Ricardo Nemirovsky
San Diego State University, Mathematics, Faculty Member
Traditionally, mathematics has been used as means for modeling aspects of the experienced world, and it is often taken as axiomatic that one can learn mathematics more effectively if one is able to apply what one already knows and can... more
Traditionally, mathematics has been used as means for modeling aspects of the experienced world, and it is often taken as axiomatic that one can learn mathematics more effectively if one is able to apply what one already knows and can do. We illustrate how we can substantially deepen the connection with everyday experience by using mathematical functions to generate phenomena as well as model them. We first provide a framework for examining relations among simulations, notations and physical phenomena. We then illustrate with a 9th grade classroom episode how students’ activity taps into their linguistic, kinesthetic and notational resources to deepen their engagement with important mathematical ideas.
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This paper examines two types of discourse in which teachers engage when discussing case studies based on classroom episodes, and the ways in which the availability of video data of these episodes may motivate a shift in the mode of... more
This paper examines two types of discourse in which teachers engage when discussing case studies based on classroom episodes, and the ways in which the availability of video data of these episodes may motivate a shift in the mode of discourse used. We interviewed two pairs of secondary school mathematics teachers after they had read a case study based on a 16-minute mathematics classroom episode taped in a secondary school in the United States. During each interview, a multimedia version of the case study, including video of the original episode, was available to the participants. We identify two modes of discourse engaged in by the teachers during the interviews: Grounded Narrative and Evaluative Discourse. We examine and identify the characteristics of the two discourse forms, drawn from both video and textual analysis. These characteristics are self-reflective talk, perspective, ethics, and linguistic patterns. The identification of two modes of discourse is relevant for researchers and teacher educators using case studies or video recordings. In addition, the findings provide insight into how teachers are “seeing” classroom events in a video case study.
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We discuss the activities of two 5th grade boys working together during two days of mathematics class, on a problem of representing a motion along a linear path Over the two days, the boys represent the motion in three different... more
We discuss the activities of two 5th grade boys working together during two days of mathematics class, on a problem of representing a motion along a linear path Over the two days, the boys represent the motion in three different mathematical environments: at the blackboard using a table of positions and stepsizes over time; at the computer using a computer
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... Chapter 6: Mathematical Conversations Jesse Solomon, City on a Hill Ricardo Nemirovsky, TERC Abstract This ... Some of the commentaries include paragraphs in gray type written in first person by the teacher, Mr. Solomon. Two ...
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The broad goal of this report is to describe a form of knowing and a way of participating in mathematics learning that contribute to and further alternative views of transfer of learning. We selected an episode with an undergraduate... more
The broad goal of this report is to describe a form of knowing and a way of
participating in mathematics learning that contribute to and further alternative views
of transfer of learning. We selected an episode with an undergraduate student
engaged in a number of different tasks involving a physical tool called “water
wheel”. The embodied cognition literature is rich with connections between
kinesthetic activity and how people qualitatively understand and interpret graphs of
motion. However, studies that examine the interplay between kinesthetic activities
and work with equations and other algebraic expressions are mostly absent. We show
through this episode that kinesthetic experience can transfer or generalize to the
building and interpretation of formal, highly symbolic mathematical expressions.
participating in mathematics learning that contribute to and further alternative views
of transfer of learning. We selected an episode with an undergraduate student
engaged in a number of different tasks involving a physical tool called “water
wheel”. The embodied cognition literature is rich with connections between
kinesthetic activity and how people qualitatively understand and interpret graphs of
motion. However, studies that examine the interplay between kinesthetic activities
and work with equations and other algebraic expressions are mostly absent. We show
through this episode that kinesthetic experience can transfer or generalize to the
building and interpretation of formal, highly symbolic mathematical expressions.
