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Manipulatives as symbols: a new perspective on the use of concrete objects to teach mathematics
, 1997
"... This article offers a new perspective on the use of concrete objects to teach mathematics. It is commonly assumed that concrete manipulatives are effective because they allow children to perform mathematics without understanding arbitrary, written mathematical symbols. We argue that the sharp distin ..."
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Cited by 14 (0 self)
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This article offers a new perspective on the use of concrete objects to teach mathematics. It is commonly assumed that concrete manipulatives are effective because they allow children to perform mathematics without understanding arbitrary, written mathematical symbols. We argue that the sharp distinction between concrete and abstract forms of mathematical expression may not be justified. We believe instead that manipulatives are also symbols; teachers intend for them to stand for or represent a concept or written symbol. Consequently, research on how young children comprehend symbolic relations is relevant to studying their comprehension of manipulatives. We review evidence that many of the problems that children encounter when using manipulatives are very similar to problems that they have using other symbol systems such as scale models. Successful use of manipulatives depends on treating them as symbols rather than as substitutes for symbols. A persistent dilemma for teachers of mathematics concerns how to help children understand abstract concepts, such as addition and multiplication, and the symbols that are used to represent these concepts (Hiebert & Carpenter, 1992; Resnick & Ford, 1984). Teachers face a double challenge. Symbols may be difficult to teach to children who have not yet grasped the concepts that they represent. At the same time, the concepts may be difficult to teach to children who have not yet mastered the symbols. Not surprisingly, both teachers and mathematics researchers have called for better techniques to help children learn mathematical concepts and symbols.
The advantage of simple symbols for learning and transfer
 Psychonomic Bulletin & Review
, 2005
"... A goal of successful learning is the transfer of learned knowledge to novel situations. However, spontaneous transfer is notoriously difficult to achieve. In this research, we argue that learning and transfer can be facilitated when knowledge is expressed in an abstract, generic form. In Experiments ..."
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A goal of successful learning is the transfer of learned knowledge to novel situations. However, spontaneous transfer is notoriously difficult to achieve. In this research, we argue that learning and transfer can be facilitated when knowledge is expressed in an abstract, generic form. In Experiments 1 and 2, undergraduate students learned two isomorphic domains, which were based on the same algebraic group, with one domain expressed in a more abstract, generic form and the other expressed in a more concrete form. In both experiments, transfer from more abstract to more concrete was greater than the reverse. In Experiment 3, undergraduate students learned the same algebraic group under varying degrees of concreteness. Our results demonstrate that the use of perceptually rich, concrete symbols may hinder learning. This research indicates that concreteness may have substantial learning and transfer costs, whereas abstractness may have benefits. A goal of successful learning is transfer, or the ability to apply acquired knowledge outside of the learned situation. For example, if one learned how to calculate the probability of heads occurring twice on two fair coin tosses in a mathematics classroom, one should be able to
Output Devices, Computation, and the Future of Mathematical Crafts
 International Journal of Computers in Mathematical Learning
, 2002
"... As I write this sentence, I am glancing over at the color printer sitting beside my screen. In the popular jargon of the computer industry, that printer is called a "peripheral"—which, upon reflection, is a rather odd way to describe it. What, precisely, is it peripheral to? If the ultimat ..."
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Cited by 3 (1 self)
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As I write this sentence, I am glancing over at the color printer sitting beside my screen. In the popular jargon of the computer industry, that printer is called a "peripheral"—which, upon reflection, is a rather odd way to describe it. What, precisely, is it peripheral to? If the ultimate
Learning Math 1 Running head: LEARNING MATH WITH MANIPULATIVES Learning Math with Manipulatives
"... A standard set of variables is extracted from a set of studies with different perspectives and different findings involving learning aids in the classroom. The variables are then used to analyze the studies in order to draw conclusions about learning aids in general and manipulatives in particular. ..."
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A standard set of variables is extracted from a set of studies with different perspectives and different findings involving learning aids in the classroom. The variables are then used to analyze the studies in order to draw conclusions about learning aids in general and manipulatives in particular. Learning Math 3 Learning Math with Manipulatives Manipulatives are a widely used part of the mathematics curriculum. The National Council for Teachers of Mathematics includes the use of manipulatives in its Principles and Standards for School Mathematics (NCTM, 2000), a document that has been used nationally for developing local mathematics standards. According to the National Center for Educational Statistics (NCES, 1993), during the 19921993 school year manipulatives were used every day for math or science in fortynine percent of all public kindergarten classrooms. In the 19941995 school year, manipulatives were used in sixteen percent of all math lessons in US public eighth grade classrooms and in thirtyfour percent of all math lessons in Japanese eighth grade classrooms (NCES, 1999). Research with manipulatives such as base ten blocks have been
A New Perspective on the Use of Concrete Objects to Teach
"... This article offers a new perspective on the use of concrete objects to teach mathematics. It is commonly assumed that concrete manipulatives are effective because they allow children to perform mathematics without understanding arbitrary, written mathematical symbols. We argue that the sharp distin ..."
