This Resource Letter provides information and guidance for those looking to incorporate computation into their courses or to refine their own computational practice. We begin with general resources, including policy documents and supportive organizations. We then survey efforts to integrate computation across the curriculum as well as provide information for instructors looking to teach a computational physics course specifically. An overview of education research into computation in physics, including materials from beyond Physics Education Research, is then provided, followed by suggestions for tools, languages, and environments. We conclude with some emerging topics for which only preliminary resources exist but represent important topics for future innovation.

## I. INTRODUCTION

This Resource Letter aims to help a diverse audience discover resources to help them in their practice and education, namely, high school and college physics instructors aiming to incorporate computing into their courses, those looking to teach a course specifically on computational physics, education researchers interested in work on computation in physics, and, finally but perhaps most importantly, those physicists *including students* interested in broadening their view on computing in physics more generally and acquiring new tools and skills for their work. As I discuss below, the word computing is used here in a broad sense to cover the many ways computers are used in physics. In prior Computational Physics Resource Letters,^{1,2} many of the resources provided (and their updated versions) are still of considerable value. Much in those earlier Letters in fact seems remarkably prescient: Published in 1995, de Vries's letter CP-1^{1} discusses the tremendous potential of neural networks for physics applications, for example. Now grouped with other machine learning methods and part of the emerging discipline of data science, their potential alluded to in CP-1 has only recently begun to be fully realized in physics as improved hardware, algorithms, and datasets have become available.

Similarly visionary, in CP-2, Rubin Landau argues that computation represents a third avenue of physics knowledge production distinct from experiment and theory, and framed computational physics as a synthesis of applied math, computer science, and physics. In highlighting its highly interdisciplinary nature, and also its epistemic role, Landau foreshadows much recent work on thinking of how computing is done in physics, i.e., the skills and practices necessary rather than simply tools and problems.

Taking into account important developments since the earlier Letters, I have broadened the scope of this new Resource Letter in a number of ways that aim to be useful to educators. There now exist significant repositories of materials and supportive communities for educators and an emerging body of education research on computational education, including physics education research (PER) specifically. Even so, integrating computation remains challenging and instructors experience many barriers that may prevent them from succeeding.^{140–142} Rather than focusing entirely on computational physics courses, I, therefore, aim to help diminish some of these barriers by providing resources that are valuable for integrating computation into any physics course and note the many recommendations to do so.^{3–7}

In arguing why integration of computation into the physics curriculum is so necessary, I will go further than Landau: Computation is so interwoven into essentially all aspects of physics knowledge production that it almost cannot be meaningfully disentangled because computers are used to build theories, design experiments, model phenomena, collect, analyze, visualize data, and much more. In this Letter, I adopt a very broad view of computing in physics, including all these uses and not just computational problems, algorithms, or coding. I believe that this enriched view of computation requires a very different kind of classroom from the traditional one: One in which students are *doing physics with computers* and where students have the opportunity to engage in the various epistemic practices of physics. It repositions the instructor as a fellow scientist who facilitates discovery rather than the sole source of knowledge and authority.

Achieving this vision will require paying attention to: *who* is doing computation necessitating considerations of diversity, equity, inclusion, and justice (DEIJ); *what* is happening and *how* including practices and active learning techniques as well as insights from math and computer science education; *why* focusing on objectives of inquiry and standards of evidence; and *for whom* inviting questions of communication, outcomes, professional preparation, and the social context of science. Certain emerging areas like quantum computing even call into question *what* the computer of the future even will be! I sincerely hope readers will, therefore, read this Letter not just for information but as an invitation to innovate further with their pedagogy and practice to achieve some of these goals.

For those entirely new to the idea of integrating computation into their courses, may I suggest a few resources that form particularly good starting points: A recent AAPT report, Ref. 3, provides general recommendations. PICUP, Ref. 14, is a supportive community of educators at various stages of integrating computation into their courses and also an impressive online repository of examples. Simple, ready to go simulations with no coding are available through PhET.^{18} Specific resources for K-12 and introductory courses are found in Secs. III B and III C, respectively. No-install tools for coding using the popular python language are found in Refs. 70 and 254. A very accessible article on some of the kinds of thinking computation opens up is Ref. 194. Finally, many accessible references for diversity equity inclusion and justice in computing are found in Sec. VII A.

To assist all readers, I have structured the Letter to group particular kinds of resources as follows: Section II contains general resources for computation, including reports that provide overall guidance best practices, review papers, and collections of resources and organizations. Section III is aimed at those interested in integrating computation into any course; it also includes resources for curricular reform. Section IV, in contrast, considers computational physics courses specifically. While resources from physics education research (PER) are embedded throughout the whole paper, additional materials that may require more education background are presented in Sec. V. Suggestions for tools, languages, and environments are provided in Sec. VI. I conclude the substantive material in Sec. VII on emerging topics in computation for which only a limited selection of resources are presently available but represent important directions for future development. As an additional aid to the reader, a glossary of acronyms is provided in Sec. VIII.

- 1.
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Resource Letter CP-1: Computational physics,”P. L.DeVries, Am. J. Phys. 64, 364–368(1996). (E) - 2.
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Resource Letter CP-2: Computational physics,”R. H.Landau, Am. J. Phys. 76, 296–306(2008). (E)

## II. GENERAL RESOURCES

### A. Reports

A number of efforts have emerged to gather and synthesize the views of interested parties on computation. The resulting reports are helpful policy documents with recommendations for good practices.

- 3.
**AAPT Recommendations for Computational Physics in the Undergraduate Physics Curriculum**, prepared by members of the AAPT Undergraduate Curriculum Task Force (AAPT, 2016). Provides guidance and recommendations for incorporating computational physics into undergraduate physics programs at all levels. Discusses learning outcomes, skills, and challenges for integration. See also the accompanying Guest Editorial: “AAPT recommendations for computational physics in the undergraduate physics curriculum, and the partnership for integrating computation into undergraduate physics,”E.BehringerandL.Engelhardt, Am. J. Phys. 85, 325–326(2017). (E) - 4.
Advancing Interdisciplinary Integration of Computational Thinking in Science, May 2–5, 2019, Conference Report,

S.Brophy,M. D.Caballero,K.Fisler,M.Hicks,R.Hilborn,C. M.Romanowicz,K.Roos, andR.Vieyra(AAPT,Maryland, 2020). Report focusing on integrating computing into K-12 science education. Emphasizes importance of disciplinary values, DEIJ considerations, the need for training, assessment, and workforce preparation. See Sec. V D 3 on computational thinking below. Available at: https://www.aapt.org/resources/upload/computational_thinking_conference_report_final_200212.pdf (E) - 5.
**Computation as a Tool for Discovery in Physics**,B.McCurdy,D.Ceperly,S.Koonin,J.Ostriker,D.Reed,N. J.Wilkins, andP.Woodward(National Science Foundation,Arlington, Virginia, 2002). Report advocating that the NSF should create a new program in computational physics. Provides a very broad and interesting look at the myriad uses computation is put to across the discipline. (I) - 6.
**Effective Practices for Physics Programs: The EP3 Guide & Communities**.Joint effort of the APS and AAPT that provides advice for physics departments and programs in the U.S. to support change and improvement. Includes a section on computational skills. https://ep3guide.org (E) - 7.
**Phys21: Preparing Physics Students for 21st-Century Careers, Joint Taskforce on Undergraduate Physics Programs**.Joint report by the APS and AAPT advocating for aligning physics curricula with workforce needs. Advises integration of computation in line with other reports. https://www.compadre.org/phys21 (E) - 8.
**2021 PICUP Virtual Capstone Conference Report**,R.Hilborn,M. D.Caballero,L.Engelhardt,A.Knaub,M.Kuchera,M. L.delPuerto,B.Lunk,K.Roos, andT.Zimmerman.Report summarizing progress of the PICUP project.^{14}Includes some resources and prospects for future work. https://www.compadre.org/picup//events/pdfs/2021_PICUP_Capstone_Report_Final_Final_220502.pdf

### B. Editorial and review articles

Review articles and previous versions of this Resource Letter provide helpful references and perspective.

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Hitting the ground running: Computational physics education to prepare students for computational physics research,” A. L. Graves and A. Light, Comput. Sci. Eng. 22, 50–60(2020). Advocates for a skills-based approach. Includes the results of a survey of*N*= 46 computational physicists to determine the type of experience undergraduate students need to do research with them. Critically examines some existing courses using the results. (E) - 10.
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The benefit of computational modeling in physics teaching: A historical overview,”J.WeberandT.Wilhelm, Eur. J. Phys. 41, 034003 (2020). Overview of modeling in physics education. Chronological approach provides a sense of evolution of the field. (I) - 11.
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Undergraduate computational physics education: Uneven history and promising future,”R. F.Martin, J. Phys.: Conf. Ser. 759, 012005 (2016). Provides a brief history of computational physics education, reflects on the success of three computational physics programs, and proposes factors for successful creation of future programs. (E) - 12.
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Preparing physics students for 21st-century careers,”L.McNeil, andP.Heron, Phys. Today 70(11), 38–43 (2017). Argues that physics programs need to better prepare students for careers beyond academia. Significant emphasis on the importance of computation. (E) - 13.
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AJP theme double-issue on computation: An introduction to the theme double-issue,”W.ChristianandBradleyAmbrose, Am. J. Phys. 76, 293–295(2008). (E)

### C. Collections of resources

A number of repositories exist specifically for computation in physics education.

