VTechWorks Repository :: Browsing by Author "Wawro, Megan"

Browsing by Author "Wawro, Megan"

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Characterizations of student, instructor, and textbook discourse related to basis and change of basis in quantum mechanics
Serbin, Kaitlyn Stephens; Wawro, Megan; Storms, Rebecah (2021-06-02)
Communities develop social languages in which utterances take on culturally specific situated meanings. As physics students interact in their classroom, they can learn the broader physics community's social language by co-constructing meanings with their instructors. We provide an exposition of a systematic and productive use of idiosyncratic, socially acquired language in two classroom communities that we consider to be subcultures of the broader community of physicists. We perform a discourse analysis on twelve quantum mechanics students, two instructors, and the course text related to statements about basis and change of basis within a spin-1/2 probability problem. We classify the utterances' grammatical constructions and situated meanings. Results show that students and instructors' utterances referred to a person, calculation, vector being in, or vector written in a basis. Utterances in these categories had similar situated meanings and were used similarly by the students and instructors. Utterances referred to change of basis as changing the form of a vector, writing the vector in another way, changing the vector into another vector, or switching bases. Utterances in these categories had varying situated meanings and were used similarly by the students and instructors. The students and instructors often switched between different discourse types in quick succession. We found similar utterance types, situated meanings, and grammatical constructions across students and instructors. The textbook's discourse sometimes differed from the discourse of the students and instructors. Within this study, the students and instructors were from two universities, yet they spoke similar utterances when referring to basis and change of basis. This gives evidence to their shared social language with a broader community of physicists. Integrating and leveraging social languages in the classroom could facilitate students' enculturation into the classroom and broader professional community.
Examining Connections among Instruction, Conceptual Metaphors, and Beliefs of Instructors and Students
Rupnow, Rachel Lynn (Virginia Tech, 2019-07-29)
In this study, I will examine the beliefs and conceptual understanding of instructors and students from two abstract algebra classes. This research takes the form of a case study in which I answer four research questions, each addressing a relationship between instruction and beliefs or conceptual understanding. Specifically, these research questions are: 1. What beliefs do the instructors have about math, teaching, and learning and what relationship exists between these beliefs and instructional practice? 2. What is the relationship between instructional practice and students' beliefs about math, teaching, and learning? 3. What conceptual metaphors do the professors use to describe isomorphisms and homomorphisms and what relationship exists between these metaphors and the mathematical content in instruction? 4. What is the relationship between the mathematical content in instruction and conceptual metaphors the students use to describe isomorphisms and homomorphisms? In terms of beliefs, the instructors articulated considered positions on the nature of math, math learning, and math teaching. These beliefs were clearly reflected in their overall approaches to teaching. However, their instruction shifted in practice over the course of the semester. Students' beliefs seemed to shift slightly as a result of the ways their instructors taught. However, their core beliefs about math seemed unchanged and some lessons students took away were similar in the two classes. In terms of conceptual understanding, the instructors provided many conceptual metaphors that related to how they understood isomorphism. They struggled more to provide an image for homomorphism, which requires thinking about a more complicated mathematical object. Their understandings of isomorphism and homomorphism were largely reflected in their instruction with some notable differences. Students took away similar understandings of isomorphism to the instructors, but did not all take away the same level of structural understanding of homomorphism. In short, relationships between instructors' beliefs and instruction and between instructors' conceptual understanding and instruction were evident. However, certain elements were not made as clear as they perhaps intended. Relationships between instruction and students' beliefs and between instruction and students' conceptual understanding were also evident. However, relationships between instruction and beliefs were subtler than between instruction and conceptual understanding.
