In the Paradigms in Physics curriculum at Oregon State University, we teach a series of spins-first quantum mechanics courses that incorporate various types of active-engagement pedagogies. In this talk, I will highlight the kinesthetic activities used in our junior-level Quantum Fundamentals course and their role in the broader context of our quantum mechanics courses. These kinesthetic activities include the Arms Activities where pairs of students use their left arms to embody quantum mechanical spin states and the Quantum Measurement Skit where the class acts out the measurement of quantum states. In addition to presenting the activities, I will discuss results from interviews with students about their understanding and experience with Arms.

Waving Arms and Skits in Quantum Mechanics

The Paradigms in Physics website provides a wide range of tested active engagement activities for upper division courses. Our team has been redesigning this site to add considerable "mortar" content between activities, to make the content more readily searchable, and to make more clear the connections between content. We have also been working to make it easier to keep the content fresh, and to give users a clearer perspective of how we use our own materials. This poster will introduce the features of the new site, and we will solicit audience feedback on how the site could better serve faculty seeking research-based content.

The new Paradigms in Physics Curriculum Website

Students in a spins-first course often have difficulties transitioning from discrete to continuous quantum systems. We believe that computation may naturally support the transition because continuous structures like wavefunctions are necessarily discretized in order to be used in operations like numerical integration. We remotely interviewed six students who were concurrently enrolled in a junior-level introductory spins-first quantum mechanics course and a computational lab course, which were designed to synergize with each other. During the interview, participants organized a set of 20 cards that contained a variety of quantum mechanics concepts and representations, including snippets of Python code. Using a phenomenographic approach, this poster will discuss the various ways that participants made sense of the cards and how they organized the cards into groups such as “discrete” and “continuous”.

How Students Organize Quantum Concepts and Representations: A Card Sorting Task

The Paradigms @ OSU team developed a kinesthetic representation for complex numbers where individual students use their arms to embody an Argand diagram. This representation can be expanded to have groups of students working together to represent complex-valued state vectors in quantum mechanics. We conducted group teaching interviews about the relative and overall phase of 2- and 3-element state vectors to uncover how students understand geometric relationships using the Arms Representation. We interviewed nine students in the second week of a junior-level, spins-first quantum mechanics course after they had been introduced to complex numbers via the Arms Representation, but before quantum mechanical state vectors had been addressed in class. This poster will discuss how the Arms Representation supported students in understanding that an overall phase does not change the relative phase. In particular, we will attend to the conceptual support offered by the Arms Representation, the geometric interpretations of phase, and the ways students translate between the Arms Representation, Argand diagrams, and algebraic notations.

Teaching Relative & Overall Phase with the Arms Representation

Using a completeness (or closure) relation is a neat math trick in quantum theory. Completeness relations capture the fundamental property that a basis set spans a vector space, and conceptually can be thought of as a special form of the identity operator. In quantum mechanics, completeness relations can be used to express a quantum state in a particular basis, change which basis a state is expressed in, as well as mathematically model the effect of a quantum interferometer. We have analyzed students' written responses to homework and final exam questions related to completeness relations from a junior-level, introduction to quantum mechanics course. In this poster, we will present student reasoning and difficulties using completeness relations.

Student Reasoning with Completeness Relations in Quantum Mechanics

Our open-source textbook, Exploring Physical Phenomena: What Happens When Light from the Sun Shines on the Earth? is now available at https://open.oregonstate.education/physicsforteachers. Each unit begins by identifying student resources for learning, developing central ideas based on evidence from exploring phenomena with everyday equipment and materials, using central ideas to explain intriguing phenomena, developing mathematical representations of the phenomena, using those representations to predict or estimate a quantity of interest, and making connections to educational policies such as the Next Generation Science Standards. All units include examples of student work by prospective elementary and middle school teachers while exploring light and thermal phenomena, considering the influence of light and thermal phenomena on local weather and global climate change, and exploring astronomical phenomena in the context of the Sun/Earth/Moon system. Supplemental materials include instructor guide for each unit, equipment lists for both on-campus and remote instruction, student handouts, and example homework assignments.

Exploring Physical Phenomena: A Physics Course for Prospective Teachers

From our experience teaching the spins-first Paradigms quantum mechanics course, students run into difficulties transitioning from discrete to continuous quantum systems. In computation, continuous structures like wavefunctions are necessarily discretized in order to be used in operations like numerical integration. Because of this feature, we believe that computation may naturally support the transition. Junior-level students at Oregon State University take a computational lab course where they numerically solve quantum mechanics problems by pair-programming in Python. This computational lab course is coordinated with the spins-first Paradigms course. We remotely interviewed six participants at the end of the courses. Participants completed a card-sorting task where they organized twenty cards with a variety of quantum mechanics content and representations. In this talk, we will discuss the ways participants organized this information, the overarching patterns, and the ways that students understand and coordinate discrete and continuous quantum systems.