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**Partnership for Integrating Computation into Undergraduate Physics (PICUP)**. PICUP is a collection of resources including peer reviewed “exercise sets,” and other contributed materials to integrate computation in a variety of undergraduate courses. Resources are categorized along various dimensions such as intro or advanced, topical area, language, and time estimated. Particularly valuable is the associated community of practice and Slack channel. Available at: https://www.compadre.org/PICUP/. PICUP hosts regular webinars with further information and recordings available at https://www.compadre.org/PICUP/community/. See also associated articles: “PICUP: A community of teachers integrating computation into undergraduate physics courses,”M. D.Caballero,NormanChonacky,LarryEngelhardt,Robert C.Hilborn,Marie Lopezdel Puerto, andKelly R.Roos, Phys. Teach. 57, 397–399(2019) and “Analysis of the PICUP collection: Strengths and areas for development,” W. B. Lane, e-print arXiv:2103.02408 (2021). (E) - 15.
**Enhancing Science Courses by Integrating Python (ESCIP)**.A collaborative dedicated to providing easy access to python and Jupyter Notebooks for chemistry, physics, and astronomy education. Also provides trainings for instructors new to python.https://escip.io (E) - 16.
**Open Source Physics**.Extensive collection of resources to engage students in physics, computation, and computer modeling, primarily in java and javascript. Available at: https://www.compadre.org/osp (E) - 17.
**The Compadre Resource for Physics Education**.Large repository of valuable resources for physics education including.^{7,16,52}http://www.compadre.org/ (E) - 18.
**PhET Interactive Simulations**. Provides ready to use interactive simulations for introductory and intermediate undergraduate classes. java and browser-based versions available. https://phet.colorado.edu. Based on an extensive research program; the website includes links to numerous research articles. See selected associated articles: “PhET: Interactive simulations for teaching and learning physics,”K.Perkins,W.Adams,M.Dubson,N.Finkelstein,S.Reid,C.Wieman, andR.LeMaster, Phys. Teach. 44, 18–23(2006) and “Teaching physics using PhET simulations,”C. E.Wieman,W. K.Adams,P.Loeblein, andK. K.Perkins, Phys. Teach.**48**, 225–227 (2010). (E) - 19.
**The Shodor Foundation**provides a national resource for computational science education including resources and workshops. (I) http://www.shodor.org (I)

### D. Organizations

Here, I include organizations whose mission includes computation as used by the physics community or with an educational focus.

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**Extreme Science and Engineering Discovery Environment (XSEDE)**.A single virtual system that scientists can use to interactively share computing resources, data, and expertise. Aims to support U.S.-based researchers wish to use high performance computing resources, including services, software, compute, etc. Provides extensive training materials, including online courses. Material for beginners also available: https://www.xsede.org. (A) - 21.
**American Physical Society Division of Computational Physics (DCOMP)**.Promotes research and development in computational physics: https://engage.aps.org/dcomp/home (A) - 22.
**International Union of Pure and Applied Physics (IUPAP)**.Includes a Commission on Computational Physics (C20) that organizes the yearly international Conference on Computational Physics (CCP): https://iupap.org/who-we-are/internal-organization/commissions/c20-computational-physics/ (A) - 23.
**NumFOCUS**.Promotes open practices in research, data, and scientific computing by sponsoring open source projects and organizing community-driven educational programshttps://numfocus.org (A) - 24.
**Centre Européen de Calcul Atomique et Moléculaire (CECAM)**.Promotes fundamental research on advanced computational methods and their applications to science and technology. Traditionally focused on condensed matter. http://www.cecam.org (A) - 25.
**National Labs**.Many National Labs have a very strong Computational Physics focus to support their activities. A few examples:**Los Alamos:**https://www.lanl.gov/org/padwp/adx/computational-physics/index.php and https://www.lanl.gov/org/ddste/aldsc/computer-computational-statistical-sciences/index.php. Codes, internships, and summer programs are available. (A)**Argonne:**https://www.anl.gov/topic/science-technology/computational-physics (A)**Lawrence Berkeley:**https://www.physics.lbl.gov (A)**Oak Ridge:**https://www.ornl.gov/group/theoretical-and-computational-physics (A)

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**Association for Computing Machinery Special Interest Group (ACMSIG)**on High Performance Computing. Promotes interest in and knowledge of applications of high performance computing. https://sighpceducation.acm.org/ (A)

## III. COMPUTATION ACROSS THE CURRICULUM

In this section, I present resources for instructors who are looking to integrate computation into non-specialized computational physics courses. For each type of course, I present a selection of papers focusing on a material that is relatively accessible to instructors and clearly document what has been done. Some of these are also education research papers and incorporate various kinds of evaluation.

Beyond these resources, I strongly recommend that any effort to integrate computation begin by seeking out a supportive community of practitioners already embarking on this journey (e.g., PICUP^{14} or ESCIP^{15}), and also by reading recent reports that provide general guidance (see Sec. II A). Having a clear understanding of your own objectives for computation, how they might connect with other elements of your course and other classes at your institution will aid you immeasurably in this rewarding process. As you try a few iterations, you may find valuable the additional resources of Sec. V that require more background in education and even find yourself contributing to this exciting literature.

### A. Curriculum reform

A number of institutions have successfully reformed their curricula to incorporate computation. Some of their experiences are documented in the below publications.

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Putting computation on a par with experiments and theory in the undergraduate physics curriculum,”R. M.Serbanescu,P. J.Kushner, andS.Stanley, Am. J. Phys. 79, 919–924(2011). Integration of computation into lecture and lab courses in the first and second year at the University of Toronto. python based. Examines student reactions from surveys. (I) - 28.
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Integrating computation into the undergraduate curriculum: A vision and guidelines for future developments,”N.ChonackyandD.Winch, Am. J. Phys. 76, 327–333(2008). Expansive vision for computationally integrated physics curricula leveraging survey data, meetings and discussion at a sequence of conferences and a visioning workshop. Proposes recommendations and future directions for reform, not all of which have been fully explored. (I) - 29.
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Computation in undergraduate physics: The Lawrence approach,”D. M.Cook, Am. J. Phys. 76, 321–326(2008). Part of AJP theme double issue.^{13}Comprehensive effort to redesign the entire physics curriculum around computation at a liberal arts college. (A) - 30.
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Integrating computational activities into the upper-level Paradigms in Physics curriculum at Oregon State University,”D. H.McIntyre,J.Tate, andC. A.Manogue, Am. J. Phys. 76, 340–346(2008). Part of AJP theme double issue.^{13}Describes paradigms in physics, an upper-level curriculum that tightly integrates research-based instruction and computation. (A) - 31.
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An incremental approach to computational physics education,”K. R.Roos, Comput. Sci. Eng. 8, 44–50(2006). Description of an effort at Bradley University to interweave computation into classical mechanics and statistical mechanics courses. Discusses opportunities and challenges for integrating computation. (I) - 32.
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Computation across the curriculum: What skills are needed?,”M. D.Caballero, PERC Proceedings (2015). Report on a 30 person conference to identify skills physics bachelor's graduates need with respect to computation. Discusses concepts, practices, computational ways of thinking, and meta-knowledge. (I)

### B. K-12 courses

Those teaching computation in K-12 courses are also advised to consult the literature on computational thinking, which has become a popular organizing framework for computational activity at this level (Subsection V D 3). A valuable recent report on these lines is to be found in Ref. 4. K-12 classes are a topic of increasing importance in PER, see Refs. 153, 154, 155, 161, 175, and 302.

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**A Framework for K-12 Science Education Practices: Crosscutting Concepts, and Core Ideas**, National Research Council (National Academies Press, Washington, D.C., 2012). Framework for K-12 that aims to incorporate scientific practices and emphasize computational modeling. See also the Next Generation Science Standards that uses this framework: https://www.nextgenscience.org/ (E) - 34.
**Coding in High School Physics and Physical Science**. Collection of resources at: https://www.compadre.org/books/hscoding (E) - 35.
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Integrating numerical computation into the modeling instruction curriculum,”M. D.Caballero,J. B.Burk,J. M.Aiken,B. D.Thoms,S. S.Douglas,E. M.Scanlon, andM. F.Schatz, Phys. Teach. 52, 38–41(2014). Aims to introduce high school physics students with no background in programming to computational problem-solving experiences. Presents a typical activity using vpython to model a kicked soccer ball with example student work. (E) - 36.
**STEM coding**.Aims to re-imagine introductory STEM courses with computer science in mind.Emerging effort to also produce data science materials. https://u.osu.edu/stemcoding/ (E) - 37.
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Numerical modeling with spreadsheets as a means to promote STEM to high school students,”J.Benacka, Eur. J. Math., Sc. Tech. Ed. 12, 947–964(2016). High school developed spreadsheet applications to simulate fall and projectile motion. Includes example spreadsheets. (E) - 38.
**Next Generation Science Standards**.Multi-state effort to create a science curriculum organized around disciplinary core ideas, concepts used across disciplines, and scientific practices. Computation feeds into both “developing and using models” and “using mathematics and computational thinking,” which are both practices https://www.nextgenscience.org (E) - 39.
**Scratch**.Visual block-based programming environment widely used in K-12 contexts. https://scratch.mit.edu (E) - 40.
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Scratch as a computational modeling tool for teaching physics,”V.LopezandM.Isabel Hernandez, Phys. Educ. 50, 310–318(2015). Illustrates use of Scratch^{39}in a physics context. Includes a discussion of other environments that may be of interest. (E) - 41.
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Computational thinking for physics: Programming models of physics phenomenon in elementary school,” H. Dwyer,B.Boe,C.Hill,D.Franklin, andD.Harlow, PERC Proceedings (2013). Scratch-based curriculum for grades 4–6. (E) - 42.
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Teachers' intended learning outcomes around computation in high school physics,”D. P.Weller,M. D.Caballero, andP. W.Irving, PERC Proceedings (2019). Examines high school teachers' objectives for computation relative to the NGSS. (A)

### C. Introductory courses

Introductory courses are a major focus of physics education research, and there are a good deal of valuable resources for courses that center computation:

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Computational physics in the introductory calculus-based course,”R.ChabayandB.Sherwood, Am. J. Phys. 76, 307–313(2008). Describes integration of computation in the context of the matter and interactions curriculum using vpython based on experiences from several institutions. Goals include improving conceptual understanding of physics principles, visualization, modeling complex situations, and curricular change. Discusses positive and negative outcomes of programming and provides example sequences of computational activities. (E) - 44.
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A model-based view of physics for computational activities in the introductory physics course,”A.Buffler,S.Pillay,F.Lubben, andR.Fearick, Am. J. Phys. 76, 431 (2008). Helpful theoretical discussion of models and modeling. Leverages vpython in a laboratory format. Focuses on modeling the translational and rotational motion of a spaceship. Analyzes student work. (E) - 45.
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Implementing and assessing computational modeling in introductory mechanics,”Marcos D.Caballero,Matthew A.Kohlmyer, andMichael F.Schatz, Phys. Rev. PER 8, 020106 (2012). Presents integration and of computational modeling homework questions into a large (1357 student) intro class with vpython. Valuable effort to assess effectiveness on a separate task, although the emphasis on student errors rather than successes would be interesting to push beyond. (E) - 46.
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Incorporating computational exercises into introductory physics courses,”G.Kortemeyer, J. Phys.: Conf. Ser. 1512, 012025 (2020). Reports integration of group vpython exercises into studio physics based intro course. Considers benefits of collaboration vis-à-vis issues like plagiarism. (E) - 47.
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Computational essays in the physics classroom,”T. O. B.OddenandJ.Burk, Phys. Teach. 58, 252–255(2020). Uses computational “essays” in an into electromagnetism class to structure student thinking. See section on Computational Literacy below. (E) - 48.
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A model scientific computing course for freshman students at liberal arts colleges,”A. K.Sharma, J. Comp. Sci. Ed. 8, 2–9(2017). A very interesting approach targeting freshman students within a liberal arts environment. Mathematica based, also using visualization software like Visual Molecular Dynamics and Avogadro. Project descriptions and extensive supplementary material provided. Reports a positive student response from survey data. (E) - 49.
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A hybrid approach for using programming exercises in introductory physics,”C. M.Orban,R. M.Teeling-Smith, andJ. R. H.Smith, Am. J. Phys. 86, 831–838(2018). Presents an approach that incorporates interactivity and coding (somewhere between PhET and a traditional coding-first approach). Promising results even for weak and beginner programmers. (E) - 50.
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P,”^{3}: A practice focused learning environmentP. W.Irving,M. J.ObsniukandM. D.Caballero, Eur. J. Phys. 38, 055701 (2017). Ambitious introductory course designed around problems and practices in physics and incorporating significant computation elements. Extensively studied by successive PER efforts. (E) - 51.
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Modeling instruction: An effective model for science education,”J.Jackson,L.Dukerich, and D. Hestenes, Sci. Educ. 17, 10–17(2008). Accessible overview of the Modeling instruction curriculum. (E) - 52.
Physlet Physics: Interactive Illustrations, Explorations and Problems for Introductory Physics, 3rd ed.,

W.Christianand M. Belloni (Google Play, 2019); 1st Ed. (Prentice Hall,New Jersey, 2004). Textbook and collection of interactive “Physlets” in javascript and run in the web browser. Latest 3rd edition is available as a free ebook. Available at: https://www.compadre.org/physlets/. (E) - 53.
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A novel approach for using programming exercises in electromagnetism coursework,”C.Orban,C. D.Porter,N. K.Brecht,R. M.Teeling-Smith, andK. A.Harper, PERC Proceedings (2017). Browser-based and student-modifiable javascript simulations for Introductory E&M. (E)

### D. Cellphones as labs

An important resource that many—but not all—students have access to are cellphones. Here, I list some resources and tools that aim to use these devices for data collection. The ready availability of such real-life data can be a powerful resource for computation.

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Resource Letter MDS-1: Mobile devices and sensors for physics teaching,”M.Monteiro, Am. J. Phys. 90, 328–343(2022). Very comprehensive Resource Letter on using cellphones for physics. (E) - 55.
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Experiments with mobile devices—A retrospective on 10 years of iPhysicsLabs,”J.KuhnandP.Vogt, Phys. Teach. 60, 88–89(2022). Overview of a recurring column in*The Physics Teacher*with numerous resources for using cellphones in labs. (E) - 56.
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Computer-based learning in an undergraduate physics course: Interfacing a mobile phone and Matlab to study oscillatory motion,”E.MomoxandC. O.DeMaio, Am. J. Phys. 88, 535–541(2020). Shows integration of cellphones with matlab to acquire and analyze accelerometer data. (E) - 57.
**Physical Phone Experiments**(PhyPhox),Free smartphone app that provides access to the cellphone's sensors, including accelerometers, magnetometers, etc. https://phyphox.org (E) - 58.
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Teaching classical mechanics using smartphones,”J.Chevrier,L.Madani,S.Ledenmat, andA.Bsiesy, Phys. Teach. 51, 376–377(2013). Describes iMecaProf software that can receive and interactively visualize sensor data from an iPhone. (I) - 59.
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Using a mobile phone acceleration sensor in physics experiments on free and damped harmonic oscillations,”J.CarlosCastro-Palacio,L.Velázquez-Abad,M. H.Giménez, andJ. A.Monsoriu, Am. J. Phys. 81, 472–475(2013). Uses a cellphone and an air track to create an oscillator and compare with theoretical predictions. (E)

### E. Laboratories

Some efforts have been made to incorporate computation into laboratories. Laboratory reform is an important contemporary topic of interest in physics education and so I also include here a few resources on that topic.

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Integrating numerical modeling into an introductory physics laboratory,”D.Sachmpazidi,M.Bautista,Z.Chajecki,C.Mendoza, and Charles Henderson, Am. J. Phys. 89, 713–720(2021). Integration of modeling with Excel spreadsheets into an introductory laboratory course. Computational tools allowed students to engage in more complex and realistic experiments than conventional labs. Students and instructors responded positively. (E) - 61.
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Introductory physics labs: We can do better,” N. G. Holmes andC. E.Wieman, Phys. Today 71, 38–45(2018). Shows that some labs are not an effective means of students learning physics content, but could be a place for scientific activity. (E) - 62.
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Model-based reasoning in the physics laboratory: Framework and initial results,”B.M. Zwickl,D.Hu,N.Finkelstein, andH. J.Lewandowski, Phys. Rev. ST Phys. Educ. Res. 11, 020113 (2015). Brings a modeling lens to labs, which focuses student attention on scientific sense making rather than technical details. Presents a framework together with results including student interviews. (I) - 63.
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Labs in building a modern physics way of thinking,”M.Michelini, in The Role of Laboratory Work in Improving Physics Teaching and Learning (Springer,New York, 2018), p. 15. K-12 oriented study that aims to ground labwork to reflect scientific practices. (I) - 64.
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Investigating complementary computational and empirical activities for students learning diffusion,”D.P. Weller,K.Hinko, andV.Sawtelle, PERC Proceedings (2018). Pairs a computational simulation of diffusion with a lab activity. (I)

### F. Upper-level courses

Upper-level courses represent significant opportunities for future work on integrating computing. A challenge is ensuring that students are adequately prepared given computation cannot be the main focus of the course; these efforts require some sort of base at the introductory level. In this section, I give a few examples of documented efforts to use computation in these more advanced courses. For those interested in doing so, many algorithms and problems suitable for upper-level courses can be found in computational physics textbooks as listed in Sec. IV B. Further project ideas are to be found in the *American Journal of Physics* and even in contemporary physics journals, e.g., Ref. 69.

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Computation in classical mechanics,”T.TimberlakeandJ. E.Hasbun, Am. J. Phys. 76, 334–339(2008). Integration of computational projects into classical mechanics. Projects included comparison of algorithms on the same problem physical motivations why their performance might differ; incorporation of phenomena that cannot be modeled analytically; approximation. Code is available in matlab, mathematica, and java. (I) - 66.
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A model for incorporating computation without changing the course: An example from middle-division classical mechanics,”M.D. Caballero andS. J.Pollock, Am. J. Phys. 82, 231–237(2014). Integration of computation into a sophomore-level course, including both Mathematica activities in homework problems and final projects. Describes course and topic-level goals for computation and provides example abbreviated example homework problems. Leveraged videos for outside class. (I) - 67.
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Shifting the learning gears: Redesigning a project-based course on soft matter through the perspective of constructionism,”E.Langbeheim,A.Abrashkin,A.Steiner,H.Edri,S.Safran, andE.Yerushalmi, Phys. Rev. PER 16, 020147 (2020). A project-based course on soft and biological materials. Includes constructing and using computational Monte Carlo and molecular dynamics models to explain experimental results. Example projects are described together with results of student interviews. Grounded in constructionist pedagogy. (I) - 68.
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Enhancing teaching and learning of fluid mechanics with interactive computational modeling,”R.G. Neves, J. Phys.: Conf. Ser. 1286, 012047 (2019). Describes Modellus activities, e.g., finding the maximum depth of water a dam can sustain, in a fluids and thermodynamics course. Student feedback was very positive about inclusion of these activities. (I) - 69.
**Physical Review E**A journal covering statistical, nonlinear, and soft matter physics. Papers in the statistical physics and computational physics sections often use computational methods that are accessible to non-experts. A valuable source of inspiration for possible project ideas.

### G. Accessible computational tools

In this section, I provide examples of computational tools that are particularly suitable for undergraduate education in non-specialized courses. vpython, for example, has been very extensively used in physics education and removes many barriers to entry. Nonetheless, there remains a need for virtual environments that offer more sophisticated feature sets. Some alternatives less widely used in physics: Scratch^{39} and Processing.^{223} At a similar level of ease of use, PhyPhox^{57} provides interactive data collection with cellphone sensors. Other tools listed elsewhere in this Resource Letter are also quite accessible: Both Google Colaboratory^{254} and Jupyter^{252} offer python in a convenient notebook form; Colaboratories run from the browser with no need for installation.