Facilitating Instructional Change: A Case Study on Diffusion of Curriculum Innovation
Mitchell, Corinne Beloved (Virginia Tech, 2023-08-15)
While much research has been conducted on train-the-trainer models for diffusing curriculum innovations at the K-12 level, not much is known about how such models play out at the undergraduate level, especially with newer curriculum innovations using student-centered instruction. I present findings from one such project: a case study on the second-generation facilitation of a professional development group focused on supporting instructors teaching with the Inquiry-Oriented Abstract Algebra (Larsen et al., 2013) curriculum materials. I investigate the relationship between the intent of the instructional support model and the facilitator's beliefs and goals for the professional development, using video data collected from a series of online meetings and from the facilitator's classroom in the year prior to his facilitation. Results indicate that the facilitator's orientations and goals around sharing authority and creating supportive learning environments, especially for women participants, both modify and stabilize the intentions of the TIMES project (NSF Awards: #1431595, #1431641, #1431393) as a whole, and the train-the-trainer model as a subsidiary.
H-Infinity Norm Calculation via a State Space Formulation
KusterJr, George Emil (Virginia Tech, 2013-01-21)
There is much interest in the design of feedback controllers for linear systems that minimize the H-infty norm of a specific closed-loop transfer function. The H-infty optimization problem initiated by Zames (1981), \\cite{zames1981feedback}, has received a lot of interest since its formulation. In H-infty control theory one uses the H-infty norm of a stable transfer function as a performance measure. One typically uses approaches in either the frequency domain or a state space formulation to tackle this problem. Frequency domain approaches use operator theory, J-spectral factorization or polynomial methods while in the state space approach one uses ideas similar to LQ theory and differential games. One of the key computational issues in the design of H-infty optimal controllers is the determination of the optimal H-infty norm. That is, determining the infimum of r for which the H-infty norm of the associated transfer function matrix is less than r. Doyle et al (1989), presented a state space characterization for the sub-optimal H-infty control problem. This characterization requires that the unique stabilizing solutions to two Algebraic Riccati Equations are positive semi definite as well as satisfying a spectral radius coupling condition. In this work, we describe an algorithm by Lin et al(1999), used to calculate the H-infty norm for the state feedback and output feedback control problems. This algorithm only relies on standard assumptions and divides the problem into three sub-problems. The first two sub-problems rely on algorithms for the state feedback problem formulated in the frequency domain as well as a characterization of the optimal value in terms of the singularity of the upper-half of a matrix created by the stacked basis vectors of the invariant sub-space of the associated Hamiltonian matrix. This characterization is verified through a bisection or secant method. The third sub-problem relies on the geometric nature of the spectral radius of the product of the two solutions to the Algebraic Riccati Equations associated with the first two sub-problems. Doyle makes an intuitive argument that the spectral radius condition will fail before the conditions involving the Algebraic Riccati Equations fail. We present numerical results where we demonstrate that the Algebraic Riccati Equation conditions fail before the spectral radius condition fails.
Instructors' Orientation on Mathematical Meaning
Chowdhury, Ahsan Habib (Virginia Tech, 2021-06-11)
Students often ask "when is this ever going to be useful?", "why are we doing this?", etc. when speaking about mathematics. If we take this as a question about 'meaningfulness', how can instructors respond and how do they even understand the terms 'meaningful' and 'meaning'? My dissertation looked at how college instructors see their instruction as meaningful or not. Drawing on social and cognitive perspectives of learning, I define four ways to think of what's 'meaningful' about mathematics. From a cognitive perspective, instructors can understand 'meaningful' as mathematical understanding versus understanding the significance of mathematics. From a social perspective where meaning is taken as the experiences of everyday life within communities, teachers can understand 'meaningful' as anything that engages students in practices the mathematics community engage in versus practices non-mathematics communities engage in (e.g. pushing computation or critical thinking as a means for maintaining social hierarchies). Using these four conceptions to categorize instructors' goals, this work focuses on how four undergraduate mathematics instructors thought of their instruction as meaningful and contextual and background factors that influenced those views.