Research Interests:
When students share and explore chemistry ideas with others, they use gestures and their bodies to perform their understanding. As a publicly visible, spatio-dynamic medium of expression, gestures and the body provide productive resources... more
When students share and explore chemistry ideas with others, they use gestures and their bodies to perform their understanding. As a publicly visible, spatio-dynamic medium of expression, gestures and the body provide productive resources for imagining the submicroscopic, three-dimensional, and dynamic phenomena of chemistry together. In this paper, we analyze the role of gestures and the body as interactional resources in interactive spaces for collaborative meaning making in chemistry. With our moment-by-moment analysis of video-recorded interviews, we demonstrate how creating spaces for, attending to, and interacting with students’ gestures and bodily performances generate opportunities for learning. Implications for teaching and assessment that are responsive to students’ ideas in chemistry are discussed.
Research Interests:
Research Interests:
This paper examines two types of discourse in which teachers engage when discussing case studies based on classroom episodes, and the ways in which the availability of video data of these episodes may motivate a shift in the mode of... more
This paper examines two types of discourse in which teachers engage when discussing case studies based on classroom episodes, and the ways in which the availability of video data of these episodes may motivate a shift in the mode of discourse used. We interviewed two pairs of secondary school mathematics teachers after they had read a case study based on a 16-minute mathematics classroom episode taped in a secondary school in the United States. During each interview, a multimedia version of the case study, including video of the original episode, was available to the participants. We identify two modes of discourse engaged in by the teachers during the interviews: Grounded Narrative and Evaluative Discourse. We examine and identify the characteristics of the two discourse forms, drawn from both video and textual analysis. These characteristics are self-reflective talk, perspective, ethics, and linguistic patterns. The identification of two modes of discourse is relevant for researchers and teacher educators using case studies or video recordings. In addition, the findings provide insight into how teachers are “seeing” classroom events in a video case study.
Research Interests:
In this article we contribute a perspective on mathematical embodied cognition consistent with a phenomenological understanding of perception and body motion. It is based on the analysis of 4 selected episodes in 1 session of an... more
In this article we contribute a perspective on mathematical embodied cognition consistent with a phenomenological understanding of perception and body motion. It is based on the analysis of 4 selected episodes in 1 session of an undergraduate mathematics class. The theme of this particular class session was the geometric interpretation of the addition and multiplication of complex numbers. On the basis of these episodes, the article examines 2 conjectures: (a) The mathematical insights developed by an individual or a group are expressed in and constituted by perceptuo-motor activity, and (b) the learning of mathematical ideas is shaped in nondeterministic ways by the setting or learning environment.
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Abstract This article reports a preliminary study of high school students making sense of the behavior displayed by a Lorenzian Water Wheel. The Lorenzian Water Wheel is a rotating disk driven by the flow of water whose motion may be... more
Abstract This article reports a preliminary study of high school students making sense of the behavior displayed by a Lorenzian Water Wheel. The Lorenzian Water Wheel is a rotating disk driven by the flow of water whose motion may be periodic or not, depending on ...
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Research Interests:
The goal of this article is to develop a new perspective on transfer of learning integrating cognition, emotion, and bodily experience. It is based on a case study with a 10-year-old girl as she explored the use of a motion detector,... more
The goal of this article is to develop a new perspective on transfer of learning integrating cognition, emotion, and bodily experience. It is based on a case study with a 10-year-old girl as she explored the use of a motion detector, allowing for the simultaneous graphing of the position versus time of 2 moving points. The paper elaborates on the notion of episodic feeling and illustrates a phenomenological approach for the study of transfer of learning whose point is not ascertaining mechanisms of transfer but elucidating within the infinite landscape of human experiences certain ones that seem amenable to characterization as transfer of learning.