- 70.
**vpython.**python module to create navigable 3D displays and animations. Widely used in Physics Education, notably in the Matter and Interactions curriculum. See also article: “VPython: 3D interactive scientific graphics for students,”D.Scherer,P.Dubois, andB.Sherwood, Comput. Sci. Eng. 25, 56–62(2000). Widely used in Physics Education, notably in the Matter and Interactions curriculum.^{43}Available at https://vpython.org. See also the web-based version Glowscript that enables the user to create and store programs in the cloud: https://glowscript.org (E) - 71.
“

Algodoo: A tool for encouraging creativity in physics teaching and learning,”B.GregorcicandM.Bodin, Phys. Teach. 55, 25–28(2017). Digital sandbox for 2D physics simulations. Compatible with digital whiteboards/touchscreen interface. (E) - 72.
**Easy Java Simulations**. Free authoring tool that helps non-programmers create interactive simulations in java or javascript. Part of Ref. 16. See article: “Modeling physics with easy Java simulations,”W.ChristianandF.Esquembre, Phys. Teach. 45, 475–480(2007). https://www.um.es/fem/EjsWiki/ (E)

## IV. COMPUTATIONAL PHYSICS COURSES

Courses that focus specifically on computation enable students to learn and use more complex tools and refine both their ability to do computation and their ability to do physics.

### A. Journal articles

This section collects a number of reports by authors who have created dedicated computational physics courses. Many of these incorporate or are entirely based on projects, or are structured as labs. These materials should be read in the context of general recommendations for computation as described in Sec. II A.

- 73.
“

A project-oriented course in computational physics: Algorithms, parallel computing, and graphics,”C.Rebbi, Am. J. Phys. 76, 314–320(2008). Describes a number of projects given in a primarily graduate level computational physics course, including Laplace's equation, stochastic simulations, the time-dependent Schrödinger equation, and molecular dynamics. (I) - 74.
“

Developing a project-based computational physics course grounded in expert practice,”C.J. Burke andT. J.Atherton, Am. J. Phys. 85, 301–310(2017). Describes the design and implementation of a project-based course, primarily for undergraduates, grounded in an analysis of computational skills required for research. Includes project descriptions and a skills-based rubric. (I) - 75.
“

The role of computational physics in the liberal arts curriculum,”R.DominguezandB.Huff, J. Phys.: Conf. Ser. 640, 012061 (2015). A computational physics course incorporating homework and a capstone project. Extensively discusses the role of such a course in the context of a liberal arts college. (E) - 76.
“

Teaching computational physics as a laboratory sequence,”R.L. Spencer, Am. J. Phys. 73, 151–153(2005). Brief report of an interesting structure: Splits computational physics instruction into three one-credit hour laboratories one for sophomores, one for juniors, and one for seniors. maple and matlab. (I) - 77.
“

Pwning level bosses in MATLAB: Student reactions to a game-inspired computational physics course,”I.D. Beatty andL. A.Harris, PERC Proceedings (2014). Report of a game-based computational physics course; course structure includes a number of very interesting design elements around feedback, deadlines, etc. (E)

### B. Computational physics textbooks

An inevitable question in course design is the question of appropriate textbooks. The choice of textbook will inevitably depend on the goals of the instructor and whether they plan to use the book as a core text to be read carefully or as a reference for students. In reviewing the available options, I have focused on those that are readily available (i.e., not out of print). I have also tried to categorize them as either *more numerical*, i.e., focusing on numerical analysis, algorithms, and tasks like root-finding, integration, etc., or *more physics*, i.e., framed around physical applications. A few textbooks take a *blended* approach incorporating both elements as separate sections. A number of books have become freely available open-source affairs mirroring code-sharing practices common in science. In each case, I note the main programming language(s) used by the book.

#### 1. More physics

- 78.
**An Introduction to Computer Simulation Methods**, 3rd ed.,H.Gould,J.Tobochnik, andW.Christian(CreateSpace, California, 2017); 2nd ed. (Addison-Wesley, Massachusetts, 2006). Open source textbook that aims to teach physical concepts using computer simulations. Primarily undergraduate level. Introduces java programming by example. As well as a print edition, the electronic edition is freely available at http://physics.clarku.edu/sip/ (I) - 79.Computational Physics: Problem Solving with Python, 3rd ed.,R.H. Landau,M. J.Páez, andC. C.Bordeianu(Wiley, New York, 2015). Very comprehensive physics-focused book starting from elementary topics and culminating in advanced material such as parallel and GPU computing. python listings provided. Upper-level undergraduate or beginning graduate level. Extensive materials including video lectures online. https://sites.science.oregonstate.edu/~landaur/Books/CPbook/eBook/Lectures/index.html (I)
- 80.
Computational Physics with Python

,E. Ayars (2013). Free, open source textbook aimed at undergraduates. Concise and interweaves numerical and physics topics. Introduction to the Linux environment. (I) - 81.
Computational Physics, 2nd ed.,

N.GiordanoandH.Nakanishi(Prentice Hall,New Jersey, 2006). Undergraduate text framed around physics problems, gradually introducing programming and algorithms in context of a particular system. Numerical methods are presented as Appendices for reference. Some programs in FORTRAN and True BASIC are available on a companion website http://www.physics.purdue.edu/~hisao/book/ (A) - 82.
Elementary Mechanics Using Python,

A.Malthe-Sørenssen(Springer,Berlin, 2015). Undergraduate text focused on teaching mechanics through python. Detailed worked examples with code included, and assumes no prior knowledge of python. (I)

#### 2. Blended

- 83.
Computational Quantum Mechanics,

J.IzaacandJ.Wang(Springer, Berlin,2019). Upper-undergraduate level text that takes an interesting approach by focusing on applications in quantum mechanics. Introduces programming and numerical methods in fortran and python. (I) - 84.
Basic Concepts in Computational Physics,

B. A.SticklerandE.Schachinger(Springer, Berlin, 2016). Aims to provide exact mathematical analysis of both deterministic and stochastic methods. Rather concise but with useful mathematical detail. Intersperses numerical methods and physical problems. c++ code online but not integrated into text. Upper-level undergraduate or early graduate level. Perhaps more valuable as a secondary text. (I) - 85.
Computational Physics: Simulation of Classical and Quantum Systems, 2nd ed., Philipp O. J. Scherer (

Springer,Berlin, 2013). Graduate-level text divided into two parts: The first introduces numerical methods; the second considers some physical applications. Electronic edition available. Does not include listings. (A) - 86.
Computation In Modern Physics, 3rd. ed.,

W. R.Gibbs(World Scientific,Singapore, 2006). fortran based. Upper undergraduate/early graduate level. Some discussion of parallel computing. Blended approach: Early chapters discuss numerical tasks and later chapters explore some physical systems. (A)

#### 3. More numerical

- 87.
Numerical Methods in Physics with Python,

A.Gezerlis(Cambridge U. P.,Cambridge, 2020). Focuses on numerical methods, motivating each one with a physics application, with a brief introduction to python. Listings provided in the text and supplementary material at https://numphyspy.org (I) - 88.
Applied Computational Physics,

J. F.BoudreauandE. S.Swanson(Oxford U. P.,Oxford, 2018). Upper-level undergraduate or graduate level (more tuned to the latter). c++ based and extensively discusses programming topics including OOP and best practices. Chapters on numerical methods, programming, and physical applications are interspersed. Parallel computing is introduced early, and interesting advanced topics include many-body physics and quantum field theory. Expensive. (A) - 89.
Computational Methods in Physics: Compendium for Students, 2nd ed.,

S.ŠircaandM.Horvat(Springer,Berlin, 2018). Graduate level guide to numerical methods that aims to cultivate careful scrutiny of numerics. Rather detailed coverage makes it useful as a reference. (A) - 90.
Numerical Methods for Physics, 2nd ed. revised,

A. L.Garcia(CreateSpace,California, 2016); 2nd. ed. (Prentice Hall,New Jersey, 2000). Two versions are available in python and matlab/c++, respectively. Undergraduate text framed around numerical tasks with physics applications. Supplementary material at: http://www.algarcia.org/nummeth/nummeth.html (I) - 91.
Computation and Problem Solving in Undergraduate Physics, 2nd ed.,

D.Cook(2016). Free, open source textbook developed at Lawrence University (see Ref. 29). Focuses on typical numerical tasks at an undergraduate level. Adopts an interesting multi-language approach: Parallel versions of the text are available for different languages including c, fortran, python, mathematica, matlab, and others. Includes an introduction to typesetting with latex. Available at https://psrc.aapt.org/curricula/cpsup/ (I) - 92.
Computational Methods for Physics, 1st ed.,

J.Franklin(Cambridge U. P., Cambridge,2013). Well-written undergraduate text framed around numerical methods but with each chapter introduced by a section on physical motivation. Uses mathematica. (I) - 93.
Computational Physics,

M.Newman(CreateSpace,Cambridge,2013). Undergraduate text framed around computational tasks, with examples from physics and problems. Provides a particularly detailed introduction to programming in python that may aid beginners. Programs and data available on companion website http://www-personal.umich.edu/~mejn/cp/index.html (I) - 94.
An Introduction to Computational Physics, 2nd ed.,

T.Pang(Cambridge U. P.,Cambridge, 2010). Intended for upper undergraduate/early graduate but appears more suitable for the latter. Some interesting advanced topics, e.g., genetic algorithm and numerical renormalization, that are not available elsewhere. Examples in java, but programs are also available in c and fortran on the companion website https://www.physics.unlv.edu/~pang/cp.html (A) - 95.
Introductory Computational Physics,

A.KleinandA.Godunov(Cambridge U. P.,Cambridge, 2010). Concise textbook on numerical methods with c++. Includes an introduction to the ROOT data analysis system used in high energy physics. (A) - 96.
Numerical Recipes: The Art of Scientific Computing, 3rd ed.,

W. H.Press,S. A.Teukolsky,W. T.Vetterling, andB. P.Flannery(Cambridge U.P.,New York, NY, 2007). Comprehensive and accessibly written reference text that aims to broadly cover many common tasks in scientific computing with example implementations and comparisons of popular algorithms. Discusses physics applications only incidentally. An online version is available at http://numerical.recipes (A) - 97.
Computational Physics: FORTRAN Edition,

S.E.KooninandD.Meredith(Avalon Publishing,New York, 1998; Taylor & Francis, London, 2018). Venerable textbook framed around numerical methods. (A)

### C. Supplementary textbooks and lecture notes

Here, I present some additional resources that are not suitable as a primary text, but could be very valuable as secondary texts or as resources for project ideas.

