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Framework for Instruction and Assessment on Elementary Inferential Statistics Thinking
 Presentation at the Second International Conference on the Teaching of Mathematics
, 2002
"... The main objective in this paper is to describe a framework to characterize and assess the learning of elementary statistical inference. The key constructs of the framework are: populations and samples and their relationships; inferential process; sample sizes; sampling types and biases. To refine a ..."
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The main objective in this paper is to describe a framework to characterize and assess the learning of elementary statistical inference. The key constructs of the framework are: populations and samples and their relationships; inferential process; sample sizes; sampling types and biases. To refine and validate this scheme we have taken data from a sample of 49 secondary students sample using a questionnaire with 12 items in three different contexts: concrete, narrative and numeric. Theoretical analysis on the results obtained in this first research phase has permitted us to establish the key constructs described below and determine levels in them. Moreover this has allowed us to determine the students’ conceptions about the inference process and their perceptions about sampling possible biases and their sources. The framework is a theoretical contribution to the knowledge of the inferential statistical thinking domain and for planning teaching in the area.
The Use of Spatial Cognition in Graph Interpretation
"... We conducted an experiment to investigate whether spatial processing is used in graph comprehension tasks. Using an interference paradigm, we demonstrate that a graph task interfered more with performance on a spatial memory task than on a visual (nonspatial) memory task. Reaction times showed ther ..."
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We conducted an experiment to investigate whether spatial processing is used in graph comprehension tasks. Using an interference paradigm, we demonstrate that a graph task interfered more with performance on a spatial memory task than on a visual (nonspatial) memory task. Reaction times showed there was no speedaccuracy tradeoff. We conclude that it was the spatial nature of the graph task that caused the additional interference in the spatial memory task. We propose that current theories of graph comprehension should be expanded to include a spatial processing component.
THE NATURE OF MULTIPLE REPRESENTATIONS IN DEVELOPING MATHEMATICAL RELATIONSHIPS1
"... This study focuses on the representations and translations of mathematical relationships that are emphasized and taught at school, and discusses two theoretical models that may explain the pattern and difficulties in translating from one form of representations to another. Data were obtained from 79 ..."
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This study focuses on the representations and translations of mathematical relationships that are emphasized and taught at school, and discusses two theoretical models that may explain the pattern and difficulties in translating from one form of representations to another. Data were obtained from 79 students of grade 6. Analyses using structural equation modeling were performed to evaluate the two theoretical models. Results provided support for the hypothesis that multiple representations of mathematical relationships constitute different entities, and thus multiple representations do not by themselves help sixth grade students develop mathematical understanding.
Implications of Using Dynamic Geometry Technology 7 Implications of Using Dynamic Geometry Technology for Teaching and Learning
"... In this talk I would like to explore the implications of using Dynamic Geometry Technology for teaching and learning geometry at different levels of education. Through example explorations and problems using the Geometer's Sketchpad I hope to provoke questions concerning how children might lear ..."
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In this talk I would like to explore the implications of using Dynamic Geometry Technology for teaching and learning geometry at different levels of education. Through example explorations and problems using the Geometer's Sketchpad I hope to provoke questions concerning how children might learn geometry with such a tool, and the implications for teaching geometry with such a tool. I shall draw on my own experiences and the experiences of other teachers and researchers using dynamic geometry technology with young children, adolescents, and college students. What is Dynamic Geometry Technology? This question is best addressed through demonstration. I include any technological medium (both handheld and desktop computing devices) that provides the user with tools for creating the basic elements of Euclidean geometry (points, lines, line segments, rays, and circles) through direct motion via a pointing device (mouse, touch pad, stylus or arrow keys), and the means to construct geometric relations among these objects. Once constructed, the objects are transformable simply by dragging any one of their constituent parts. Examples of dynamic geometry technology include, but are not limited to the following:
Reconceptualizing procedural knowledge: Flexibility and innovation in equation solving. Unpublished manuscript, under review
, 2002
"... Kaye for their help coding, and to Valerie Mills and Ellen Hopkins for their assistance in the recruitment of participants. Financial support was provided by a grant to the second author from the Office of Naval Research. RECONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 1 Reconceptualizing procedural kno ..."
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Kaye for their help coding, and to Valerie Mills and Ellen Hopkins for their assistance in the recruitment of participants. Financial support was provided by a grant to the second author from the Office of Naval Research. RECONCEPTUALIZING PROCEDURAL KNOWLEDGE p. 1 Reconceptualizing procedural knowledge: Flexibility and innovation in equation solving
Designing Knowledge Representations for Learning Epistemic Practices of Science
 Practices of Science, in The Annual Meeting of the American Educational Research Association. 2000
, 2000
"... This paper presents initial findings from our collaborative effort to understand the roles various kinds of scientific representations play in supporting students' epistemological learning in science, through their development of epistemic practices. We present concrete design principles for th ..."
