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The role of children in the design of new technology
 Behaviour and Information Technology
, 2002
"... This paper suggests a framework for understanding the roles that children can play in the technology design process, particularly in regards to designing technologies that support learning. Each role, user, tester, informant, and design partner has been defined based upon a review of the literature ..."
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Cited by 175 (33 self)
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This paper suggests a framework for understanding the roles that children can play in the technology design process, particularly in regards to designing technologies that support learning. Each role, user, tester, informant, and design partner has been defined based upon a review of the literature and my lab’s own research experiences. This discussion does not suggest that any one role is appropriate for all research or development needs. Instead, by understanding this framework the reader may be able to make more informed decisions about the design processes they choose to use with children in creating new technologies. This paper will present for each role a historical overview, research and development methods, as well as the strengths, challenges, and unique contributions associated with children in the design process.
Knowing, doing, and teaching multiplication
 Cognition and Instruction
, 1986
"... This investigation analyzes the structure and process of multidigit multiplication. It includes a review of recent theories of mathematical knowledge and a description of several fourthgrade math lessons conducted in a regular classroom setting. Four types of mathematical knowledge are identified ..."
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Cited by 82 (1 self)
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This investigation analyzes the structure and process of multidigit multiplication. It includes a review of recent theories of mathematical knowledge and a description of several fourthgrade math lessons conducted in a regular classroom setting. Four types of mathematical knowledge are identified: intuitive, concrete, computational, and principled knowledge. The author considers each type in terms of its relation to instructional issues and suggests that instruction should focus on strengthening the connections among the four types. Illustrations from instructional sessions show children generating and testing hypotheses when salient connections are made between concrete materials and principled, computational practices. Implications for teaching are discussed along with suggestions for future research. Ever since there have been schools in this country, Americans have been debating what children should be learning in them. A significant part of that debate has addressed the subject of mathematics and pits the proponents of teaching computational skill against the advocates of fostering conceptual understanding. Computation is an aspect of mathematical knowledge that is familiar to most teachers and parents; hence, they are likely to support its place in the school curriculum. Most mathematicians, in contrast, see computation as an almost insignificant branch of their subject and thus are likely to believe that it is less important to be skilled in computation than in understanding how abstract mathematical principles can be used to analyze and solve problems. Two different views of what it means to know mathematics underlie this disagreement. Developments in curriculum and instruction have exacerbated the conflicts between these views, and teachers are often left to figure out acRequests for reprints should be sent to Magdalene Lampert, Department of Teacher Education
Culturally relevant mathematics teaching in a MexicanAmerican context
 Joumal for Research in Math Education
, 1997
"... This article examines mathematics instruction and its intersection with culturally relevant teaching in an elementary/middle school in a Mexican American community. The findings are based on a collaborativeresearch and schoolchange project involving university researchers, teachers, and the school ..."
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Cited by 47 (4 self)
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This article examines mathematics instruction and its intersection with culturally relevant teaching in an elementary/middle school in a Mexican American community. The findings are based on a collaborativeresearch and schoolchange project involving university researchers, teachers, and the school’s principal. On the basis of ethnographic data and an interdisciplinary theoretical framework, we propose a threepart model of culturally relevant mathematics instruction. The 3 components are (a) building on students ’ informal mathematical knowledge and building on students ’ cultural and experiential knowledge, (b) developing tools of critical mathematical thinking and critical thinking about knowledge in general, and (c) orientations to students ’ culture and experience. I was 15 [when I came to the U.S.] The first thing I learned was that I was different. Even with my Latino peers. There are levels of being Mexican. I didn’t know how bad it was to be who I was. There were so many pressures from name calling, insults in the street, said aloud because I was so Mexican … I had a lot of anger. It was this anger, and anger at the experiences of my brother in school. We all did not do as well because of the school experiences. That made me want to be a teacher.
A Framework for Research and Curriculum Development in Undergraduate Mathematics Education
 Research in Collegiate Mathematics Education
, 1996
"... Over the past several years, a community of researchers has been using and refining a particular framework for research and curriculum development in undergraduate mathematics education. The purpose of this paper is to share the results of this work with the mathematics education community at large ..."
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Cited by 47 (6 self)
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Over the past several years, a community of researchers has been using and refining a particular framework for research and curriculum development in undergraduate mathematics education. The purpose of this paper is to share the results of this work with the mathematics education community at large by describing the current version of the framework and giving some examples of its application. Our framework utilizes qualitative methods for research and is based on a very specific theoretical perspective that is being developed through attempts to understand the ideas of Piaget concerning reflective abstraction and reconstruct them in the context of college level mathematics. Our approach has three components. It begins with an initial theoretical analysis of what it means to understand a concept and how that understanding can be constructed by the learner. This leads to the design of an instructional treatment that focuses directly on trying to get students to make the constructions cal...
THE ROLE OF VISUAL REPRESENTATIONS IN THE LEARNING of Mathematics
, 2003
"... Visualization, as both the product and the process of creation, interpretation and reflection upon pictures and images, is gaining increased visibility in mathematics and mathematics education. This paper is an attempt to define visualization and to analyze, exemplify and reflect upon the many diff ..."
