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Applying covariational reasoning while modeling dynamic events: A framework and a study
 Journal for Research in Mathematics Education
, 2002
"... The article develops the notion of covariational reasoning and proposes a framework for describing the mental actions involved in applying covariational reasoning when interpreting and representing dynamic function events. It also reports on an investigation of highperforming second semester calcul ..."
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The article develops the notion of covariational reasoning and proposes a framework for describing the mental actions involved in applying covariational reasoning when interpreting and representing dynamic function events. It also reports on an investigation of highperforming second semester calculus students ' ability to reason about covarying quantities in dynamic situations. The study revealed that these students were able to construct images of a function's dependent variable changing in tandem with the imagined change of the independent variable, and in some situations, were able to construct images of rate of change for contiguous intervals of a function's domain. However, students appeared to have difficulty forming images of continuously changing rate and were unable to accurately represent and interpret increasing and
Mathematics and virtual culture: An evolutionary perspective on technology and mathematics education
 Educational Studies in Mathematics
, 1999
"... ABSTRACT. This paper suggests that from a cognitiveevolutionary perspective, computational media are qualitatively different from many of the technologies that have promised educational change in the past and failed to deliver. Recent theories of human cognitive evolution suggest that human cogniti ..."
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Cited by 12 (4 self)
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ABSTRACT. This paper suggests that from a cognitiveevolutionary perspective, computational media are qualitatively different from many of the technologies that have promised educational change in the past and failed to deliver. Recent theories of human cognitive evolution suggest that human cognition has evolved through four distinct stages: episodic, mimetic, mythic, and theoretical. This progression was driven by three cognitive advances: the ability to “represent ” events, the development of symbolic reference, and the creation of external symbolic representations. In this paper, we suggest that we are developing a new cognitive culture: a “virtual ” culture dependent on the externalization of symbolic processing. We suggest here that the ability to externalize the manipulation of formal systems changes the very nature of cognitive activity. These changes will have important consequences for mathematics education in coming decades. In particular, we argue that mathematics education in a virtual culture should strive to give students generative fluency to learn varieties of representational systems, provide opportunities to create and modify representational forms, develop skill in making and exploring virtual environments, and emphasize mathematics as a fundamental way of making sense of the world, reserving most exact computation and formal proof for those who will need those specialized skills.
Rethinking covariation from a quantitative perspective: Simultaneous continuous variation
 North Carolina State University
, 1998
"... We hypothesize that students ’ engagement in tasks which require them to track two sources of information simultaneously are propitious for their envisioning graphs as composed of points, each of which record the simultaneous state of two quantities that covary continuously. We investigated this hyp ..."
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We hypothesize that students ’ engagement in tasks which require them to track two sources of information simultaneously are propitious for their envisioning graphs as composed of points, each of which record the simultaneous state of two quantities that covary continuously. We investigated this hypothesis in a teaching experiment involving one 8thgrade student. Details of the student’s experience and an analysis of his development are presented. Confrey and Smith (1994, 1995) explicate a notion of covariation that entails moving between successive values of one variable and coordinating this with moving between corresponding successive values of another variable (1994, p.33). They also explain, “in a covariation approach, a function is understood as the juxtaposition of two sequences, each of which is generated independently through a pattern of data values ” (1995, p. 67). Coulombe and Berenson build on these definitions, and on ideas discussed by Thompson and Thompson (1994b, 1996), to describe a concept of covariation that entails these properties: “(a) the identification of two data sets, (b) the coordination of two data patterns to form associations between increasing, decreasing, and constant patterns, (c) the linking of two data patterns to establish specific connections between data values, and (d) the generalization of the link to predict unknown data values. ” (p. 88) Thinking of covariation as the coordination of sequences fits well with employing tables to present successive states of a variation. We find it useful to extend this idea, to consider possible imagistic foundations for someone’s ability to “see ” covariation. In this regard, our notion of covariation is of someone holding in mind a sustained image of two quantities ’ values (magnitudes) simultaneously. It entails coupling the two quantities, so that, in one’s understanding, a
Designing to see and share structure in number sequences
 the International Journal for Technology in Mathematics Education
, 2006
"... This paper reports on a design experiment in the domain of number sequences conducted in the course of the WebLabs project. We iteratively designed and tested a set of activities and tools in which 1014 year old students used the ToonTalk programming environment to construct models of sequences and ..."