An Integrated Methodological Approach for Documenting Individual and Collective Mathematical Progress: Reinventing the Euler Method Algorithmic Tool
Rasmussen, Chris; Wawro, Megan; Zandieh, Michelle (MDPI, 2024-03-21)
In this paper we advance a methodological approach for documenting the mathematical progress of learners as an integrated analysis of individual and collective activity. Our approach is grounded in and expands the emergent perspective by integrating four analytic constructs: individual meanings, individual participation, collective mathematical practices, and collective disciplinary practices. Using video data of one small group of four students in an inquiry-oriented differential equations classroom, we analyze a 10 min segment in which one small group reinvent Euler’s method, an algorithmic tool for approximating solutions to differential equations. A central intellectual contribution of this work is elaborating and coordinating the four methodological constructs with greater integration, cohesiveness, and coherence.
A Mixed Methods Study of Chinese Students' Construction of Fraction Schemes: Extending the Written Test with Follow-Up Clinical Interviews
Xu, Cong Ze (Virginia Tech, 2023-01-12)
Understanding fractions is fundamental for expanding number knowledge from the whole number system to the rational number system. According to the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics (2000), learning fractions is an important mathematical goal for students in grades three through five in the U.S. Moreover, the NCTM suggests that fraction instruction start in Pre-K and continue through 8th grade. At the same time, the Common Core State Standards for Mathematics (CCSSM) suggests that fraction instruction should occur from Grade 3 to 7. In contrast to the time spent on learning fractions in the U.S., students in China spend a relatively short time learning fractions (Zhang and Siegler, 2022). According to the Chinese national curriculum standards, the Chinese National Mathematics Curriculum Standards (CNMCS) for five-four system, the fundamental fraction concepts are taught in grades 3 and 5 only. However, Chinese students continue to have higher performance on fraction items in international assessments when compared with American students (Fan and Zhu, 2004). Consequently, over the last several years, researchers have investigated subject content knowledge and pedagogical content knowledge of Chinese in-service teachers and pre-service teachers via fraction division (e.g., Li and Huang, 2008; Ma, 1999). There are also studies exploring Chinese written curricula of fraction division (e. g., Li, Zhang, and Ma, 2009). Recently, a quantitative study from Norton, Wilkins, and Xu (2018) investigated the process of Chinese students' construction of fraction knowledge through the lens of fraction schemes, a model established by western scholars Steffe (2002) and his colleague Olive (Steffe and Olive, 2010). However, there is a lack of qualitative research that attempts to use fraction schemes as an explanatory framework to interpret the process of Chinese students' construction of fraction knowledge. The main purpose of this study was to investigate Chinese students' understanding of the fundamental fraction knowledge in terms of their understanding of the "fraction unit," referred to as a "unit fraction" in the U.S., using Steffe and Olive's (2010) fraction schemes as the conceptual framework. A sequential mixed methods design was used in this study. The design included two consecutive phases, namely a quantitative phase followed by a qualitative phase (Creswell and Plano Clark, 2011). During the quantitative phase, five hundred and thirty-four Chinese fourth and fifth grade students were administered an assessment. The quantitative data was first analyzed using a Cochran's Q test to determine if the Chinese participants in this study follow the same progression of fraction schemes as their American peers. Results indicate that the development of fractional schemes among Chinese 4th and 5th grade participants in this study is similar to their U.S. counterparts and the Chinese participants in Norton et al.'s (2018) study regardless of the curricula differences across countries or areas in the same country, the textbook differences, and the language differences. Next, two different analysis of variances (ANOVA), a three-way mixed ANOVA and a two-way repeated measures ANOVA were conducted. The three-way mixed ANOVA was used to inform the researcher as to the fraction schemes these students had constructed before the concept of fraction unit is formally introduced and after the concept of fraction unit is formally introduced. The results showed that the fraction knowledge of the students in this study developed from 4th grade to 5th grade. The analysis of clinical interview data confirmed this conclusion. The two-way repeated measures ANOVA was used to determine which model (i.e., linear, circular, or rectangular) is more or less problematic for Chinese students when solving fraction tasks. The results suggest that generally students' performance on linear model tasks was better than their performance on circular model tasks, but there was no statistically significant difference between performance on circular model and its corresponding rectangular model tasks. The