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Research-based multiple-choice questions implemented in class with peer instruction have been shown to be an effective tool for improving students' engagement and learning outcomes. Moreover, multiple-choice questions that are carefully sequenced to build on each other can be particularly helpful for students to develop a systematic understanding of concepts pertaining to a theme. Here, we discuss the development, validation, and implementation of a multiple-choice question sequence (MQS) on the topic of quantum measurement in the context of wave functions in the infinite-dimensional Hilbert space. This MQS was developed using students' common difficulties with quantum measurements as a guide and was implemented in a junior-/senior-level quantum mechanics course at a large research university in the U.S. We compare student performance on assessment tasks focusing on quantum measurement before and after the implementation of the MQS and discuss how different difficulties were reduced and how to further improve students' conceptual understanding of quantum measurement in infinite-dimensional Hilbert space.

Quantum information science and engineering (QISE) is a rapidly developing field that leverages the skills of experts from many disciplines to utilize the potential of quantum systems in a variety of applications. It requires talent from a wide variety of traditional fields, including physics, engineering, chemistry, and computer science, to name a few. To prepare students for such opportunities, it is important to give them a strong foundation in the basics of QISE, in which quantum computing plays a central role. In this study, we discuss the development, validation, and evaluation of a QuILT, or Quantum Interactive Learning Tutorial, on the basics and applications of quantum computing. These include an overview of key quantum mechanical concepts relevant for quantum computation (including ways a quantum computer is different from a classical computer), properties of single- and multi-qubit systems, and the basics of single-qubit quantum gates. The tutorial uses guided inquiry-based teaching-learning sequences. Its development and validation involved conducting cognitive task analysis from both expert and student perspectives and using common student difficulties as a guide. The inquiry-based learning sequences in the tutorial provide scaffolding support to help students develop a functional understanding. The final version of the validated tutorial was implemented in two distinct courses offered by the physics department with slightly different student populations and broader course goals. Students' understanding was evaluated after traditional lecture-based instruction on the requisite concepts, and again after engaging with the tutorial. We analyze and discuss their improvement in performance on concepts covered in the tutorial., Comment: 39 pages

Quantum information science is a rapidly growing interdisciplinary field that is attracting the attention of academics and industry experts alike. It requires talent from a wide variety of traditional fields, including physics, engineering, chemistry, and computer science, to name a few. To prepare students for such opportunities, it is important to give them a strong foundation in the basics of quantum information science, in which quantum computing plays a central role. In this study, we discuss the development, validation, and evaluation of a tutorial on the Bloch sphere, a useful visual tool for developing intuition about single quantum bits (qubits), which are the basic building block of any quantum computer. Students' understanding was evaluated after they received traditional lecture-based instruction on the requisite topics, and again after engaging with the tutorial. We observe, analyze, and discuss their improvement in performance on concepts covered in the tutorial., Comment: 19 pages, 1 figure

Research-validated multiple-choice questions comprise an easy-to-implement instructional tool that serves to scaffold student learning and formatively assess students knowledge. We present findings from the implementation, in consecutive years, of research-validated multiple-choice question sequence on measurement uncertainty as it applies to two-state quantum systems. This study was conducted in an advanced undergraduate quantum mechanics course, in an online and in-person learning environments in consecutive years. Student learning was assessed after receiving traditional lecture-based instruction in relevant concepts, and their performance was compared with that on a similar assessment given after engaging with the multiple-choice question sequence. We analyze and discuss the similar and differing trends observed in the two modes of instruction., Comment: 23 pages, 0 figures

Research-validated multiple-choice questions comprise an easy-to-implement instructional tool that serves to scaffold student learning and formatively assess students' knowledge. We present findings from the implementation, in consecutive years, of a research-validated multiple-choice question sequence [referred to in this study as a Clicker Question Sequence (CQS)] on quantum measurement as it applies to two-state quantum systems. This study was conducted in an advanced undergraduate quantum mechanics course, in both online and in-person learning environments across three years. Student learning was assessed after traditional lecture-based instruction in relevant concepts, and their performance was compared with that on a similar assessment given after engaging with the CQS. We analyze, compare, and discuss the trends observed in the three implementations., Comment: 30 pages, 3 figures

Research-validated clicker questions as instructional tools for formative assessment are relatively easy to implement and can provide effective scaffolding when developed and implemented in a sequence. We present findings from the implementation of a research-validated clicker question sequence (CQS) on student understanding of the time-development of two-state quantum systems. This study was conducted in an advanced undergraduate quantum mechanics course for two consecutive years in virtual and in-person classes. The effectiveness of the CQS discussed here in both modes of instruction was determined by evaluating students' performance after traditional lecture-based instruction and comparing it to their performance after engaging with the CQS., Comment: 18 pages, 1 figure

