- Michelle Zandieh is a professor at Arizona State University whose research focuses on the teaching and learning of un... moreMichelle Zandieh is a professor at Arizona State University whose research focuses on the teaching and learning of undergraduate mathematics courses such as linear algebra, geometry and calculus. She is an author of over 30 journal articles and book chapter and has been funded as a principle investigator on National Science Foundation grants since 2001. Since 2007, most of her funding has contributed to the development of the Inquiry-Oriented Linear Algebra (IOLA) curriculumedit
The Arizona Collaborative for Excellence in Preparation of Teachers is a large National Science Foundation funded project aimed at revising science and mathematics pre-service courses at a large public university in the South-western... more
The Arizona Collaborative for Excellence in Preparation of Teachers is a large National Science Foundation funded project aimed at revising science and mathematics pre-service courses at a large public university in the South-western United States. This chapter describes the collaborations of a community of university faculty in reforming a block of five pre-service mathematics and mathematics education courses. Through a series of workshops and ongoing dialogue, both the instructional delivery and curriculum for these pre-service courses has shifted to student-centred classrooms with inquiry, concept development and problem solving as central themes. The chapter provides information about the process and products of these reforms, with a major focus on providing specific insights into the role of research in guiding the curricular and instructional philosophies and decisions.
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Los sistemas de ecuaciones lineales (SEL) corresponden a un concepto fundamental del álgebra lineal, pero hay relativamente poca investigación, pero hay relativamente poca investigación acerca de la enseñanza y el aprendizaje de los SEL,... more
Los sistemas de ecuaciones lineales (SEL) corresponden a un concepto fundamental del álgebra lineal, pero hay relativamente poca investigación, pero hay relativamente poca investigación acerca de la enseñanza y el aprendizaje de los SEL, particularmente de las concepciones de los estudiantes acerca de sus soluciones. Se ha encontrado que la resolución de sistemas con un número infinito de soluciones o sin solución tiende a ser menos intuitivo para los estudiantes, lo cual indica la necesidad de más investigación en la enseñanza y aprendizaje de este tema. Entrevistamos a dos estudiantes de matemáticas que eran también maestros en formación a través de un experimento de enseñanza por parejas para mirar cómo razonaban acerca de las soluciones de SEL en ℝ3. Presentamos los resultados enfocando en la progresión del razonamiento de los estudiantes sobre las soluciones de los SEL a través del lento de simbolización. Documentamos la progresión de su razonamiento como una acumulación de sig...
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We present an innovative task sequence for an introductory linear algebra course that supports students' reinvention of eigentheory and diagonalization. Grounded in the instructional design theory of Realistic Mathematics Education,... more
We present an innovative task sequence for an introductory linear algebra course that supports students' reinvention of eigentheory and diagonalization. Grounded in the instructional design theory of Realistic Mathematics Education, the task sequence builds from students' experience with linear transformations in to introduce the idea of stretch factors and stretch directions. This is leveraged towards defining eigenvalues and eigenvectors, reinventing methods to determine them, and connecting them to change of basis and diagonalization. In the poster, we discuss the development of the task sequence and analyses of students' work on the tasks, specifically on characterizing their approaches for developing eigentheory methods.
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Researchers and mathematicians have argued that students should be engaged in the activity of defining mathematical concepts. This report looks at the role of proving in students’ defining activity. A preliminary framework is offered to... more
Researchers and mathematicians have argued that students should be engaged in the activity of defining mathematical concepts. This report looks at the role of proving in students’ defining activity. A preliminary framework is offered to account for the ways in which proving can contribute to the process of defining. Three categories of contribution (motivation, guidance, and assessment) are illustrated in the context of two classroom episodes (one from a geometry course and another from a group theory course) in which students are engaged in defining.
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In his 1976 book, Proofs and Refutations, Lakatos presents a collection of case studies to illustrate methods of mathematical discovery in the history of mathematics. In this paper, we reframe these methods in ways that we have found make... more
In his 1976 book, Proofs and Refutations, Lakatos presents a collection of case studies to illustrate methods of mathematical discovery in the history of mathematics. In this paper, we reframe these methods in ways that we have found make them more amenable for use as a framework for research on learning and teaching mathematics. We present an episode from an undergraduate abstract algebra classroom to illustrate the guided reinvention of mathematics through processes that strongly parallel those described by Lakatos. Our analysis suggests that the constructs described by Lakatos can provide a useful framework for making sense of the mathematical activity in classrooms where students are actively engaged in the development of mathematical ideas and provide design heuristics for instructional approaches that support the learning of mathematics through the process of guided reinvention.