results from the quantitative analyses were also used to screen students to form groups based on their highest available fraction scheme for a clinical interview in the second phase, the qualitative phase. In the qualitative phase, a clinical interview using a think-aloud method was used to gain insight into the role of students' conceptual understanding of the fraction unit in their construction of fraction knowledge. In this phase, students were asked to solve the tasks in the clinical interview protocol using the think aloud method. Two main findings were revealed analyzing the clinical interview data. First, a conceptual understanding of fraction units as well as a conceptual understanding of a unit whole play a critical role in the construction of Chinese students' fraction knowledge. Second, the lack of the understanding of a fraction unit as an iterable unit may be one of the reasons that obstructs students move from part-whole concept of fractions to the measurement concept of fractions. This study also demonstrates that a conceptual understanding of fraction units and the unit whole are a necessary condition for constructing of a conceptual understanding of fraction knowledge. Thus, implications of this study suggest that teachers not only should help students build a conceptual understanding of fraction units, but also need to confirm that students have constructed the concept of what the unit whole is before asking students to identify the fraction units for the referent whole. On the other hand, the tasks used in the present study only include continuous but not discrete wholes. Therefore, future research may focus on investigating how students identify fraction units and in what way the iterating operation could be used when students encounter a discrete whole.
Modeling Middle Grade Students' Algebraic and Covariational Reasoning using Unit Transformations and Working Memory
Kerrigan, Sarah Therese (Virginia Tech, 2023-02-07)
Quantitative reasoning permeates mathematical thinking, and mathematics education researchers have taken a quantitative reasoning approach to examining and modeling students' mathematical thinking and development in various domains. From this approach, secondary and post-secondary researchers have focused on students' ability to reason about how two quantities vary together (covariational reasoning). However, little is known about how covariational reasoning develops from, or connects with, arithmetic and algebraic reasoning. This study begins to bridge the gap in this knowledge. Originally this study was designed to examine middle grade students' units coordination in covariational reasoning across stages and consider the cognitive limiting factor of working memory. In this case study of Daniel, an advanced Stage 2 middle-grade algebra student, I examined the role his units coordinating structures played in his covariational reasoning in non-graphing and algebra tasks. I considered three main components in covariational reasoning (type of quantity, modality of change, and role of time) when analyzing covariational reasoning and capturing the underlying mental units and actions. I found type of quantity and time were the two biggest factors when determining Daniel's covariational reasoning. Daniel also used his units coordinating structures in various ways in the different covariation tasks, generating three different types of change units that were cognitively structurally different. These findings suggest cognitive connections between the types of units a student assimilates with, and the types of covariational reasoning they engage in, are interconnected and warrant future study.
Modeling Students' Units Coordinating Activity
Boyce, Steven James (Virginia Tech, 2014-08-29)
Primarily via constructivist teaching experiment methodology, units coordination (Steffe, 1992) has emerged as a useful construct for modeling students' psychological constructions pertaining to several mathematical domains, including counting sequences, whole number multiplicative conceptions, and fractions schemes. I describe how consideration of units coordination as a Piagetian (1970b) structure is useful for modeling units coordination across contexts. In this study, I extend teaching experiment methodology (Steffe and Thompson, 2000) to model the dynamics of students' units coordinating activity across contexts within a teaching experiment, using the construct of propensity to coordinate units. Two video-recorded teaching experiments involving pairs of sixth-grade students were analyzed to form a model of the dynamics of students' units coordinating activity. The modeling involved separation of transcriptions into chunks that were coded dichotomously for the units coordinating activity of a single student in each dyad. The two teaching experiments were used to form 5 conjectures about the output of the model that were then tested with a third teaching experiment. The results suggest that modeling units coordination activity via the construct of propensity to coordinate units was useful for describing patterns in the students' perturbations during the teaching sessions. The model was moderately useful for identifying sequences of interactions that support growth in units coordination. Extensions, modifications, and implications of the modeling approach are discussed.