Research-validated multiple-choice questions comprise an easy-to-implement instructional tool for scaffolding student learning and providing formative assessment of students' knowledge. We present findings from the implementation of a research-validated multiple-choice question sequence on the basics of two-state quantum systems, including inner products, outer products, translation between Dirac notation and matrix representation in a particular basis, and change of basis. This study was conducted in an advanced undergraduate quantum mechanics course, in both online and in-person learning environments, across three years. For each cohort, students had their learning assessed after traditional lecture-based instruction in relevant concepts before engaging with the multiple-choice question sequence. Their performance was evaluated again afterwards with a similar assessment and compared to their earlier performance. We analyze, compare, and discuss the trends observed in the three implementations., Comment: 25 pages, 0 figures; added references

Structural equation modeling (SEM) is a statistical method widely used in educational research to investigate relationships between variables. SEM models are typically constructed based on theoretical foundations and assessed through fit indices. However, a well-fitting SEM model alone is not sufficient to verify the causal inferences underlying the proposed model, as there are statistically equivalent models with distinct causal structures that equally well fit the data. Therefore, it is crucial for researchers using SEM to consider statistically equivalent models and to clarify why the proposed model is more accurate than the equivalent ones. However, many SEM studies did not explicitly address this important step, and no prior study in physics education research has delved into potential methods for distinguishing statistically equivalent models with differing causal structures. In this study, we use physics identity model as an example to discuss the importance of considering statistically equivalent models and how other data can help to distinguish them. Previous research has identified three dimensions of physics identity: perceived recognition, self-efficacy, and interest. However, the relationships between these dimensions have not been thoroughly understood. In this paper, we specify a model with perceived recognition predicting self-efficacy and interest, which is inspired by individual interviews with students in physics courses to make physics learning environments equitable and inclusive. We test our model with fit indices and discuss its statistically equivalent models with different causal inferences among perceived recognition, self-efficacy, and interest. We then discuss potential experiments that could further empirically test the causal inferences underlying the models, aiding the refinement to a more accurate causal model for guiding educational improvements., Comment: We note that a corresponding paper for physics 2 was published in the International Journal of Innovation in Science and Mathematics Education (IJISME) based on the conference proceedings of the International Conference on Physics Education that one author attended

Students' self-efficacy, interest, and perceived recognition from others have been shown to be very important for the development of their identity in a given field, which is a critical predictor of students' career decisions. Prior research suggests that students' self-efficacy and interest play an important role in their performance and persistence in STEM fields. However, very little has been investigated about the role of perceived recognition and validation by instructors on students' self-efficacy and interest. Moreover, prior quantitative studies show that women often report a lower level of physics perceived recognition, self-efficacy and interest. In this study, we analyzed data from individual interviews with 38 female students to investigate their learning experiences in physics courses in order to obtain a qualitative understanding of the factors that shape their self-efficacy and interest. We find that female students' negative and positive perceived recognition from instructors and teaching assistants (TAs) greatly influenced their self-efficacy and interest and even impacted their desire to persist in STEM majors. We categorize different types of perceived recognition that women reported in our interviews and how they influenced them. For example, many women reported that they felt belittled for their questions or efforts in physics courses, which often negatively influenced their self-efficacy. These findings can help physics educators develop better ways to interact with students in order to provide positive recognition and validation, such as acknowledging students' efforts and questions, expressing faith in students' ability to excel, and being careful not to give unintended messages to students. Our research also suggests that it is important for instructors/TAs to note that it is not their intentions that matter but the impact they are having on their students.

In this study, we adapt prior identity framework to investigate the effect of learning environment (including perceived recognition, peer interaction and sense of belonging) on students' physics self-efficacy, interest and identity by controlling for their self-efficacy and interest at the beginning of a calculus-based introductory physics course. We surveyed 1203 students, 35% of whom were women. We found that female students' physics self-efficacy and interest were lower than male students' at the beginning of the course, and the gender gaps in these motivational constructs became even larger by the end of the course. Analysis revealed that the decrease in students' physics self-efficacy and interest were mediated by the learning environment and ultimately affected students' physics identity. Our model shows that perceived recognition played a major role in explaining students' physics identity, and students' sense of belonging in physics played an important role in explaining the change in students' physics self-efficacy.

We discuss a study in a first year college introductory physics course for physical science and engineering majors that shows that women, on average, feel less recognized by their physics instructors than men as students who can excel in physics. We also discuss how this lack of perceived positive recognition pertaining to physics can adversely affect their self-efficacy and performance in the course. We recommend that physics instructors not be parsimonious in their praise of students and make a conscious effort to positively recognize their students for their effort and progress whenever an opportunity arises. Interviews with female students suggest that instructors should be careful not to give unintended messages to students, e.g., by praising some students for brilliance or intelligence as opposed to their effort because praising a student for brilliance can convey to other students that they do not have what is required to excel in physics. Interviews also suggest that when students ask instructors for help on physics problems, if instructors inadvertently label those problems as "easy", "trivial" or "obvious", it can also make students feel disparaged. The perception of being belittled by these kinds of unintended comments by instructors as well as a lack of positive recognition for good effort and progress have the potential to most adversely impact students from underrepresented groups including women.