Matching Items (12)

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College Students’ and Inservice Teachers’ Evoked Concept Images and Ways of Understanding Congruence

Description

Eleven years after being put into practice, the Common Core State Standards for Mathematics still take a back seat as traditional approaches drive many secondary geometry classrooms, specifically in regard

Eleven years after being put into practice, the Common Core State Standards for Mathematics still take a back seat as traditional approaches drive many secondary geometry classrooms, specifically in regard to congruence. This thesis explores how university students reason about congruence based on their high school learning experience, as well as how in-service geometry teachers reason about and teach congruence. During the Summer of 2020, two distinct surveys were distributed to 33 undergraduate students at Arizona State University and two in-service geometry teachers in Arizona to characterize the ways they understand congruence and reflect on their experiences in secondary geometry classrooms. The results of the survey indicate that students who understood congruence either in terms of corresponding measurements or transformations were successful in identifying congruent shapes, while only students who understood congruence in terms of transformations were successful in constructing congruent shapes. Transformational reasoning was both the most productive and the least prominent way of understanding congruence among students. Their responses to activities and reflections on their experiences also suggested that deductive reasoning is not practiced or prioritized in many secondary geometry classrooms. Teacher understandings of congruence varied, and reflections suggested that development of materials and training that are aligned with the goals of CCSSM for both pre-service and in-service teachers would help teachers create an environment conducive to a transformational understanding of congruence and that promotes deductive reasoning.

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Created

Date Created
  • 2020-12

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Effects of a digital environment on students' construction of geometric shapes and their reasoning about them

Description

Construction is a defining characteristic of geometry classes. In a traditional classroom, teachers and students use physical tools (i.e. a compass and straight-edge) in their constructions. However, with

Construction is a defining characteristic of geometry classes. In a traditional classroom, teachers and students use physical tools (i.e. a compass and straight-edge) in their constructions. However, with modern technology, construction is possible through the use of digital applications such as GeoGebra and Geometer’s SketchPad.
Many other studies have researched the benefits of digital manipulatives and digital environments through student completion of tasks and testing. This study intends to research students’ use of the digital tools and manipulatives, along with the students’ interactions with the digital environment. To this end, I conducted exploratory teaching experiments with two calculus I students.
In the exploratory teaching experiments, students were introduced to a GeoGebra application developed by Fischer (2019), which includes instructional videos and corresponding quizzes, as well as exercises and interactive notepads, where students could use digital tools to construct line segments and circles (corresponding to the physical straight-edge and compass). The application built up the students’ foundational knowledge, culminating in the construction and verbal proof of Euclid’s Elements, Proposition 1 (Euclid, 1733).
The central findings of this thesis are the students’ interactions with the digital environment, with observed changes in their conceptions of radii and circles, and in their use of tools. The students were observed to have conceptions of radii as a process, a geometric shape, and a geometric object. I observed the students’ conceptions of a circle change from a geometric shape to a geometric object, and with that change, observed the students’ use of tools change from a measuring focus to a property focus.
I report a summary of the students’ work and classify their reasoning and actions into the above categories, and an analysis of how the digital environment impacts the students’ conceptions. I also briefly discuss the impact of the findings on pedagogy and future research.

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Created

Date Created
  • 2020-05

Interactive Learning of Euclid's Elements

Description

Today, there is a gap between the effectiveness of learning online and learning in person. Online educational videos such as ones found on Youtube mimic more of a lecture style

Today, there is a gap between the effectiveness of learning online and learning in person. Online educational videos such as ones found on Youtube mimic more of a lecture style of learning, which is easy ignore without a teacher nearby to engage the viewer. Furthermore, there is a lack of educational videos on the topic of Euclid’s Elements geometry proofs. This project remedies both accounts by offering a new approach on interactive online learning videos and exercises for the topic of Euclid’s Elements Book One, Propositions One and Two. This is accomplished by combining interactive videos, exercises, questions, and sketchpads into one online worksheet. The interactive videos are made using traditional methods of audio and visual elements, with an emphasis on having more dynamic visuals to engage with the viewer. The exercises are made using a program called Geogebra, and consist in having a question to solve, and diagram the use can manipulate to help solve the question. The questions consist in ensuring the viewer understands the material, as well as potential questions to gauge general understanding before and after using the worksheet. The sketchpads consist in stating the proposition being proved, and giving the user all the tools they need to construct or prove the Euclidean proposition in the online interactive environment offered by Geogebra. All of these components are then ordered into the worksheet to make an interactive online learning experience for the viewer. This way the viewer may both watch and actively use the material being presented to promote learning through engagement in a teacher-less environment.

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Agent

Created

Date Created
  • 2019-12

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Investigating Students' Thought Processes While Engaging in Proof-Related Activities in Mathematical Contexts

Description

This thesis is an extension of previous research done by Roh and Lee (2018). Their research involved the design and implementation of a survey to analyze students’ cognitive inconsistencies. This

This thesis is an extension of previous research done by Roh and Lee (2018). Their research involved the design and implementation of a survey to analyze students’ cognitive inconsistencies. This thesis expands upon this research to interview students who demonstrated logical inconsistencies and evaluates the kinds of struggles students faced while evaluating statements and validating arguments. Three students who demonstrated logical inconsistencies were interviewed and asked to answer questions originally pulled from Roh and Lee’s (2018) survey. This thesis found that there were many aspects of each section of the survey that students had struggled with, including use of intuition, analyzing a proof-by-contradiction that utilized a negated statement, and distrust of alternate proving methods. Overall, these techniques the students used while evaluating statements and validating arguments gives interesting insight into the pedagogy of teaching proofs.