On the role of student understanding of function and rate of change in learning differential equations
Kuster Jr, George Emil (Virginia Tech, 2016-07-22)
In this research, I utilize the theoretical perspective Knowledge In Pieces to identify the knowledge resources students utilize while in the process of completing various differential equations tasks. In addition I explore how this utilization changes over the course of a semester, and how resources related to the concepts of function and rate of change supported the students in completing the tasks. I do so using data collected from a series of task-based individual interviews with two students enrolled in separate differential equations courses. The results provide a fine-grained description of the knowledge students consider to be productive with regard to completing various differential equations tasks. Further the analysis resulted in the identification of five ways students interpret differential equations tasks and how these interpretations are related to the knowledge resources students utilize while completing the various tasks. Lastly, this research makes a contribution to mathematics education by illuminating the knowledge concerning function and rate of change students utilize and how this knowledge comes together to support students in drawing connections between differential equations and their solutions, structuring those solutions, and reasoning with relationships present in the differential equations.
Positive attitudinal shifts and a narrowing gender gap: Do expertlike attitudes correlate to higher learning gains for women in the physics classroom?
Robinson, Alma; Simonetti, John H.; Richardson, Kasey; Wawro, Megan (2021-01-13)
A large body of research shows that using interactive engagement pedagogy in the introductory physics classroom consistently results in significant student learning gains; however, with a few exceptions, those learning gains tend not to be accompanied by more expertlike attitudes and beliefs about physics and learning physics. In fact, in both traditionally taught and active learning classroom environments, students often become more novicelike in their attitudes and beliefs following a semester of instruction. Furthermore, prior to instruction, men typically score higher than women on conceptual inventories, such as the Force Concept Inventory (FCI), and more expertlike on attitudinal surveys, such as the Colorado Learning Attitudes about Science Survey (CLASS), and those gender gaps generally persist following instruction. In this paper, we analyze three years of pre-post matched data for physics majors at Virginia Tech on the FCI and the CLASS. The courses were taught using a blended pedagogical model of peer instruction, group problem solving, and direct instruction, along with an explicit focus on the importance of conceptual understanding and a growth mindset. We found that the FCI gender gap decreased, and both men and women showed positive, expertlike shifts on the CLASS. Perhaps most surprisingly, we found a meaningful correlation between a student's post-CLASS score and normalized FCI gain for women, but not for men.
Professional Mathematicians' Level of Understanding: An Investigation of Pseudo-Objectification
Flanagan, Kyle Joseph (Virginia Tech, 2023-12-20)
This research study investigated how professional mathematicians understand and operate with highly-abstract, advanced mathematical concepts in their own work. In particular, this study examined how professional mathematicians operated with mathematical concepts at different levels of understanding. Moreover, this study aimed to capture what factors influence professional mathematicians' level of understanding for particular mathematical concepts. To frame these research goals, three theoretical levels of understanding were proposed, process-level, pseudo-object-level, object-level, leveraging two ways that Piaget (1964) described what it meant to know or understand a mathematical concept. Specifically, he described understanding an object as being able to "act on it," and also as being able to "understand the process of this transformation" (p. 176). Process-level understanding corresponds to only understanding the underlying processes of the concept. Pseudo-object-level understanding corresponds to only being able to act on the concept as a form of object. Object-level understanding corresponds to when an individual has both of these types of understanding. This study is most especially concerned with how professional mathematicians operate with a pseudo-object-level understanding, which is called pseudo-objectification. For this study, six professional mathematicians with research specializing in a subfield of algebra were each interviewed three times. During the first interview, the participants were given two mathematical tasks, utilizing concepts in category theory which were unfamiliar to the participants, to investigate how they operate with mathematical concepts. The second interview utilized specific journal publications from each participant to generate discussion about influences on their level of understanding for the concepts in that journal article. The third interview utilized stimulated recall to triangulate and support the findings from the first two interviews. The findings and analysis revealed that professional mathematicians do engage in pseudo-objectification with mathematical concepts. This demonstrates that pseudo-objectification can be productively leveraged by professional mathematicians. Moreover, depending on their level of understanding for a given concept, they may operate differently with the concept. For example, when participants utilized pseudo-objects, they tended to rely on figurative material, such as commutative diagrams, to operate on the concepts. Regarding influences on understanding, various factors were shown to influence professional mathematicians' level of understanding for the concepts they use in their own work. These included factors pertaining to the mathematical concept itself, as well as other sociocultural or personal factors.