- 98.
A Student's Guide to Python for Physical Modeling, 2nd ed.,

J. M.KinderandP.Nelson(Princeton U. P.,Princeton, 2021). Fairly brief book on python aimed at the beginning programmer: Focuses on basic programming constructs but with scientific applications in mind. Potentially very useful as a secondary text. Inexpensive paperback edition. (E) - 99.
Computational Problems for Physics: With Guided Solutions Using Python,

R. H.LandauandM. J.Páez (Taylor & Francis,London, 2018). Extensive sourcebook of problems from across the undergraduate physics curriculum that could be incorporated into courses. Includes complete python listings. (A) - 100.
Classical Mechanics with Matlab Applications, 2nd ed., Javier E. Hasbun (

CreateSpace, California, 2018). Classical mechanics textbook with examples in matlab. (A) - 101.
Computational Physics,

R.Fitzpatrick(updated 2014). Upper-division undergraduate. Polished and includes an extensive introduction to the c language followed by discussion of several physical systems. Available at: https://farside.ph.utexas.edu/teaching/329/329.html (I) - 102.
Computational Physics, 2nd. ed.,

J.Thijssen(Cambridge U. P.,Cambridge, 2007). Graduate-level text framed around condensed matter physics applications, including density functional theory, fluid dynamics, and molecular dynamics. (A) - 103.
Monte Carlo Methods in Statistical Physics,

M. E.J. Newman andG. T.Barkema(Oxford U. P.,Oxford, 1999). Narrow in focus but extremely readable book. Particularly valuable for enriching discussions of the Ising model. (I) - 104.
Introduction to Computational Science: Modeling and Simulation for the Sciences,

A. B.ShifletandG. W.Shiflet(Princeton U. P,Princeton, 2014). Introductory textbook broadly focused on modeling and simulation in science. (E)

### D. Valuable pedagogical papers

In this subsection, I collect a few papers specially intended to be pedagogical; the Computational Physics section of AJP contains many other examples. I tend to give papers like this directly to students to enable them to cultivate the important scientific practice of reading scientific papers.

- 105.
“

Six textbook mistakes in computational physics,”A.GezerlisandM.Williams, Am. J. Phys. 89, 51–60(2021). Discusses some common misconceptions about numerical methods, including floating-point numbers, linear algebra, interpolation, and initial value problems. Provocative in tone but valuable. (A) - 106.
“

Stable solutions using the Euler approximation,”A.Cromer, Am. J. Phys. 49, 455–457(1981). Discusses the Backwards Euler algorithm that has improved stability over the regular Euler method. (I) - 107.
“

Symplectic integrators: An introduction,”D.DonnellyandE.Rogers, Am. J. Phys. 73, 938–945(2005). Describes a class of algorithms that preserve energy and, hence, provide superior performance. (I) - 108.
“

Adaptive stepsize Runge-Kutta integration,”W.H. Press andS. A.Teukolsky, Comp. Phys. 6, 188–191(1992). Witty article on adaptive stepsize. (I) - 109.
“

Computational physics with particles,”W.G. Hoover andC. G.Hoover, Am. J. Phys. 76, 481–492(2008). Introduction to particle based simulation methods. (I) - 110.
“

Introduction to molecular dynamics simulations,”K.Vollmayr-Lee, Am. J. Phys. 88, 401–422(2020). Outline of molecular dynamics algorithms and concepts. (I)

### E. Applied math textbooks

Applied mathematicians bring a very different set of lenses to computation than physicists: Among other things, they aim to design and analyze algorithms and prove their properties rather than use them to study physical phenomena. In this section, I list a few popular and established textbooks used in applied math that could be useful supplementary texts to make available to students; they all discuss issues like stability, order, error propagation, etc., from a mathematical perspective. I highly recommend (from personal experience) consulting with applied mathematicians and computer scientists for production implementations of scientific codes.

- 111.
A First Course in Numerical Methods,

U. M.AscherandC.Greif(SIAM,Philadelphia, 2011). Clear exposition, more contemporary feel and includes matlab scripts. (I) - 112.
Numerical Analysis,

10th ed.,R. L.Burden,D. J.Faires, andA. M.Burden(Cengage Learning,Massachusetts, 2015). Numerous exercises. Algorithms in pseudocode with listings available in multiple languages as the supplementary material. (I) - 113.
Accuracy and Stability of Numerical Algorithms, 2nd ed.,

N. J.Higham(SIAM,Philadelphia, 2002). Focused on numerical linear algebra. (A) - 114.
Numerical Linear Algebra, 1st ed.,

L. N.TrefethenandD.BauIII (SIAM,Philadelphia, 1997). A very interesting approach that provides considerable insight into how factorizations like the singular value decomposition (SVD), QR decomposition, etc., work. (I) - 115.
Approximation Theory and Approximation Practice,

L. N.Trefethen(SIAM,Philadelphia, 2013). Highly readable with great insight into approximation theory. matlab code integrated into the text (each chapter is a matlab file). (I) - 116.
A First Course in the Numerical Analysis of Differential Equations, 2nd ed.,

A.Iserles(Cambridge U. P.,Cambridge, 2008). Oriented to PDEs. Introduces many ideas, including topics such as finite differences, finite elements, and multigrid through the Poisson equation with other exemplar equations. (A) - 117.
Numerical Analysis, 2nd ed.,

K. E.Atkinson(Wiley,New York, 1991). Venerable but clear. (A)

### F. Computer science books

In the same spirit as Subsection IV E, I present a selection of popular Computer Science books that are less well read than they might be in the physics community and are very useful to assist physicists in refining their practice. Topics include program design, algorithms and data structures, implementation, documentation, and engineering practices.

- 118.
Introduction to Algorithms, 3rd ed.,

T. H.Cormen,C. E.Leiserson,R. L.Rivest, andC.Stein(MIT Press,Massachusetts, 2009). Undergraduate textbook that introduces many common and useful algorithms and provides tools for analyzing their performance such as how they scale with problem size, typical vs. worst-case estimates, etc. (I) - 119.
The Pragmatic Programmer: Your Journey To Mastery, 2nd ed.,

D.ThomasandA.Hunt(Addison Wesley Professional,Massachusetts, 2019). Discusses best practices and major pitfalls of software engineering. (I) - 120.
Code: The Hidden Language of Computer Hardware and Software, 1st ed.,

C.Petzold(Microsoft Press,Unterschleissheim, 2000). Highly readable book about how computers work. (E) - 121.
Code Complete, 2nd ed.,

S.McConnell(Microsoft Press,Unterschleissheim, 2004). Discusses programming strategies for high quality software as “software craftsmanship.” (I) - 122.
Design Patterns: Elements of Reusable Object-Oriented Software, 1st ed.,

E.Gamma,R.Helm,R.Johnson, andJ.Vlissides(Addison-Wesley Professional,Massachusetts, 1994). Describes commonly occurring problems that occur and provides “design patterns,” solution templates to solve them. Provides powerful insight into how to use OOP effectively. Examples are in c++ but applicable to most OOP languages. (A) - 123.
Art of Computer Programming, Vol. 1–4A.,

D.Knuth(Addison-Wesley Professional,Massachusetts, 1997–2011). The Feynmann lectures of Computer Science. Deep insights into algorithms and data by the creator of the TEXtypesetting system. His article “Literate programming,”D.Knuthin Literate Programming (CSLI,California, 1992), p. 99, is also highly influential for advocating a form of self-documenting program design. (A)

## V. EDUCATION RESEARCH

Computation has emerged as an important topic within physics education in the last decade or so. Much of this work has been in conjunction with various course-level and curricular efforts aiming to integrate computation. I already presented a portion of this literature earlier in Sec. III, focusing there on resources that are relatively accessible to those who are simply looking for ways to integrate computation into their classes. Here, I build on this by presenting additional resources that may require more background in PER to fully utilize.

Because computation is, as I argued in the Introduction, a fundamental but complex disciplinary practice that cuts across many other aspects of physics, I believe that multiple theoretical perspectives will be valuable in designing and evaluating computational interventions. I have, therefore, tried to provide a number of resources, especially theoretical perspectives on computational education, that have been developed *outside* the physics community but are being adopted by the computational PER community. There nonetheless remains significant room for further theorizing of computation in physics specifically. The multifaceted nature of computation also means that it naturally connects to many other important topics *within* PER; group work, scientific agency, laboratories, to name a few.