Computation to Support Understanding of Discrete and Continuous Quantum Systems

In order to help our students better understand the conceptual connections between spin states and wavefunctions in our junior-level, spins-first quantum course, we have recently begun to emphasize the use of completeness relations to write a state in a particular basis. We describe completeness relations as “special forms of 1” that can be used to project a quantum state vector onto a basis. Using a discrete spin bases, the expansion coefficients are interpreted as probabilities amplitudes. Using a continuous position basis, these probability amplitudes get “smeared out” to become a “probability amplitude density” (a.k.a., wavefunction), similar to how discrete point masses go to mass densities. In this talk, we will share the in-class activities and homework tasks we’ve developed to help students understand make these connections.


Using Completeness Relations to Help Students Understand Wavefunctions

The Paradigms in Physics website provides a wide range of tested active engagement activities for upper division courses. Our team has been redesigning this site to add considerable "mortar" content between activities and to make the content more readily searchable. We have also been working to make it easier to keep the content fresh, and to give users a clearer perspective of how we use our own materials. The features of the new site will be introduced, and we will solicit audience feedback on how the site could better serve you.

Introducing the new Paradigms in Physics Curriculum Website

External representations play a vital role in doing and learning physics. At this poster, we will discuss a new framework for describing external representations and implications for teaching. First, we identify three aspects that distinguish one representation from another: the conceptual referent, the medium of the representation, and how information is organized in the representation. We then identify structural features of representations that take into account both the form and the function of the representation. We will illustrate this framework of structural features with some examples, including plastic “surface" models of 2D functions and an “arms” representation of quantum states. Based on this framework, we suggest strategies for helping students learn to interpret and use external representations. &nbsp;

Structural Features of External Representation in Physics

The Paradigms @ OSU team has developed a series of activities that supports students in reasoning about complex numbers by having students act out those numbers with their arms. The Arms representation is versatile and can be expanded, by grouping students, to include complex-valued vectors. This expansion enables groups of students to represent quantum mechanical state vectors with their arms. The Arms Sequence of activities parallels the progression of OSU’s spins-first approach by starting with complex numbers, advancing through two- and three-state systems, considering time-dependence, and eventually extending to approximate wavefunctions. In this poster we will present Arms as a representation and discuss some of the advantages and disadvantages of using Arms to teach quantum mechanics. We will discuss the development of mathematical and physics ideas, cognitive and pedagogical affordances, movement logistics, and the implicit elements of the representation.

Embodying Complex Numbers and Quantum States

The Paradigms in Physics program often incorporates activities that encourage&nbsp;exploration of multiple representations simultaneously. In this poster, we discuss an electrostatics activity that occurs near the beginning of the&nbsp;Static Fields Paradigm. We detail one group's use of multiple representations, including a plastic surface model of the potential due to an electric quadrupole. Our analysis examines the features and functions of the external&nbsp;representations, as well as the order in which the representations were provided to the students. We specifically answer the question of how the representations supported inquiry during this activity.

Surface Models and Multiple External Representations in Paradigms

Sensemaking is a skill valued in the physics classroom as it aids students and expert physicists in drawing connections between varying representations and concepts in order to make sense of what they are learning or teaching. To promote students’ use of sensemaking practices throughout The Paradigms in Physics program we designed a course to explicitly teach students sensemaking skills early in their physics curriculum, Techniques of Theoretical Mechanics. This course covers both classical mechanics and special relativity, giving students experience with sensemaking in two different contexts. By teaching sensemaking explicitly and early, students have a tool box of sensemaking skills to draw on when solving novel physics problems. In this poster we explain the different pedagogical strategies used in Theoretical Mechanics to expand students’ views about sensemaking and help them develop habits that persist throughout The Paradigms in Physics program.

A Sensemaking Foundation for Paradigms in Physics at OSU

Exploring Physical Phenomena: What happens when light from the Sun shines on the Earth?&nbsp; &nbsp;is now available at&nbsp;https://open.oregonstate.education/physicsforteachers/. We developed this open source textbook in a laboratory-based physics course for prospective elementary and middle school teachers. Explorations mostly involve everyday equipment so portions also are appropriate for use in general science courses,&nbsp;hallway exhibits, and outreach activities.&nbsp;Units include the nature of light phenomena, nature of thermal phenomena, influence of light and thermal phenomena on local weather, influence of light and thermal phenomena on global climate change, and nature of astronomical phenomena such as phases of the moon. Emphasis is upon questioning, predicting, exploring, and discussing what one thinks and why.&nbsp;This open source textbook also emphasizes the integration of science and literacy learning. Each unit ends by making connections to the&nbsp; Next Generation Science Standards. &nbsp;Supplementary materials include equipment lists suitable for both on-campus and&nbsp;remote learning contexts. &nbsp; &nbsp;

Elizabeth Gire Gire

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