Prospective Teachers' Knowledge of Secondary and Abstract Algebra and their Use of this Knowledge while Noticing Students' Mathematical Thinking
Serbin, Kaitlyn Stephens (Virginia Tech, 2021-08-03)
I examined the development of three Prospective Secondary Mathematics Teachers' (PSMTs) understandings of connections between concepts in Abstract Algebra and high school Algebra, as well as their use of this understanding while engaging in the teaching practice of noticing students' mathematical thinking. I drew on the theory, Knowledge of Nonlocal Mathematics for Teaching, which suggests that teachers' knowledge of advanced mathematics can become useful for teaching when it first helps reshape their understanding of the content they teach. I examined this reshaping process by investigating how PSMTs extended, deepened, unified, and strengthened their understanding of inverses, identities, and binary operations over time. I investigated how the PSMTs' engagement in a Mathematics for Secondary Teachers course, which covered connections between inverse functions and equation solving and the abstract algebraic structures of groups and rings, supported the reshaping of their understandings. I then explored how the PSMTs used their mathematical knowledge as they engaged in the teaching practice of noticing hypothetical students' mathematical thinking. I investigated the extent to which the PSMTs' noticing skills of attending, interpreting, and deciding how to respond to student thinking developed as their mathematical understandings were reshaped. There were key similarities in how the PSMTs reshaped their knowledge of inverse, identity, and binary operation. The PSMTs all unified the additive identity, multiplicative identity, and identity function as instantiations of the same overarching identity concept. They each deepened their understanding of inverse functions. They all unified additive, multiplicative, and function inverses under the overarching inverse concept. They also strengthened connections between inverse functions, the identity function, and function composition. They all extended the contexts in which their understandings of inverses were situated to include trigonometric functions. These changes were observed across all the cases, but one change in understanding was not observed in each case: one PSMT deepened his understanding of the identity function, whereas the other two had not yet conceptualized the identity function as a function in its own right; rather, they perceived it as x, the output of the composition of inverse functions. The PSMTs had opportunities to develop these understandings in their Mathematics for Secondary Teachers course, in which the instructor led the students to reason about the inverse and identity group axioms and reflect on the structure of additive, multiplicative, and compositional inverses and identities. The course also covered the use of inverses, identities, and binary operations used while performing cancellation in the context of equation solving. The PSMTs' noticing skills improved as their mathematical knowledge was reshaped. The PSMTs' reshaped understandings supported them paying more attention to the properties and strategies evident in a hypothetical student's work and know which details were relevant to attend to. The PSMTs' reshaped understandings helped them more accurately interpret a hypothetical student's understanding of the properties, structures, and operations used in equation solving and problems about inverse functions. Their reshaped understandings also helped them give more accurate and appropriate suggestions for responding to a hypothetical student in ways that would build on and improve the student's understanding.