### A. Entry points

- 124.
“

Synthesis of discipline-based education research in physics,”J.L. Docktor andJ. P.Mestre, Phys. Rev. ST Phys. Educ. Res. 10, 020119 (2014). Very helpful entry point into the PER literature that synthesizes main themes and approaches in the field up to the date of publication. (E) - 125.
“

Physics computational literacy: What, why, and how?,”T. O. B.OddenandM. D.Caballero, manuscript in preparation. Forthcoming review article that covers a good deal of important work in the field from the view of the computational literacy framework. (E)

### B. Journals

- 126.
Physical Review Physics Education Research (formerly

Physical Review Special Topics-Physics Education Research). Physics education research journal. (I) - 127.
**European Journal of Physics**. Physics education research journal. (I) - 128.
**Physics Education Research Conference Proceedings.**Associated with the Physics Education Research Conference (PERC) and contains many valuable papers, including cutting edge and preliminary reports of emerging work. In the below section, I have selected a few particularly interesting resources; many more are to be found in the archive online at https://www.compadre.org/per/perc/proceedings.cfm (I) - 129.
**The Physics Teacher**.Practitioner-focused journal publishing shorter articles primarily for high school teachers and introductory physics. (E) - 130.
**Physics Education**.IOP journal with similar audience to The Physics Teacher. (E) - 131.
**American Journal of Physics**.Practitioner-focused journal but with more in-depth articles and covering all levels of physics. Contains many examples PER, particularly before Phys. Rev. ST: PER and Phys. Rev PER were established. (E) - 132.
**Journal of Computational Science Education**.General journal for computation across STEM, but includes some articles for physics. (I) - 133.
**Journal of Physics: Conference Series**. General publisher of conference proceedings that has published some papers related to computation. (A) - 134.
Annual Conference on Innovation and Technology in Computer Science Education. Computer Science Education conference proceedings. Note that computer science, as a discipline, uses conference proceedings as their top-tier publication location. (I)

- 135.
ACM International Computing Education Research Conference. Another source of Computer Science Education articles. (I)

### C. Publications

#### 1. Pioneering work

Early efforts on using computers for physics education are interesting not least because of the creativity of those working with very limited hardware and the visionary ideas they put forward.

- 136.
“

Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,”A.Goldberg,H. M.Schey, andJ. L.Schwartz, Am. J. Phys. 35, 177–186(1967). Early example of using computer graphics to visualize 1D solutions of the time-dependent Schrödinger equation. (I) - 137.Computers in Undergraduate Science Education Conference Proceedings, edited by R. Blum,Commission on College Physics (1970). Report of a conference sponsored by the Commission on College Physics that collects many early examples of practitioners using computers in physics education. (A)
- 138.
“

Computers as an aid to increasing physical intuition,”A.Bork, Am. J. Phys. 46, 796–800(1978). Pioneering paper arguing that computation can aid students build physical intuition in ways complementary to but distinct from experiments. Importantly, Bork is concerned with how to use these materials with “all students” even “poorly motivated ones.” (E) - 139.
“

Student programming in the introductory physics course: M.U.P.P.E.T.,”E.F. Redish andJ. M.Wilson, Am. J. Phys. 61, 222–232(1993). Early modular programming environment for physics. Particularly interesting because of a detailed exposition of the rationale for integrating computation, including mimicking scientific practice. Included open-ended projects intended as a proxy for scientific research. (A)

#### 2. Policy and curricular reform

- 140.
“

Identifying features predictive of faculty integrating computation into physics courses,”N.T. Young,G.Allen,J. M.Aiken,R.Henderson, andM. D.Caballero, Phys. Rev. PER 15, 010114 (2019). Applies a ML classifier to data from a national survey of faculty in physics departments. Finds four factors that predict whether faculty integrate computation into their courses: Using computation in their research, not personally using computation, believing computation allows them to bring new physics into their course, believing computation allows them to bring new problems into their course. (A) - 141.
“

Prevalence and nature of computational instruction in undergraduate physics programs across the United States,”M.D. Caballero andL.Merner, Phys. Rev. PER 14, 020129 (2018). Large scale survey of N = 1246 people from 357 distinct institutions to study the level of computational integration in physics courses. Finds that while many faculty do use computing in some form, combining active learning with computation remains a significant opportunity. (I) - 142.
“

Numerical computations in US undergraduate physics courses,”R.G. Fuller, Comp. Sci. Eng. 8(5), 16–21 (2006). Survey of physics faculty attitudes to integrating computation into courses across the US. Finds that while some faculty are excited and committed to developing computation in their courses, others are frustrated by the lack of resources. (I) - 143.
“

Computational practices in introductory science courses,”C.FracchiollaandM.Meehan, PERC Proceedings (2021). Aimed to identify practices used by faculty across the sciences; found little consensus on baseline knowledge of computation across disciplines. (I) - 144.
“

Impact of industry experience on faculty teaching practices in STEM,”D.Hu,B. M.Zwickl, andC.Carusone, PERC Proceedings (2019). Interviews with instructors who have significant industry experience; computing and programming are key topics. (I) - 145.
“

Understanding the PICUP community of practice,”P.W. Irving andM. D.Caballero, PERC Proceedings (2017). Studies evolution of PICUP and the roles that participants can engage in through interviews. (I) - 146.
“

Preliminary efforts to evaluate an initiative introducing computation across the undergraduate physics curriculum,”A.GavrinandG.Vemuri, PERC Proceedings (2021). Early report of a project to develop an assessment instrument to measure students' attitudes and self-efficacy towards computation over the course of their degree. (A) - 147.
“

The difficulties associated with integrating computation into undergraduate physics,”A.Leary,P. W.Irving, andM. D.Caballero, PERC Proceedings (2018). Identifies a number of challenges experienced by faculty seeking to integrate computation. (I) - 148.
“

Characterizing analytical and computational mathematics use during PhD research,”B.Zwickl,K.Chen,J.Deslongchamps,A.LeakandK.Martin, PERC Proceedings (2017). Snapshot of some ways in which PhD students use computation to advance their research. (I) - 149.
“

Challenges of preparing secondary STEM pre-service teachers in computational thinking,”H. C.Sabo,T. O. B.Odden, andT. F.Gregers, PERC Proceedings (2022). Interview based study of secondary pre-service teachers. Participants found it challenging to translate computational knowledge and computational thinking into their practice of teaching.

#### 3. Modeling and practices

*Computational modeling*—using computers to build physics models—has been an important thread running through PER. See also Ref. 40. Another theme running through some of these studies is the idea of *practices*, things physicists *do* as they create physics knowledge, and hence I combine the two in this section. See also Ref. 74 for a practices based approach to a computational physics course.

- 150.
“

Mapping university students' epistemic framing of computational physics using network analysis,”M.Bodin, Phys. Rev. PER 8, 010115 (2012). Students investigate a numerical particle-spring model. Uses interviews and a network analysis method to compare students' epistemic framing before and after the task. Shows that students change their epistemic framing from a modeling task to a physics task. (A) - 151.
“

Physics students' performance using computational modeling activities to improve kinematics graphs interpretation,”I. S.Araujo,E. A.Veit, andM. A.Moreira, Comp. Ed. 50, 1128–1140(2008). Examines student performance with complementary computational modeling activities to improve physics learning using the Modellus software. (I) - 152.
“

Physicality, modeling and making in a computational physics class,”A. M.Phillips,E. J.Gouvea,B. E.Gravel,P.-H.BeaucheminandT. J.Atherton, e-print arXiv:2203.04134 (2022). https://arxiv.org/abs/2203.04134 Provides a theoretical analysis of computational modelling in physics and operationalizes this in an educational environment, computational making, that centers practices and facilitates connections between experiment and modeling. (I) - 153.
“

C2STEM: A system for synergistic learning of physics and computational thinking,”N.M. Hutchins,G.Biswas, Miklós Maróti, Ákos Lédeczi,S.Grover,R.Wolf,K.PilnerBlair, D Chin,L.Conlin,S.Basu, and Kevin McElhaney. J. Sci. Ed. Tech. 29, 83–100(2020). Describes a high school learning environment for model building. Approach and theory are particularly interesting. (A) - 154.
“

Conceptual understanding of Newtonian dynamics in a comparative study of computational modeling and video motion analysis,”J.WeberandT.Wilhelm, PERC Proceedings (2021). Finds that both computational modeling and video motion improve high school students' learning of mechanics according to the FCI. (I) - 155.
“

Programming and its affordances for physics education: A social semiotic and variation theory approach to learning physics,”K.Svensson,U.Eriksson, and A.-M. Pendrill, Phys. Rev. PER 16, 010127 (2020). Qualitative study of high school students using python and processing to model phenomena. Very rich use of theoretical frameworks including social semiotics for investigating group meaning making, and affordances, what different objects offer a person who interacts with them. These are used to illuminate the interplay between coding, interacting with the simulation and visualization. (A) - 156.
“

A framework for understanding physics students' computational modeling practices,”B.R. Lunk, PhD thesis,North Carolina State University(2012). Very interesting exploration of computational modeling in the Matter and Interactions curriculum; numerous ideas around resources, programming and group work that could be applied more broadly. (I) - 157.
“

Using conceptual blending to model how we interpret computational models,”B.R. Lunk, PERC Proceedings (2019). Sees code as a representation of a physical system; unpacks a single students' reading of a code. (A) - 158.
“

Student performance in computer modeling and problem solving in a modern introductory physics course,”M. A.Kohlmyer, PhD thesis,Carnegie-Mellon University(2005). Early exploration of computational modeling in Matter and Interactions. (I) - 159.
“

Model-based inquiry vs. traditional computer simulation-based instruction: Which can better help students construct the quantum-mechanical model of an atom?,”T.Yuksel,N. S.Rebello, andLynn A.Bryan, PERC Proceedings (2017). Small but interesting study showing that a model-based learning environment including simulations can help students construct a quantum-mechanical model of the atom. (I) - 160.
“

Scientific practices in minimally working programs,”D.P. Oleynik andP. W.Irving, PERC Proceedings (2019). Examines students working in*P*^{3}(see Ref. 50) to modify existing codes in introductory physics through a practices and task analysis framework. (I) - 161.
“

How do you eat an elephant? How problem solving informs computational instruction in high school physics,”J.Willison,J.Christensen,S.Byun,D.Stroupe, andM. D.Caballero, PERC Proceedings (2022). Case study of a high school teacher participating in the integrating computation in science across Michigan program. Reports on the ways this teacher integrated computation in their classroom, along with the challenges they faced. (I)

#### 4. Programming and code

There are a few very interesting examples of deep analysis of students coding.