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This article offers a new perspective on the use of concrete objects to teach mathematics. It is commonly assumed that concrete manipulatives are effective because they allow children to perform mathematics without understanding arbitrary, written mathematical symbols. We argue that the sharp distinction between concrete and abstract forms of mathematical expression may not be justified. We believe instead that manipulatives are also symbols; teachers intend for them to stand for or represent a concept or written symbol. Consequently, research on how young children comprehend symbolic relations is relevant to studying their comprehension of manipulatives. We review evidence that many of the problems that children encounter when using manipulatives are very similar to problems that they have using other symbol systems such as scale models. Successful use of manipulatives depends on treating them as symbols rather than as substitutes for symbols. A persistent dilemma for teachers of mathematics concerns how to help children understand abstract concepts, such as addition and multiplication, and the symbols that are used to represent these concepts (Hiebert & Carpenter, 1992; Resnick & Ford, 1984). Teachers face a double challenge. Symbols may be difficult to teach to children who have not yet grasped the concepts that they represent. At the same time, the concepts may be difficult to teach to children who have not yet mastered the symbols. Not surprisingly, both teachers and mathematics researchers have called for better techniques to help children learn mathematical concepts and symbols.
children’s relational reasoning
, 2012
"... The importance of being interpreted: grounded words and ..."
Preparing Elementary Preservice Teachers to Use Mathematics Curriculum Materials
"... Learning how to use mathematics curriculum materials to create learning opportunities is an important part of the work of teaching. This paper presents findings from a study involving 15 elementary preservice teachers enrolled in, first, a content and, then, a methods course, and discusses the exten ..."
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Learning how to use mathematics curriculum materials to create learning opportunities is an important part of the work of teaching. This paper presents findings from a study involving 15 elementary preservice teachers enrolled in, first, a content and, then, a methods course, and discusses the extent to which three curriculum interventions influenced their conceptions of how math curriculum materials are used. Additionally, this paper discusses the implications of this research for mathematics teacher education programs and proposes a framework for integrating work around curriculum materials into mathematics content and methods courses in order to prepare preservice teachers for using these materials effectively. Recent efforts by the National Council of Teachers of Mathematics (NCTM) to improve the way that K12 mathematics is taught and learned have implications for mathematics teacher education. As teacher education programs aim to develop teachers’ knowledge of mathematics and their knowledge of students as learners, these programs “should [also] develop teachers ’ knowledge of and ability to use and
MIND, BRAIN, AND EDUCATION Developing Children’s Understanding of Fractions: An Intervention Study
"... ABSTRACT—Fractions constitute a stumbling block in mathematics education. To improve children’s understanding of fractions, we designed an intervention based on learningbydoing activities, which focused on the representation of the magnitude of fractions. Participants were 292 Grade 4 and 5 childre ..."
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ABSTRACT—Fractions constitute a stumbling block in mathematics education. To improve children’s understanding of fractions, we designed an intervention based on learningbydoing activities, which focused on the representation of the magnitude of fractions. Participants were 292 Grade 4 and 5 children. Half of the classes received experimental instruction, while the other half pursued their usual lessons. For 10 weeks, they played five different games using cards representing fractions (e.g., Memory and Blackjack). Wooden disks helped them represent and manipulate fractions while playing games. Our results showed an improvement in the conceptual understanding of fractions. The findings confirmed that the usual practice in teaching fractions is largely based on procedural knowledge and provides only minimal opportunities for children to conceptualize the meaning and magnitude of fractional notations. Furthermore, our results demonstrate that a short intervention inducing children to manipulate, compare, and evaluate fractions improves their ability to associate fractional notations with numerical magnitude. 1
Early Years Learning with Digital Technologies: The Relationship Between Research and Design
"... Abstract: The overall aim of this symposium is to focus on the relationship between research and design of technologies for early years learning. Presentations will centre around two studies, one concerned with understanding the role that digital technologies play in shaping interactions between par ..."
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Abstract: The overall aim of this symposium is to focus on the relationship between research and design of technologies for early years learning. Presentations will centre around two studies, one concerned with understanding the role that digital technologies play in shaping interactions between parents and young children in the home and the other with understanding the role of digital manipulatives in early years learning of numeracy in school. It will be argued that whereas a common theme emerging from these studies is the importance of shared physical and social interaction within early years learning, digital technologies for this age group tend to be designed for individual use. Participants will be invited to offer explanations for this emphasis on individual learning and discussion will focus on alternative approaches to design that might take into account the intimacy of the interaction between young children and adults. Background and Aims of Symposium This symposium draws on the work of an ESRC 1 and Futurelab 2 funded network that aims to develop an understanding of early years learning with digital technologies. The work of the symposium will centre around two studies carried out by members of the network. The first study is concerned with understanding the