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This study examines students’ reasoning about eigenvalues and eigenvectors as evidenced by their written responses to two open-ended response questions. This analysis draws on data taken from 126 students whose instructors received a set... more
This study examines students’ reasoning about eigenvalues and eigenvectors as evidenced by their written responses to two open-ended response questions. This analysis draws on data taken from 126 students whose instructors received a set of supports to implement a particular inquiry-oriented instructional approach and 129 comparable students whose instructors did not use this instructional approach. In this chapter, we offer examples of student responses that provide insight into students’ reasoning and summarize broad trends observed in our quantitative analysis. In general, students in both groups performed better on the procedurally oriented question than on the conceptually oriented question. The group of students whose instructors received support to implement the inquiry-oriented approach outperformed the other group of students on the conceptually oriented question and performed equally well on the procedurally oriented question.
Linear algebra poses a number of significant challenges for students that need to be better understood in order to improve instruction and student understanding. At the time the tenth conference on Research in Undergraduate Mathematics... more
Linear algebra poses a number of significant challenges for students that need to be better understood in order to improve instruction and student understanding. At the time the tenth conference on Research in Undergraduate Mathematics Education, we had just begun a study intended to explore these challenges. Our preliminary report was given in a “working group session ” format, in which we brought together participants to examine and discuss the potential for a specific modeling task designed to develop, explore, and reveal students ’ thinking about ideas relating to eigenvalues, eigenvectors, and eigenspaces. This brief report provides background information on our study and summarizes the discussion from the working group session.
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We explore ways that university students handle proving statements that have the overall structure of a conditional implies a conditional, i.e., (p # q) ! (r # s). We structure our analysis using the theory of conceptual blending. We find... more
We explore ways that university students handle proving statements that have the overall structure of a conditional implies a conditional, i.e., (p # q) ! (r # s). We structure our analysis using the theory of conceptual blending. We find conceptual blending useful for describing the creation of powerful new ideas necessary for proof construction as well as for describing the creation of blends that slow or hinder student efforts at proof construction.
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... open-ended questions, are the primary instruments for collecting data on each student'sunderstanding. ... Dubinsky and colleagues), and notions of multiple representations for function,limit, and derivative. I describe the... more
... open-ended questions, are the primary instruments for collecting data on each student'sunderstanding. ... Dubinsky and colleagues), and notions of multiple representations for function,limit, and derivative. I describe the concept of derivative as three layers of process-objects: the ...
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This paper looks at proof production in the midst of classroom interaction. The setting is a collegelevel geometry course in which students are working on the following task: Prove that two paralleltransported lines in the plane are... more
This paper looks at proof production in the midst of classroom interaction. The setting is a collegelevel geometry course in which students are working on the following task: Prove that two paralleltransported lines in the plane are parallel in the sense that they do not intersect. A proof of this statement istraced from a student's idea, through a small group discussion, to a large class discussion moderated by ateacher. As the proof emerges through a series of increasingly public settings we see ways in which the keyidea of the proof serves to both open and close class discussion. We look at several examples of openingand closing, showing how not only the key idea, but the warrants and justifications connected to it, play animportant role in the proof development.
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In this paper we develop the notion of a hypothetical collective progression (HCP). We offer this construct as an alternative to the construct of hypothetical learning trajectory in order to (a) foreground the mathematical development of... more
In this paper we develop the notion of a hypothetical collective progression (HCP). We offer this construct as an alternative to the construct of hypothetical learning trajectory in order to (a) foreground the mathematical development of the collective rather than that of individuals, and (b) highlight the integral role of the teacher within this development. We offer an abbreviated example of an HCP from introductory linear algebra based on the “Italicizing N” task sequence, in which students work to generate and combine matrices that correspond to geometric transformations specified within the problem context. In particular, we describe the ways in which the HCP supports students in developing and extending local “matrix acting on a vector” views of matrix multiplication (focused on individual mappings of input vectors to output vectors) to more global views in which matrices are conceptualized in terms of how they transform a space in a coordinated way.