- 162.
“

Comparing student conceptions and construction of while loops in modeling motion,”G.Mackessy,P. W.Irving,M. D.Caballero, andL.Doughty, PERC Proceedings (2021). Deep dive into connections between programming and disciplinary thinking in the context of a simple integration loop to simulate motion. (A) - 163.
“

Programming as a semiotic system to support physics students' construction of meaning: A pilot study,”K.Svensson,U.Eriksson, A.-M. Pendrill, andL.Ouattara, J. Phys. Conf. Ser. 1512, 012026 (2020). Qualitative study that interprets programming as a way of communicating disciplinary knowledge with video analysis of student work. (I) - 164.
“

Student predictions of functional but incomplete example programs in introductory calculus-based physics,” S.Weatherford andR.Chabay, PERC Proceedings (2012). Analysis of a minimal working program design in Matter & Interactions. (I) - 165.
“

Coding expediently: A computationally situated epistemic game,”A.Anderson,P.Pressler, andW. B.Lane, PERC Proceedings (2022). Observational study of students programming in python. Interprets their decision through the lens of “epistemic games” to contrast two students of differing backgrounds learning python. (I)

#### 5. Support mechanisms

These works consider how students work together with one another as well as with computers and support mechanisms that may facilitate that.

- 166.
“

The nature of collaborations on programming assignments in introductory physics courses: A case study,”G.KortemeyerandA. F.Kortemeyer, Eur. J. Phys. 39, 055705 (2018). Network analysis of student collaboration on programming assignments. Finds collaboration networks are very extensive, while copying large sections of code was rare. Potentially important implications for design of activities and assessment. (A) - 167.
“

Communities of practice as a curriculum design theory in an introductory physics class for engineers,”P.W. Irving,D.McPadden, andM. D.Caballero, Phys. Rev. PER 16, 020143 (2020). Describes the role of communities of practice in shaping the*P*^{3}course (see Ref. 50). (I) - 168.
“

Student representations of computation in the physics community,”W.B. Lane andC.Headley, e-print arXiv:2112.05581 (2021), https://arxiv.org/abs/2112.05581 Uses the communities of practice framework to understand how students represent the use of computation in their mental models of the global physics community. (I) - 169.
“

Learning assistant approaches to teaching computational physics problems in a problem-based learning course,”A.Pawlak,P. W.Irving, andM. D.Caballero, Phys. Rev. PER 16, 010139 (2020). Learning assistants are valuable in computation physics settings but may have different perceptions of computation than faculty. (I) - 170.
“

Comparison of student and instructor reasons for using computation,”W.B. Lane andC.Headley, PERC Proceedings (2021). Finds students and instructors identify many of the same reasons to use computation, but instructors view this more broadly and use examples from their research. (I) - 171.“Investigating peer and teaching assistant interactions of physics students working on computational coding,”Z.Rowatt,R.Rosenblatt, andR.Zich, PERC Proceedings (2017). Video analysis of a computational lab. Interesting ramifications for how TAs can productively engage with students. Collaborative groupwork is also suggested as a solution. (I)
- 172.
“

A case study: Novel group interactions through introductory computational physics,”M.J. Obsniuk,P. W.Irving, andM. D.Caballero, PERC Proceedings (2015). Nice example of students debugging and making physical sense of a simulation in the*P*^{3}course at MSU. (I) - 173.
“

Disciplinary Inq[ee]ry in computational physics,”A.M. Phillips,E.Gouvea,B.Gravel, andT. J.Atherton, PERC Proceedings (2022). Analyzes moments from an inquiry-centered computational physics class that resonate with queer theory as a possible lens for DEIJ. Argues that disciplinary practices and inclusion can be aligned goals.

#### 6. How computation shifts thinking

A number of works have considered how computation and computational environments may change students' thinking about physics.

- 174.
“

Operationalizing relevance in physics education: Using a systems view to expand our conception of making physics relevant,”A.NairandV.Sawtelle, Phys. Rev. PER 15, 020121 (2019). Explores how students transfer knowledge between courses with a detailed example of a student and physics, computational modeling, and biochemistry. (I) - 175.
“

The effect of computer science on physics learning in a computational science environment,”R.Taub,M.Armoni,E.Bagno, and M. Ben-Ari, Comp. Ed. 87, 10–23(2015). Qualitative study that explores ways in which high school students programming simulations affected the students' conceptual understanding of physics. (I) - 176.
“

Factors promoting engaged exploration with computer simulations,”N.S. Podolefsky,K. K.Perkins, andW. K.Adams, Phys. Rev. ST PER 6, 020117–020128(2010). Qualitative study of students exploring the wave interference PhET simulation (see Ref. 18). (I) - 177.
“

An analysis of the ability to apply student problem solving concepts and algorithms in online and offline learning systems based on realistic mathematical approaches,”T.N. Sipayung,S. D.Simanjuntak,A.Wijayaet al, J. Phys. Conf. Ser. 1836, 012053 (2021). Compares problem solving between two groups, one in an online and the other an offline environment. Finds that students in the offline environment performed better on a test because of group discussion and the ability of the instructor to respond to cues about learning. However, the online environment facilitated re-watching of materials. Obvious implications for the design of digital environments to overcome these limitations. (I) - 178.
“

Attitudes of life science majors towards computational modeling in introductory physics,”B.R. Lunk andR. J.Beichner, PERC Proceedings (2016). Suggests life-science majors may be more receptive to spreadsheet-based approaches. (I) - 179.
“

How computation can facilitate sensemaking about physics: A case study,”O.P. Sand,T. O.B. Odden,C.Lindstrom, andM. D.Caballero, PERC Proceedings (2018). Neat qualitative studying following a single student's sensemaking of a computational model of radioactive decay. (I) - 180.
“

Examining thematic variation in a phenomenographical study on computational physics,”N.Hawkins,M. J.Obsniuk,P. W.Irving, andM. D.Caballero, PERC Proceedings (2017). Examines student responses with the theme “Computation Helps to Learn Physics” in the*P*^{3}course at MSU. (I) - 181.
“

How do previous coding experiences influence undergraduate physics students?,”J.N. Bumler,P. C.Hamerski,M. D.Caballero, andP. W.Irving, PERC Proceedings (2019). Examines how the EMP-Cubed activities changed students' coding identities relative to prior coding experiences including competence, performance, and interest factors. (I) - 182.
“

Students' development of a logarithm function in Python using taylor expansions: A teaching design case study,” O. P. Sand,E.Lockwood,M. D.Caballeroand K. MÃžrken, Digital Experiences Mathematics Educ. 8, 213–255(2022). Interesting example from math education of the iterative design of a tutorial where students implement their own logarithm function with detailed analysis of student work. (I)

### D. Theoretical perspectives

In this section, I present a number of theoretical perspectives on computational education emerging from outside PER as well as examples of how they are being used in various ways by physics education researchers. Theory works both as a lenses for design and to evaluate learning, but also organizes communities of researchers working on a common endeavor. I include these perspectives, therefore, because they greatly enrich the space of design possibilities for what *could* occur in a physics classroom, and also provide powerful lenses to help interpret what *does* occur. I note that the perspectives I have chosen to present have considerable overlap with one another both in their content and their associated communities. Indeed, significant interactions have historically occurred between their various proponents.

#### 1. Constructionism

Constructionism is a pedagogical approach originally articulated by Seymour Papert that involves students creating things, including both digital artifacts like code and documentation, as well as physical objects. Grounded in but distinct from Piaget's *constructivism*, a theoretical framework for pedagogy that argues learners construct knowledge rather than simply absorb information. Extremely influential in the teaching of computing, and with important implications for equity,^{186} I note it here because while a number of physics efforts (VPython, PhET) absorb some of its ideas around virtual environments, other aspects remain under-explored. Indeed, despite constructionism's considerable influence on computing education, surprisingly few studies within PER explicitly connect themselves to constructionist ideas (a few that do: Refs. 18, 67, and 192).

- 183.
Mindstorms Children, Computers, and Powerful Ideas,

S.Papert(Basic Books, New York, 2020). Highly influential book originally published in 1980 on the possibilities afforded by computational environments in education. Grounded in Piaget's theory of learning. Focuses on the LOGO language. (E) - 184.
“

The significance of constructionism as a distinctive pedagogy,”D.Laurillard, in Constructionism 2020 Conf. Proc. (2020), p. 29. A very helpful contemporary elucidation of constructionist ideas; the conference proceedings represents a wealth of constructionist approaches to many areas. (E) - 185.
Constructionism: Research reports and Essays, edited by

I.HarelandS.Papert(Ablex Publishing,New York, 1991). Collection of constructionist literature addressing both theoretical foundations and realizations. (A) - 186.
“

Epistemological pluralism and the revaluation of the concrete,”S.TurkleandS.Papert, J. Math. Behavior 11, 3–33(1992). Considers implications of constructionist design for equity. (I) - 187.
“

Dynaturtle revisited: Learning physics through collaborative design of a computer model,”B.Sherin,A.diSessa, andD.Hammer, Int. L. Env. 3, 91–118(2006). Advocates collaborative design as an activity to support student inquiry in physics. (A) - 188.
“

Rethinking the microworld idea,”C.Hoyles,R.Noss, andR.Adamson, J. Ed. Comp. Res. 27, 29–53(2002). Exploration of virtual environments. (A)

#### 2. Computational literacy

Emerging from the work of Andrea DiSessa, who theorized computation as a new literacy, computational literacy has gained traction in the PER community. Computational literacy is particularly interesting because, like constructionism, it pays significant attention to social processes such as communication and group interactions. See also Refs. 47 and 192.