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This paper presents initial findings from our collaborative effort to understand the roles various kinds of scientific representations play in supporting students' epistemological learning in science, through their development of epistemic practices. We present concrete design principles for the development of representational tools that support students' inquiry and their development of scientific epistemic practices; and we sketch a framework for using such tools to support students' collaborative inquiry, both facetoface and online. These principles elucidate what we have learned about the ways in which representational tools support students' articulation of their knowledge, evaluation and negotiation of those ideas with their peers, collaboration around the knowledge representations, and instructional practices that support such complex forms of inquiry. We first present a general overview of our meaning of epistemic practices and general design principles to promote them. Subsequent sections briefly describe how our various research efforts instantiate these design principles within knowledge representations and activities designed to guide students' use of these representations. EPISTEMIC PRACTICES IN SCIENCE
WHAT IS THE PHILOSOPHY OF MATHEMATICS EDUCATION?
"... This question (what is the philosophy of mathematics education?) provokes a number of reactions, even before one tries to answer it. Is it a philosophy of mathematics education, or is it the philosophy of mathematics education? Use of the preposition ‘a ’ suggests that what is being offered is one o ..."
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This question (what is the philosophy of mathematics education?) provokes a number of reactions, even before one tries to answer it. Is it a philosophy of mathematics education, or is it the philosophy of mathematics education? Use of the preposition ‘a ’ suggests that what is being offered is one of several such perspectives, practices or areas of study. Use of the definite article ‘the ’ suggests to some the arrogation of definitiveness to the account given. 1 In other words, it is the dominant or otherwise unique account of philosophy of mathematics education. However, an alternative reading is that ‘the ’ refers to a definite area of enquiry, a specific domain, within which one account is offered. So the philosophy of mathematics education need not be a dominant interpretation so much as an area of study, an area of investigation, and hence something with this title can be an exploratory assay into this area. This is what I intend here. Moving beyond the first word, there is the more substantive question of the reference of the term ‘philosophy of mathematics education’. There is a narrow sense that can be applied in interpreting the words ‘philosophy ’ and ‘mathematics education’. The philosophy of some area or activity can be understood as its aims or rationale. Mathematics education understood
Geometric Explorations with Dynamic Geometry Applications based on van Hiele Levels
"... ABSTRACT: The purpose of this paper is to present classroomtested geometry activities based on the van Hiele geometric thinking levels using dynamic geometry applications. The other ideas behind the activities include teacher questioning, active student participation, and studentcentered decisionm ..."
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ABSTRACT: The purpose of this paper is to present classroomtested geometry activities based on the van Hiele geometric thinking levels using dynamic geometry applications. The other ideas behind the activities include teacher questioning, active student participation, and studentcentered decisionmaking. During the lessons student teachers engaged in selfexploration and reinvention of geometric relations. It was evident from the episodes that students raised their level of geometric thinking by building on their current geometric understanding. Key Words: dynamic geometry, van Hiele levels, teacher questioning
An examination of educational practices and assumptions regarding algebra instruction in the United
 Department of Science and Mathematics Education, The University of Melbourne
, 2001
"... Two potential roadblocks to successful implementation of early algebra in the United States are examined. Systematic algebra instruction has traditionally been reserved for students who have achieved arithmetic proficiency and have shown evidence of having crossed into the formal operational stage a ..."
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Two potential roadblocks to successful implementation of early algebra in the United States are examined. Systematic algebra instruction has traditionally been reserved for students who have achieved arithmetic proficiency and have shown evidence of having crossed into the formal operational stage as defined by Piaget’s stage theory. Current theories regarding these assumptions are discussed. Three developmental models are contrasted with Piaget’s theory. Several cognitive obstacles to algebra are analysed in relation to the arithmeticbefore algebra expectation. Finally, the paper highlights the importance of dialogue regarding the mental models math educators hold about the appropriateness of early algebra. Consider two children in 8th grade “Algebra 1, ” both intelligent and
Enabling teachers to perceive the affordances of a technologically rich learning environment for linear functions in order to design units of work incorporating best practice
 in Yang WC, Chu AC, de Alwis T, Ang KC (Eds.) Technology in Mathematics
, 2004
"... Abstract: A technology rich teaching and learning environment affords new ways of engaging students in learning mathematics. Teachers and students equally have to learn to become attuned to the affordances of technologically rich learning environments. This paper reports preliminary findings from an ..."
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Abstract: A technology rich teaching and learning environment affords new ways of engaging students in learning mathematics. Teachers and students equally have to learn to become attuned to the affordances of technologically rich learning environments. This paper reports preliminary findings from an Australian Research Council funded linkage project. This three year project aims to enhance mathematics achievement and engagement by using technology to support real problem solving and lessons of high cognitive demand. A design research methodology is being used to develop lesson sequences incorporating best practice. Findings from the first cycle of design research for the development and implementation of units for teaching linear functions in Year 9 will be reported. These are from classes in two schools where quite different approaches were taken to the teaching. A selection of electronic technologies was available to the teachers including graphing calculators and laptops with the image digitiser, GridPic. Student responses are presented to a hidden function task. These responses are discussed together with the results of document analysis of teacher work programs and student workbooks, and analyses of student and teacher interviews, lesson observational data, video tapes of student task solving, and verbal reports by teachers of their practice. The purpose of the research is to establish what it is that enables teachers to perceive, attend to, and exploit affordances of the technology salient to their teaching practice and likewise for students in their learning about function. 1.