Shirley Atzmon, the Hebrew University <shirlyazmon@walla.com>, Rina Hershkowitz (organizer), the Weizmann Institute <rina.hershkovitz@weizmann.ac.il> Chris Rasmussen, San Diego State University... more
Shirley Atzmon, the Hebrew University <shirlyazmon@walla.com>, Rina Hershkowitz (organizer), the Weizmann Institute <rina.hershkovitz@weizmann.ac.il> Chris Rasmussen, San Diego State University <chrisraz@sciences.sdsu.edu>, Baruch Schwarz (organizer and chair), the Hebrew University <msschwar@mscc.huji.ac.il>, Gerry Stahl, Drexel University <gs47@drexel.edu>, Megan Wawro, San Diego State University <meganski110@hotmail.com>, Michelle Zandieh, Arizona State University zandieh@asu.edu, Mitchell Nathan (discussant), University of Wisconsin-Madison <mnathan@wisc.edu>
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A prominent problem in the teaching and learning of undergraduate mathematics is how to build on students' current ways of reasoning to develop more generalizable and abstract ways of reasoning. A promising aspect of linear algebra is... more
A prominent problem in the teaching and learning of undergraduate mathematics is how to build on students' current ways of reasoning to develop more generalizable and abstract ways of reasoning. A promising aspect of linear algebra is that it presents instructional designers with an array of applications from which to motivate the development of mathematical ideas. The purpose of this talk is to report on student reasoning as they reinvented the concepts of span and linear independence. The reinvention of these concepts was guided by an innovative instructional sequence known as the Magic Carpet Ride problem, whose creation was framed by the emergent models heuristic (Gravemeijer, 1999). During our talk we will: explain how this instructional sequence differs from a popular "systems of equations first" approach, present the instructional sequence via the framing of the emergent models heuristic; and provide samples of students' sophisticated thinking and reasoning.
Linear algebra poses a number of significant challenges for students that need to be better understood in order to improve instruction and student understanding. At the time the tenth conference on Research in Undergraduate Mathematics... more
Linear algebra poses a number of significant challenges for students that need to be better understood in order to improve instruction and student understanding. At the time the tenth conference on Research in Undergraduate Mathematics Education, we had just begun a study intended to explore these challenges. Our preliminary report was given in a "working group session" format, in which
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Linear algebra poses a number of significant challenges for students that need to be better understood in order to improve instruction and student understanding. At the time the tenth conference on Research in Undergraduate Mathematics... more
Linear algebra poses a number of significant challenges for students that need to be better understood in order to improve instruction and student understanding. At the time the tenth conference on Research in Undergraduate Mathematics Education, we had just begun a study intended to explore these challenges. Our preliminary report was given in a "working group session" format, in which
Research Interests:
Linear algebra poses a number of significant challenges for students that need to be better understood in order to improve instruction and student understanding. At the time the tenth conference on Research in Undergraduate Mathematics... more
Linear algebra poses a number of significant challenges for students that need to be better understood in order to improve instruction and student understanding. At the time the tenth conference on Research in Undergraduate Mathematics Education, we had just begun a study intended to explore these challenges. Our preliminary report was given in a "working group session" format, in which
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As part of a larger study of student understanding of concepts in linear algebra, we interviewed 10 university linear algebra students as to their conceptions of functions from high school algebra and linear transformation from their... more
As part of a larger study of student understanding of concepts in linear algebra, we interviewed 10 university linear algebra students as to their conceptions of functions from high school algebra and linear transformation from their study of linear algebra. An overarching goal of this study was to examine how linear algebra students see linear transformation as related to high school function. Analysis of these data led to a characterization of student responses into three categories of mathematical structures used to discuss function: properties, computations, and a series of five interrelated clusters of metaphorical expressions. In this paper, we use this analytic framing for exploring the question: to what extent does each of the students in this study have a unified concept image of function across two mathematical contexts, high school algebra and their study of linear algebra? We found that students who expressed a unified notion of function used metaphorical language to bridge any gaps between the notion of function from high school and the notion of linear transformation from linear algebra. We conjecture that the framing of computations, properties, and metaphorical clusters could be extended to discussions of functions in contexts with other mathematical domains. Future research that further examines the extent to which undergraduate students develop a unified concept image of function could then lead to various design research efforts aimed at explicitly fostering such a unified understanding.
Research and surveys continue to perpetuate deficit narratives about women of color, particularly regarding their participation in and contribution to mathematics. Following the broader call for more research concerning STEM learning... more
Research and surveys continue to perpetuate deficit narratives about women of color, particularly regarding their participation in and contribution to mathematics. Following the broader call for more research concerning STEM learning experiences of women of color, this study focuses on the sense making of eight women of color regarding their understanding of basis in linear algebra. We documented diverse ways that these women creatively explained the concept of basis using intuitive ideas from their everyday lives. These examples revealed important nuances and aspects of understanding of basis that are rarely discussed in instruction. These students’ ideas can also serve as potentially productive avenues to access the topic. Our results also challenge the existing broader narrative about academic underachievement of women of color in mathematics.