Relating Understanding of Inverse and Identity to Engagement in Proof in Abstract Algebra
Plaxco, David Bryant (Virginia Tech, 2015-09-05)
In this research, I set out to elucidate the relationships that might exist between students' conceptual understanding upon which they draw in their proof activity. I explore these relationships using data from individual interviews with three students from a junior-level Modern Algebra course. Each phase of analysis was iterative, consisting of iterative coding drawing on grounded theory methodology (Charmaz, 2000, 2006; Glaser and Strauss, 1967). In the first phase, I analyzed the participants' interview responses to model their conceptual understanding by drawing on the form/function framework (Saxe, et al., 1998). I then analyzed the participants proof activity using Aberdein's (2006a, 2006b) extension of Toulmin's (1969) model of argumentation. Finally, I analyzed across participants' proofs to analyze emerging patterns of relationships between the models of participants' understanding of identity and inverse and the participants' proof activity. These analyses contributed to the development of three emerging constructs: form shifts in service of sense-making, re-claiming, and lemma generation. These three constructs provide insight into how conceptual understanding relates to proof activity.
Students' Conceptions of Normalization
Watson, Kevin L. (Virginia Tech, 2020-10-13)
Improving the learning and success of students in undergraduate science, technology, engineering, and mathematics (STEM) courses has become an increased focus of education researchers within the past decade. As part of these efforts, discipline-based education research (DBER) has emerged within STEM education as a way to address discipline-specific challenges for teaching and learning, by combining expert knowledge of the various STEM disciplines with knowledge about teaching and learning (Dolan et al., 2018; National Research Council, 2012). Particularly important to furthering DBER and improving STEM education are interdisciplinary studies that examine how the teaching and learning of specific concepts develop among and across various STEM disciplines...
Students' metarepresentational competence with matrix notation and Dirac notation in quantum mechanics
Wawro, Megan; Watson, Kevin; Christensen, Warren (2020-08-25)
This article shares analysis regarding quantum mechanics students' metarepresentational competence (MRC) that is expressed as they engaged in solving an expectation value problem, which involves linear algebra concepts. The particular characteristic of MRC that is the focus of this analysis is students' critiquing and comparing the adequacy of representations, specifically matrix notation and Dirac notation, and judging their suitability for various tasks. With data of students' work during semistructured individual interviews, components of students' MRC were analyzed and categorized according to value-based preferences, problem-based preferences, and purpose and utility awareness. Detail is provided on two students who serve as paradigmatic examples of students' power and flexibility within different notation systems, and detail of a third student is given as a point of contrast. In addition to adapting MRC as a helpful construct for characterizing student understanding at the intersection of undergraduate mathematics and physics, we aim to demonstrate how students' rich understanding of linear algebra and quantum mechanics includes and is aided by their understanding and flexible use of different notational systems. For example, the problem-based preference aspects of MRC highlight that any particular problem-solving approach is itself intrinsically tied to a notational system. We suggest that any instruction with the goal of helping students develop a deep understanding of quantum mechanics and linear algebra should provide opportunities for students to use and improve their MRC.
Teachers' Reflection on Inquiry-Oriented Instruction in Online Professional Development
Kelley, Marilin Annie (Virginia Tech, 2021-01-11)
In light of the expansion of student-centered instructional approaches in mathematics education and a brightening spotlight on virtual teacher supports, I look to Inquiry-Oriented Instruction (IOI) and explore how instructors reflect on and plan for their implementation of IOI in online professional development. I focus specifically on two teachers' comments on their implementation of IOI materials covering Abstract Algebra topics in online work groups developed to support teachers in implementing IOI. I analyze both reflection and enactment through the components of IOI characterized through the Instructional Triangle. Analysis of the teachers' reflections, viewed through their participation in the roles of sense maker, inquirer, and builder, revealed interesting differences in the teachers' approaches to IOI. I detail these two teachers' approaches to IOI and ultimately shed light on the intricacies of IOI and online professional development. Such findings support the growing bodies of research centered around IOI and corresponding professional development.

Browsing by Author "Wawro, Megan"

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