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This paper reports on a design experiment in the domain of number sequences conducted in the course of the WebLabs project. We iteratively designed and tested a set of activities and tools in which 1014 year old students used the ToonTalk programming environment to construct models of sequences and series, and then shared their models and their observations about them utilising a webbased collaboration system. We report on the evolution of a design pattern (programming method) called ‘Streams’ which enables students to engage in the process of summing and ‘hold the series in their hand’, and consequently make sophisticated arguments regarding the mathematical structures of the sequences without requiring the use of algebra. While the focus of this paper is mainly on the design of activities, and in particular their epistemological foundations, some illustrative examples of one group of students ’ work indicate the potential of the activities and tools for expressing and reflecting on deep mathematical ideas. 1
Teachers ’ Ways of Listening and Responding to Students ’ Emerging Mathematical Models
"... Abstract: In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how a ..."
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Abstract: In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students ' ways of thinking about exponential functions and (b) how they responded to the students’ thinking in their classroom practice. Through an analysis of the teachers ' actions in the classroom, I describe the teachers ' developing knowledge when using modeling tasks with secondary students. The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models. Having a welldeveloped schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth. Implications for the knowledge needed to teach mathematics through modeling are discussed. ZDMClassification: I20, M13 More than a decade of research on teachers’ professional development would suggest that, among other things, effective teachers need to attend to students ’ ways of thinking about mathematical
Early Notions of Functions in a TechnologyRich Teaching and Learning Environment (TRTLE)
"... This paper focuses on notions of function Year 9 students hold as they begin to study functions. As these notions may be fragile, the questions, tasks, and ways of interacting orchestrated by the teacher to elicit depth of understanding, or allow observation of changing notions, are of interest. Ext ..."
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This paper focuses on notions of function Year 9 students hold as they begin to study functions. As these notions may be fragile, the questions, tasks, and ways of interacting orchestrated by the teacher to elicit depth of understanding, or allow observation of changing notions, are of interest. Extended tasks where students were required to make choices about solution paths provided opportunities for students to develop and consolidate their concept images. Discussion between small groups provided the best evidence of developing and stable conceptions held by students in contrast to written scripts where the strength of these understandings was not evident.
UNDERSTANDING OF RATE OF CHANGE AND ACCUMULATION IN MULTIAGENT
"... Our everyday world is characterized by quantitative change – from fluctuating global temperatures and shifting medical insurance costs to the changes in a car’s tire pressure from winter to spring. Often, these quantities do not reflect only a single entity or action, but many different interactions ..."
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Our everyday world is characterized by quantitative change – from fluctuating global temperatures and shifting medical insurance costs to the changes in a car’s tire pressure from winter to spring. Often, these quantities do not reflect only a single entity or action, but many different interactions and behaviors. This paper investigates how students think and talk about patterns of quantitative change over time while they interact with a computational agentbased model of population growth, which represents change in population as the result of many entities (simulated people) contributing individually to a single changing quantity (population). We found that students often mixed not only mathematical, but also scientific and everyday explanations to make sense of patterns of change. These combinations of explanations led some students to experience (or resolve) difficulties in describing what rate of change reflects in the specific case of population growth; and in understanding and interpreting quantitative change in terms of the individual and populationlevel behavior of a dynamic system.
oro.open.ac.uk Designing To See And Share Structure In Number Sequences
, 2006
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Running Head: Students, Functions, and Curriculum
"... recommendations stated here are those of the author and do not necessarily reflect official positions of NSF. Thompson Students, Functions, and Curriculum Someone, I cannot remember who, paraphrased Winston Churchill by saying that mathematics and mathematics education are two disciplines separated ..."
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recommendations stated here are those of the author and do not necessarily reflect official positions of NSF. Thompson Students, Functions, and Curriculum Someone, I cannot remember who, paraphrased Winston Churchill by saying that mathematics and mathematics education are two disciplines separated by a common subject. The mathematician is primarily concerned with doing mathematics at a high level of abstraction. The mathematics educator is primarily concerned with what it is that one does when doing mathematics and what kinds of experiences are propitious for a person’s later successes. Until recently mathematics education research has focused predominantly on the learning and teaching of early mathematics in the school curriculum, so it is natural that practicing mathematicians have found it difficult to relate to mathematics education research. I suspect that the current interest in calculus reform [21, 63] and the broader rethinking of the undergraduate curriculum, together with the advent of the AMS/MAA Joint Committee on Research in Undergraduate Mathematics Education, will lead to a wider recognition that mathematics and mathematics education