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Cited by 42 (0 self)
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Visualization, as both the product and the process of creation, interpretation and reflection upon pictures and images, is gaining increased visibility in mathematics and mathematics education. This paper is an attempt to define visualization and to analyze, exemplify and reflect upon the many different and rich roles it can and should play in the learning and the doing of mathematics. At the same time, the limitations and possible sources of difficulties visualization may pose for students and teachers are considered.
Knowledge construction and diverging thinking in elementary & advanced mathematics
 Educational Studies in Mathematics
, 1999
"... ABSTRACT: This paper begins by considering the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum. We base our theoretical development on fundamental cognitive activities, namely, perception of the world, action up ..."
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Cited by 16 (5 self)
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ABSTRACT: This paper begins by considering the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum. We base our theoretical development on fundamental cognitive activities, namely, perception of the world, action upon it and reflection on both perception and action. We see an emphasis on one or more of these activities leading not only to different kinds of mathematics, but also to a spectrum of success and failure depending on the nature of the focus in the individual activity. For instance, geometry builds from the fundamental perception of figures and their shape, supported by action and reflection to move from practical measurement to theoretical deduction and euclidean proof. Arithmetic, on the other hand, initially focuses on the action of counting and later changes focus to the use of symbols for both the process of counting and the concept of number. The evidence that we draw together from a
Concept Maps & Schematic Diagrams as Devices for Documenting the Growth of Mathematical Knowledge
 In O. Zaslavsky (Ed.), Proceedings of the 23 rd Conference of PME
, 1999
"... The major focus of this study is to trace the cognitive development of students throughout a mathematics course and to seek the qualitative differences between those of different levels of achievement. The aspect of the project described here concerns the use of concept maps constructed by the stude ..."
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Cited by 14 (6 self)
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The major focus of this study is to trace the cognitive development of students throughout a mathematics course and to seek the qualitative differences between those of different levels of achievement. The aspect of the project described here concerns the use of concept maps constructed by the students at intervals during the course. From these maps, schematic diagrams were constructed which strip the concept maps of detail and show only how they are successively built by keeping some old elements, reorganising, and introducing new elements. The more successful student added new elements to old in a structure that gradually increased in complexity and richness. The less successful had little constructive growth, building new maps on each occasion.
Handling pupils’ misconceptions
 Pythagoras
, 1989
"... This paper will briefly delineate a theory for learning mathematics as a basis to reflect on (some particular) misconceptions of pupils in mathematics. Such a theory should enable us • to predict what errors pupils will typically make • to explain how and why children make (these) errors • to help p ..."
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Cited by 11 (0 self)
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This paper will briefly delineate a theory for learning mathematics as a basis to reflect on (some particular) misconceptions of pupils in mathematics. Such a theory should enable us • to predict what errors pupils will typically make • to explain how and why children make (these) errors • to help pupils to resolve such misconceptions. 1. THE ROLE OF THEORY Teachers are often wary of theory they want something practical. Yet, as Dewey has said, “in the end, there is nothing as practical as a good theory. ” How come? Theory is like a lens through which one views the facts; it influences what one sees and what one does not see. “Facts ” can only be interpreted in terms of some theory. Without an appropriate theory, one cannot even state what the “facts ” are. Let me illustrate with a story, taken from Davis (1984). It is said that in Italy in the 1640’s, the water table had receded so far that a very deep well had to be sunk in order to reach water. This was done. Then pumps were fitted to the pipes, and... disaster!... no water poured out of the spigot. It was clear that something was wrong, but what? Their understanding of the situation, i.e. their theory of pumping water, was that it was the pumps that pulled (sucked) the water to the surface. So the fault had to be with
Creative Problem Solving
, 1991
"... Findings from prior research are drawn together to create a learning model for elementary school mathematics in the cognitiveconstructivist tradition. A potential teaching/learning process consistent with the model was developed and applied in a longitudinal collaborative arrangement between univer ..."
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Cited by 9 (1 self)
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Findings from prior research are drawn together to create a learning model for elementary school mathematics in the cognitiveconstructivist tradition. A potential teaching/learning process consistent with the model was developed and applied in a longitudinal collaborative arrangement between university personnel and a local elementary school using a conceptually based curriculum that posed problems requiring active student involvement with physical materials to model mathematical situations, defined symbols, and developed solution strategies. As children used these materials, they actively construed the operations and principaes of arithmetic. In another phase children sketched the materials and situations in a move toward abstraction. They then constructed mental images through imagining actions on physical materials. Experiences with the mental images allowed for student construction of arithmetic generalizations
D.: 2000, ‘Objects, Actions and Images: A Perspective on Early Number Development
 Journal of Mathematical Behavior
"... is the purpose of this paper to present a review of research evidence that indicates the existence of qualitatively different thinking in elementary number development. In doing so the paper summarises empirical evidence obtained over a period of ten years. This evidence first signalled qualitative ..."
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Cited by 8 (0 self)
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is the purpose of this paper to present a review of research evidence that indicates the existence of qualitatively different thinking in elementary number development. In doing so the paper summarises empirical evidence obtained over a period of ten years. This evidence first signalled qualitative differences in numerical processing (Gray, 1991), and was seminal in the development of the notion of procept (Gray & Tall, 1994). More recently it examines the role of imagery in elementary number processing (Pitta and Gray, 1997). Its conclusions indicate that in the abstraction of numerical concepts from numerical processes qualitatively different outcomes may arise because children concentrate on different objects or different aspects of the objects which are components of numerical processing. The notion that numerical concepts are formed from actions with physical