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In this article, we will describe the results of a study of 6th grade students learning about the mathematics of change. The students in this study worked with software environments for the computer and the graphing calculator that... more
In this article, we will describe the results of a study of 6th grade students learning about the mathematics of change. The students in this study worked with software environments for the computer and the graphing calculator that included a simulation of a moving elevator, linked to a graph of its velocity vs. time. We will describe how the students and their teacher negotiated the mathematical meanings of these representations, in interaction with the software and other representational tools available in the classroom. The class developed ways of selectively attending to specific features of stacks of centimeter cubes, hand-drawn graphs, and graphs (labeled velocity vs. time) on the computer screen. In addition, the class became adept at imagining the motions that corresponded to various velocity vs. time graphs. In this article, we describe this development as a process of learning to see mathematical representations of motion. The main question this article addresses is: How do students learn to see mathematical representations in ways that are consistent with the discipline of mathematics?
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This paper is a case study of how a high school student, whom we call Karen, used a computer-based tool, the Contour Analyzer, to create graphs of height vs. distance and slope vs. distance for a flat board that she positioned with... more
This paper is a case study of how a high school student, whom we call Karen, used a computer-based tool, the Contour Analyzer, to create graphs of height vs. distance and slope vs. distance for a flat board that she positioned with different slants and orientations. With the Contour Analyzer one can generate, on a computer screen, graphs representing functions of height and slope vs. distance corresponding to a line traced along the surface of a real object. Karen was interviewed for three one-hour sessions in an individual teaching experiment. In this paper, our focus is on how Karen came to recognize by visual inspection the mathematical behavior of the slope vs. distance function corresponding to contours traced on a flat board. Karen strove to organize her visual experience by distinguishing which aspects of the board are to be noticed and which ones are to be ignored, as well as by determining the point of view that one should adopt in order to ‘see’ the variation of slope along an object. We have found it inspiring to use Winnicott's (1971) ideas about transitional objects to examine the role of the graphing instrument for Karen. This theoretical background helped us to articulate a perspective on mathematical visualization that goes beyond the dualism between internal and external representations frequently assumed in the literature, and focuses on the lived-in space that Karen experienced which encompassed at once physical attributes of the tool and human possibilities of action.
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This case study focuses on how a high school student, Laura, learned the meaning of the velocity sign. By moving a toy car she created many real-time graphs on a computer screen. The study strives to show that her learning was not just an... more
This case study focuses on how a high school student, Laura, learned the meaning of the velocity sign. By moving a toy car she created many real-time graphs on a computer screen. The study strives to show that her learning was not just an acknowledgment of a rule, but a broad questioning and revision of her thinking about graphs and motion. Laura's process exemplifies what is involved in the learning of a way of symbolizing situations of physical change.
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In this article we contribute a perspective on mathematical embodied cognition consistent with a phenomenological understanding of perception and body motion. It is based on the analysis of 4 selected episodes in 1 session of an... more
In this article we contribute a perspective on mathematical embodied cognition consistent with a phenomenological understanding of perception and body motion. It is based on the analysis of 4 selected episodes in 1 session of an undergraduate mathematics class. The theme of this particular class session was the geometric interpretation of the addition and multiplication of complex numbers. On the basis of these episodes, the article examines 2 conjectures: (a) The mathematical insights developed by an individual or a group are expressed in and constituted by perceptuo-motor activity, and (b) the learning of mathematical ideas is shaped in nondeterministic ways by the setting or learning environment.
Research Interests:
Case studies of major creative figures who were active in different domains can help to indicate commonalities and distinctive features in the creative process. With this goal in mind, a comparison is made between the mathematician Georg... more
Case studies of major creative figures who were active in different domains can help to indicate commonalities and distinctive features in the creative process. With this goal in mind, a comparison is made between the mathematician Georg Cantor's study of various orders of infinity and the psychologist Sigmund Freud's exploration of the operation of the unconscious. In both cases, similar processes can be discerned: (a) articulations of a new intuition; (b) construction of local coherences; (c) the reworking of standard symbol systems, giving way to the creation of a new, more adequate symbolic system; and (d) the articulation of a new thema (Holton, 1988). The study also describes a number of contrasts, among them the criteria by which formulations are judged in the two domains, the contrasting cosmological stances assumed by the investigators toward their projects, and the differing needs for a formal symbol system.