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Agent

Created

Date Created
  • 2021-05

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Understanding Informal to Formal Comparisons and Proof Comprehension: a Replication Study

Description

This study sought to replicate previous work in student conceptions of formal proofs based on informal arguments, originally explored by Zazkis et al. (2016). Additional tasks were added to the

This study sought to replicate previous work in student conceptions of formal proofs based on informal arguments, originally explored by Zazkis et al. (2016). Additional tasks were added to the experiment to produce new data that could further verify the analysis of Zazkis et al. (2016) as well as provide more insight into how students comprehend proofs, what types of mistakes occur, and why. Results from one-on-one interviews confirmed that some students were not able to make accurate informal to formal comparisons because they were not considering multiple facets of the problem. Additionally, patterns in the students’ analysis introduced more questions concerning the motivations behind what students choose to think about when they read and dissect proofs.

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Created

Date Created
  • 2020-05

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Students’ Quantifications, Interpretations, and Negations of Complex Mathematical Statements from Calculus

Description

This study investigates several students’ interpretations and meanings for negations of various mathematical statements with quantifiers, and how their meanings for quantified variables impact their interpretations and denials of these

This study investigates several students’ interpretations and meanings for negations of various mathematical statements with quantifiers, and how their meanings for quantified variables impact their interpretations and denials of these quantified statements. Eight students participated in three separate exploratory teaching interviews and were selected from Transition-to-Proof and advanced mathematics courses beyond Transition-to-Proof. In the first interview, students were asked to interpret mathematical statements from Calculus contexts and provide justifications and refutations for why these statements are true or false in particular situations. In the second interview, students were asked to negate the same set of mathematical statements. Both sets of interviews were analyzed to determine students’ meanings for the quantified variables in the statements, and then these meanings were used to determine how students’ quantifications influenced their interpretations, denials, and evaluations for the quantified statements. In the final interview, students were also be asked to interpret and negation statements from different mathematical contexts. All three interviews were used to determine what meanings comprised students’ interpretations and denials for the given statements. Additionally, students’ interpretations and negations across different statements in the interviews were analyzed and then compared within students and across students to determine if there were differences in student denials across different moments.

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Agent

Created

Date Created
  • 2020

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Conceptualizing and Reasoning with Frames of Reference in Three Studies

Description

This dissertation reports three studies about what it means for teachers and students to reason with frames of reference: to conceptualize a reference frame, to coordinate multiple frames of reference,

This dissertation reports three studies about what it means for teachers and students to reason with frames of reference: to conceptualize a reference frame, to coordinate multiple frames of reference, and to combine multiple frames of reference. Each paper expands on the previous one to illustrate and utilize the construct of frame of reference. The first paper is a theory paper that introduces the mental actions involved in reasoning with frames of reference. The concept of frames of reference, though commonly used in mathematics and physics, is not described cognitively in any literature. The paper offers a theoretical model of mental actions involved in conceptualizing a frame of reference. Additionally, it posits mental actions that are necessary for a student to reason with multiple frames of reference. It also extends the theory of quantitative reasoning with the construct of a ‘framed quantity’. The second paper investigates how two introductory calculus students who participated in teaching experiments reasoned about changes (variations). The data was analyzed to see to what extent each student conceptualized the variations within a conceptualized frame of reference as described in the first paper. The study found that the extent to which each student conceptualized, coordinated, and combined reference frames significantly affected his ability to reason productively about variations and to make sense of his own answers. The paper ends by analyzing 123 calculus students’ written responses to one of the tasks to build hypotheses about how calculus students reason about variations within frames of reference. The third paper reports how U.S. and Korean secondary mathematics teachers reason with frame of reference on open-response items. An assessment with five frame of reference tasks was given to 539 teachers in the US and Korea, and the responses were coded with rubrics intended to categorize responses by the extent to which they demonstrated conceptualized and coordinated frames of reference. The results show that the theory in the first study is useful in analyzing teachers’ reasoning with frames of reference, and that the items and rubrics function as useful tools in investigating teachers’ meanings for quantities within a frame of reference.

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Created

Date Created
  • 2019

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Students’ Interpretations of Expressions in the Graphical Register and Its Relation to Their Interpretation of Points on Graphs when Evaluating Statements about Functions from Calculus

Description

Functions represented in the graphical register, as graphs in the Cartesian plane, are found throughout secondary and undergraduate mathematics courses. In the study of Calculus, specifically, graphs of functions are

Functions represented in the graphical register, as graphs in the Cartesian plane, are found throughout secondary and undergraduate mathematics courses. In the study of Calculus, specifically, graphs of functions are particularly prominent as a means of illustrating key concepts. Researchers have identified that some of the ways that students may interpret graphs are unconventional, which may impact their understanding of related mathematical content. While research has primarily focused on how students interpret points on graphs and students’ images related to graphs as a whole, details of how students interpret and reason with variables and expressions on graphs of functions have remained unclear.

