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Higher Order Thinking in Young Children’s Engagements with a Fraction Machine
Abstract
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 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.
... Teaching the concept of fractions has been of interest to many researchers in mathematics education owing to its complexity (Hunting, Davis, & Bigelow, 1991;Kieren, 1980;Mack, 1990;Nesher, 1989;Steffe & Olive, 1993;Streefland, 1991;. This thesis tested the general hypothesis that learning fractions should be facilitated using realistic, familiar materials. ...
... Research that deals with rational numbers indicates several schemas central to learning fractions (Hunting et al., 1991;Kieren, 1993;Nesher, 1989;Steffe & Olive, 1993). ...
... Not only can the fraction concept be presented and perceived in many different meanings and sub-constructs, but also the interactions between these sub-constructs must be understood in order to gain a holistic understanding (Chinnappan, 2005;Hunting et al., 1991;Pitkethly & Hunting, 1996). Researchers identified five subconstructs (Behr, Wachsmuth, Post, & Lesh, 1984;Kieren, 1981;Nesher, 1989). ...
... Similarly, "halving" involved a focus on the act of splitting a shape rather than on the quantity ½. This is particularly notable given that even children as young as four years old have been shown to have an intuitive understanding of the fraction ½ (Hunting, Davis, & Bigelow, 1991;. Therefore, the persistent understandings can be thought of as reflecting (and possibly caused by) the student's inability to mentally represent and manipulate numbers. ...
Mathematical learning disability (MLD) research often conflates low achievement with disabilities and focuses exclusively on deficits of students with MLDs. In this study I adopt an alternative approach using a response-to-intervention MLD classification model and identify the resources students draw upon rather than the skills they lack. The intervention model involved videotaped one-on-one fraction tutoring sessions implemented with students with low mathematics achievement. This article presents case studies of two students who did not benefit from the tutoring sessions. Detailed diagnostic analyses of the sessions revealed that the students understood mathematical representations in atypical ways and that this directly contributed to the persistent difficulties they experienced. Implications for screening and remediation approaches are discussed.
Mathematical learning disability (MLD) research often conflates low achievement with disabilities and focuses exclusively on deficits of students with MLDs. In this study, the author adopts an alternative approach using a response-to-intervention MLD classification model to identify the resources students draw on rather than the skills they lack. Detailed diagnostic analyses of the sessions revealed that the students understood mathematical representations in atypical ways and that this directly contributed to the persistent difficulties they experienced. Implications for screening and remediation approaches are discussed.
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.
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.
This book espouses a theory of scientific realism in which due weight is given to mathematics and logic. The authors argue that mathematics can be understood realistically if it is seen to be the study of universals, of properties and relations, of patterns and structures, the kinds of things which can be in several places at once. Taking this kind of scientific platonism as their point of departure, they show how the theory of universals can account for probability, laws of nature, causation and explanation, and explore the consequences in all these fields. This will be an important book for all philosophers of science, logicians and metaphysicians, and their graduate students. The readership will also include those outside philosophy interested in the interrelationship of philosophy and science.
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.
To probe certain logical aspects of preschoolers' dealing and counting we showed edited highlights of three preschool children engaged in sharing activities to 17 second-grade children in an interview situation. In the videotaped episodes, the preschoolers routinely counted or checked the heights of the stacks when asked how they knew there was a fair share; the second graders, with one notable, and another possible, exception, expressed the view that counting after dealing was essential to tell if there was a fair share. One second grader gave us articulate explanations about the procedures used by the preschoolers, and another apparently changed her mind during the course of the interview about the need to count after dealing out.
Children and adolescents tested to determine the development of the operator construct of rational numbers employed different problem-solving strategies depending on test presentation. Three major phases in the rational number construct appear to be a primitive fractional construct, a unit operator phase, and a general operator phase. (SB)
Used a theory of rational numbers, which sees the rational number construct of a person as potentially built on a number of subconstructs as the basis for research which explored the view of rational numbers as operators as held by children and adolescents. Ss aged 8–14 yrs were faced with 6 task settings involving operator representations of the rational numbers ½, ⅓, ⅔, ¾, ½ × ¾. Significant changes in performance level and quality appeared at ages 11–22 yrs and particularly at ages 12–23. Based on the findings, it is hypothesized that 3 levels of "operator" development exist, and that Ss use such constructive mechanisms as partitioning to control the operator situation. (8 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
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 Rachel's schemes are examined in the dynamic setting of a teaching experiment. An analysis of interactions between certain schemes is provided in the context of Rachel's efforts to learn about fifths.
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.
The fraction machines: A research and classroom tool for fraction learning. Written in GFA compiled Basic for Atari ST machines
- G E Davis
- GE Davis
Arithmetica universalis
- J Newton
A preliminary report on the development of the operator construct of rational number in Papua New Guinea
- B Southwell
The reality of numbers
- J C Bigelow
- JC Bigelow
Some remarks on the homology and dynamics of rational number learning
- J C Bigelow
- G E Davis
- R P Hunting
- JC Bigelow
How children account for fraction equivalence
- R P Hunting
- RP Hunting
The trouble with fractions: Can computer-based learning environments help?
- R P Hunting
- J C Bigelow
- G E Davis
- RP Hunting
Elements of algebra (trans
- L Euler
Understanding equivalent fractions
- R P Hunting
- RP Hunting