, Wachsmuth, Post, & Lesh, 1984; Hunting, 1986; Streefland, 1984). Hunting (1986) suggested that a reason why children seem to have difficulty learning stable and appropriate meanings for fractions is because instruction on fractions, if delayed too long, allows whole number knowledge to become the predominant scheme to which fraction language and symbolism is then related. There is some evidence which suggests that children can successfully complete fraction-related tasks earlier than when these procedures are taught in school. Polkinghome (1935) concluded from a study of 266 kindergarten, first, second, and third grade children that considerable knowledge of fractions is held prior to formal instruction in this topic, and Gunderson and Gunderson (1957) demonstrated that second graders had concepts and ideas about fractions that could be developed subsequently.

A Representational Communication Approach to the Development of Inductive and Deductive Logic

Publisher Summary This chapter is taken up with presentation of some detailed work on deductive reasoning with syllogisms. It is followed by a brief digest of work on inductive reasoning using conditionals and other logical expressions. Approach to the deductive logic of syllogisms arose from a number of sources: the philosophy of mathematics and logic contained in the later writings of Wittgenstein; discontent about the way psychologists have construed the learning of mathematical concepts by mathematics educators, especially Freudenthal. In addition, socialization also influences aspects of representations and their use. The chapter draws a clear distinction between mathematics and logic as conceived by most mathematicians and logicians and performance in the deductive reasoning tasks used to study deductive reasoning by psychologists. The most notable developmental trends in the interpretation of logical expressions during adolescence occur for conditionals. Insofar as adolescents and adults are able to give anything like a correct interpretation of conditionals, this tends to resemble the biconditional interpretation, “X is B if and only if X is A.” One explanation for such conversion relies on discourse presuppositions.

Preschoolers’ Spontaneous Partitioning of Discrete Items

Does the dealing strategy that is widely used by young children in clinical interviews occur in less structured situations? Does dealing to establish fair shares occur, for example, when young children perceive a need in their play activities to distribute discrete items evenly? Or is the dealing strategy largely an artifact of structured clinical interviews? In this short chapter we will give some evidence that spontaneous sharing by dealing is, like the bark of the Hound of the Baskervilles, conspicuous by its absence.

Higher Order Thinking in Young Children’s Engagements with a Fraction Machine

A group of six second grade children (average age 7 years 5 months) and their class teacher, were sitting around a table. On the table were arrangements of small wooden objects - mostly representations of animals There were two rabbits, four bears, six roosters, eight trees, 10 peacocks, and 12 worms. Starting with the two rabbits, the teacher had introduced each larger set in turn, asking a child to divide the set of objects in half, and to write symbols for the number of objects in each subset. The children offered their own interpretations for each partition, but through discussion and assistance most children, in the end, had written the sequence of fractions \( \frac{1}{2},\frac{2}{4},\frac{3}{6},\frac{4}{8},\frac{5}{{10}},\frac{6}{{12}}\) The teacher then said: “You have told me that each is a half of their combined total. How can these all be halves when we have got different numbers in each group?” In this instant those children were brought face to face with the possibility of developing a deeper interpretation of the fraction one half: that of a meta-relation.

Context and process in fraction learning

Solution processes of two groups of 9‐10 year old children to a common set of fraction problems were contrasted at the conclusion of a teaching experiment. One group was taught meanings for fractions using popsticks as discrete quantity material; the other group used paper strips as continuous quantity material. The study suggests that a knowledge of fractions based upon discrete quantities may have certain advantages over fraction knowledge based on continuous quantities. The Continuous group children were less successful with problems cast in discrete quantity settings. They also displayed unstable strategies when using continuous material. Children of both groups preferred discrete material to solve problems cast in either discrete or continuous contexts. Discrete quantity materials could have a valuable role in the development of fraction knowledge in the elementary school.

A TEST OF A TWO-STAGE MODEL OF THE EVALUATION OF HYPOTHESES FROM QUANTIFIED FIRST-ORDER PREDICATE LOGIC IN INFORMATION USE TASKS ',*

480 adolescents and young adults between the ages of 12 and 29 years participated in an experiment in which they were asked to evaluate hypotheses from quantified first-order predicate logic specifying that certain classes of event were necessarily, possibly, or certainly not included within a universe of discourse. Results were used to test a two-stage model of performance on hypothesis evaluation tasks that originated in work on the evaluation of conditionals. The two-stage model, unlike others available, successfully predicted the range of patterns of reply observed. In dealing with very simple hypotheses subjects in this age range tended not to make use of alternative hypotheses unless these were explicitly or implicitly suggested to them by the task. This tells against complexity of hypothesis as an explanation of the reluctance to use alternative hypotheses in evaluating standard conditionals.