This dissertation reports a study characterizing undergraduate students’ interpretations of expressions in the graphical register with statements from Calculus, its association with their evaluations of these statements, its relation to the mathematical content of these statements, and its relation to their interpretations of points on graphs. To investigate students’ interpretations of expressions on graphs, I conducted 150-minute task-based clinical interviews with 13 undergraduate students who had completed Calculus I with a range of mathematical backgrounds. In the interviews, students were asked to evaluate propositional statements about functions related to key definitions and theorems of Calculus and were provided various graphs of functions to make their evaluations. The central findings from this study include the characteristics of four distinct interpretations of expressions on graphs that students used in this study. These interpretations of expressions on graphs I refer to as (1) nominal, (2) ordinal, (3) cardinal, and (4) magnitude. The findings from this study suggest that different contexts may evoke different graphical interpretations of expressions from the same student. Further, some interpretations were shown to be associated with students correctly evaluating some statements while others were associated with students incorrectly evaluating some statements.

I report the characteristics of these interpretations of expressions in the graphical register and its relation to their evaluations of the statements, the mathematical content of the statements, and their interpretation of points. I also discuss the implications of these findings for teaching and directions for future research in this area.

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Created

Date Created
  • 2019

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Examining the development of students' covariational reasoning in the context of graphing

Description

Researchers have documented the importance of seeing a graph as an emergent trace of how two quantities’ values vary simultaneously in order to reason about the graph in terms of

Researchers have documented the importance of seeing a graph as an emergent trace of how two quantities’ values vary simultaneously in order to reason about the graph in terms of quantitative relationships. If a student does not see a graph as a representation of how quantities change together then the student is limited to reasoning about perceptual features of the shape of the graph.

This dissertation reports results of an investigation into the ways of thinking that support and inhibit students from constructing and reasoning about graphs in terms of covarying quantities. I collected data by engaging three university precalculus students in asynchronous teaching experiments. I designed the instructional sequence to support students in making three constructions: first imagine representing quantities’ magnitudes along the axes, then simultaneously represent these magnitudes with a correspondence point in the plane, and finally anticipate tracking the correspondence point to track how the two quantities’ attributes change simultaneously.

Findings from this investigation provide insights into how students come to engage in covariational reasoning and re-present their imagery in their graphing actions. The data presented here suggests that it is nontrivial for students to coordinate their images of two varying quantities. This is significant because without a way to coordinate two quantities’ variation the student is limited to engaging in static shape thinking.

I describe three types of imagery: a correspondence point, Tinker Bell and her pixie dust, and an actor taking baby steps, that supported students in developing ways to coordinate quantities’ variation. I discuss the figurative aspects of the students’ coordination in order to account for the difficulties students had (1) constructing a multiplicative object that persisted under variation, (2) reconstructing their acts of covariation in other graphing tasks, and (3) generalizing these acts of covariation to reason about formulas in terms of covarying quantities.

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Agent

Created

Date Created
  • 2017

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The Relationships Between Meanings Teachers Hold and Meanings Their Students Construct

Description

This dissertation reports three studies of the relationships between meanings teachers hold and meanings their students construct.

The first paper reports meanings held by U.S. and Korean secondary mathematics teachers

This dissertation reports three studies of the relationships between meanings teachers hold and meanings their students construct.

The first paper reports meanings held by U.S. and Korean secondary mathematics teachers for teaching function notation. This study focuses on what teachers in U.S. and Korean are revealing their thinking from their written responses to the MMTsm (Mathematical Meanings for Teaching secondary mathematics) items, with particular attention to how productive those meanings would be if conveyed to students in a classroom setting. This paper then discusses how the MMTsm serves as a diagnostic instrument by sharing a teacher’s story. The data indicates that many teachers name rules instead of constructing representations of functions through function notation.

The second paper reports the conveyance of meaning with eight Korean teachers who took the MMTsm. The data that I gathered was their responses to the MMTsm, what they said and did in the classroom lessons I observed, pre- and post-lesson interviews with them and their students. This paper focuses on the relationships between teachers’ mathematical meanings and their instructional actions as well as the relationships between teachers’ instructional actions and meanings that their students construct. The data suggests that holding productive meanings is a necessary condition to convey productive meanings to students, but not a sufficient condition.

The third paper investigates the conveyance of meaning with one U.S. teacher. This study explores how a teacher’s image of student thinking influenced her instructional decisions and meanings she conveyed to students. I observed 15 lessons taught by a calculus teacher and interviewed the teacher and her students at multiple points. The results suggest that teachers must think about how students might understand their instructional actions in order to better convey what they intend to their students.

The studies show a breakdown in the conveyance of meaning from teacher to student when the teacher has no image of how students might understand his or her statements and actions. This suggests that it is crucial to help teachers improve what they are capable of conveying to students and their images of what they hope to convey to future students.

Contributors

Agent

Created

Date Created
  • 2019