- 189.
**Computational Literacy and “The Big Picture” Concerning Computers in Mathematics Education**,A. A.DiSessa,Theorizes computation as a new type of literacy with examples from students learning motion.Interesting material on the limitations of other theoretical perspectives such as computational thinking. (E) - 190.
Changing Minds: Computers, Learning, and Literacy,

A. A.DiSessa(MIT Press,Massachusetts, 2000). Envisions computing as the basis for a new literacy that fundamentally changes how students think and learn. (E) - 191.
“

Boxer: A reconstructible computational medium,”A.A. diSessa andH.Abelson, Comms. ACM 2, 859–868(1986). Pioneering programmable computational environment that emphasized a spatial metaphor, making elements of the environment (in effect, objects) visible as concrete “boxes.” Compare with Smalltalk and LOGO. (I) - 192.
“

Physics computational literacy: An exploratory case study using computational essays,”T.O. B. Odden,E.Lockwood, andM. D.Caballero, Phys. Rev. PER 15, 020152 (2019). Implements and analyzes DiSessa's idea of a computational “essay”: a notebook that combines code, text, and figures to communicate an idea. Extensively theorizes computational literacy as a framework for undergraduate physics. (I) - 193.Using computational essays to scaffold professional physics practice,T. O.B. Odden andA.Malthe-Sørenssen, Eur. J. Phys. 42, 015701 (2021). Detailed study connecting computational literacy with practices. (A)

#### 3. Computational thinking

The framework of Computational Thinking has become very popular in the STEM and CS education literature in recent years. While it captures only part of physics computational practice, a significant body of work is emerging that is highly relevant to teaching computation in physics. See also the report in Ref. 4.

- 194.
“

Computational thinking in introductory physics,”C. M.OrbanandR. M.Teeling-Smith, Phys. Teach. 58, 247–251(2020). Highly readable introductory paper firmly grounded in a physics context. Describes CT through four components: Dealing with data, modelling/simulating, problem-solving via computers, and thinking about a complex system the systems level. (E) - 195.
“

On computational thinking and STEM education,”Y.Li,A. H.Schoenfeld,A. A.diSessa,A. C.Graesser,L. C.Benson,L. D.English, andR. A.Duschl, J.STEM Ed. Res. 3, 147–166(2020). Extensive review article focusing on efforts to integrate computational thinking into disciplinary education. (I) - 196.
“

Defining computational thinking for mathematics and science classrooms,”D.Weintrop,E.Beheshti,M.Horn,K.Orton,K.Jona,L.Trouille, andU.Wilensky, J. Sci. Educ. Technol. 25, 127–147(2016). Defines computational thinking through practices, including data practices, modeling and simulation practices, problem solving practices, and systems thinking. (I) - 197.
“

Computational thinking,”J.Wing, Comms. ACM 49, 33–35. (2006). Highly influential opinion piece that argues computational thinking is a fundamental skill for everyone, not just for computer scientists. (E) - 198.
“

Computational thinking from a disciplinary perspective: Integrating computational thinking in K-12 science, technology, engineering, and mathematics education,”I.Lee,S.Grover,F.Martin, Sarita Pillai, andJ.Malyn-Smith, J. Sci. Ed. Tech. 29, 1–8(2020). Introduction to a special issue of*J. Sci. Ed. Tech.*on computational thinking. A good editorial and overview. (I) - 199.
“

Missing in measurement: Why identifying learning in integrated domains is so hard,”W.W. Bortz,A.Gautam,D.Tatar, andK.Lipscomb, J. Sci. Ed. Tech. 29, 120–135(2020). Argues that assessing computational thinking in disciplinary contexts inherently requires mixed-methods approaches. (A) - 200.
“

Computational thinking in higher education: A review of the literature,”J.A. Lyon andA. J.Magana, Comp. Appl. Eng. Ed. 28, 1174–1189(2020). Valuable literature review for applications to higher education. (I) - 201.
“

A conceptual framework for computational pedagogy in STEAM education: Determinants and perspectives,”S.Psycharis,K.Kalovrektis, andA.Xenakis, Hellenic J. STEM Ed. 1, 17–23(2020). Considers connections between computational thinking and other STEM practices including computational experiments. (I) - 202.
“

Computational thinking, between papert and wing,”M.LodiandS.Martini, Sci. Educ. 30, 883–908(2021). Thoughtful piece that argues for a combination of computational thinking and constructionist approaches and examines the links between these ideas. (I) - 203.
“

Development and illustration of a framework for computational thinking practices in introductory physics,”D.P. Weller,T. E.Bott,M. D.Caballero, andP. W.Irving, Phys. Rev. Phys. Educ. Res. 18, 020106 (2022). Identifies a number of practices for computational thinking from a literature review and physics classroom data. (A) - 204.
“

Examining how problem design relates to computational thinking practices,”T.E. Bott,T.Stump,M. D.Caballero,D. R.McPadden, andP. W.Irving, PERC Proceedings (2022). Reports a task analysis to examine how computational thinking practices relate to specific design features of an in-class problem. Ramifications for design of material provided for students. (A)

## VI. TOOLS, LANGUAGES, AND ENVIRONMENTS

### A. Languages

Languages facilitate human expression of tasks for the computer to perform. There is a wide and ever growing selection of languages that vary in the objectives of their design, and many have been used successfully for scientific computing. While scientists may develop certain language preferences based on their own experience and objectives, it is worth learning about the available possibilities. A number of new languages have emerged and become popular since CP-1 and CP-2, and I hope to provide an updated perspective. To that end, in this section, I compare a number of popular languages and offer suggestions for emerging or lesser-known but interesting languages. For advice and examples of how to incorporate particular languages into a course, the reader is advised to consult a supportive community, e.g., Ref. 14.

Like their human counterparts, computer languages evade a precise taxonomy, but a number of valuable considerations that assist in selecting one for a particular application are as follows:

**Purpose.** Some languages are *general purpose*, intended for a broad range of applications; others are *domain specific*, intended to express certain tasks easily (text processing or querying a database are common examples) or to solve a particular type of problem.

**Performance.** Languages fall into several performance categories. The languages with fastest performance are *compiled* to machine code that can be executed in hardware by the processor. In contrast, some are *interpreted* from the source code directly; the act of interpreting incurs a performance overhead so these languages are typically slower. Between these categories are some intermediate cases: Some languages are compiled to an intermediate representation (often referred to as a *bytecode*) for faster execution and portability; some may be converted to machine code before or during execution. (These are referred to as Just-In-Time or *JIT* compiled languages.) High quality JITs achieve performance close to compiled languages. Nonetheless, the above are all broad categories: Different compilers or interpreters may yield significantly different performance even on the same program, for example. Similarly, particular language features may incur penalties and the efficiency of language implementations differs significantly. I remind the reader that, very often, the time spent programming and debugging a program exceeds the time that the computer spends running it! Programming in a slower language that facilitates writing correct code quickly is quite a rational decision, which explains the popularity of languages like python.

**Programming paradigm.** Languages fit into families based on their syntax and constructs they provide. *Procedural* languages provide functions, loops, etc., *Object-oriented* languages enable greater modularity, ease of design and code reuse by organizing data and *methods* that act on the data into packages called objects. *Functional programming* languages centralize functions and lack things like explicit control flow; these have been historically very important in computer science but have been relatively underutilized in physics.

**Typing.** Programs work on different *types* of data including integers, floating point numbers, strings, composite structures, collections, etc. In a *strongly* typed language, the user must specify type information explicitly so that the compiler can check for conflicts or illegal conversions that are often caused by programming errors. *Weakly* or *dynamically* typed languages do not require this information, gaining some flexibility at the expense of losing out on such automated verification.

Table I compares a number of programming languages popular in physics.

Language . | Purpose . | Performance . | Paradigm . | Typing . | Physics use case . |
---|---|---|---|---|---|

Python | General | Bytecode | Multi | Weak | General purpose; data analysis; Simulation; Visualization; Machine Learning and Data Science. |

C | General | Compiled | Procedural | Strong | Implementation of fast specialized codes. |

C++ | General | Compiled | Object-oriented | Strong | Implementation of large, complex codes. |

FORTRAN | Numerical computing | Compiled | Procedural | Strong | Leveraging existing numerical libraries or legacy codes. |

Java | General | JIT | Object-oriented | Strong | Implementation of cross-platform programs with graphical components. |

Matlab/Octave | Numerical computing | Interpreted | Procedural | Weak | Matrices; Prototyping; Visualization; Applications leveraging new algorithms from Applied Math |

Mathematica/Maple | Symbolic and numerical computing | Interpreted | Multi | Weak | Visualization; prototyping; symbolic manipulation; analytical calculations; computer algebra |

R | Statistics | Interpreted | Object-oriented | Weak | Statistical analysis of data |

Julia | Numerical computing | JIT | Multi | Weak | Visualization and prototyping but with high performance; significant interest from Applied Math and CS community. |

Language . | Purpose . | Performance . | Paradigm . | Typing . | Physics use case . |
---|---|---|---|---|---|

Python | General | Bytecode | Multi | Weak | General purpose; data analysis; Simulation; Visualization; Machine Learning and Data Science. |

C | General | Compiled | Procedural | Strong | Implementation of fast specialized codes. |

C++ | General | Compiled | Object-oriented | Strong | Implementation of large, complex codes. |

FORTRAN | Numerical computing | Compiled | Procedural | Strong | Leveraging existing numerical libraries or legacy codes. |

Java | General | JIT | Object-oriented | Strong | Implementation of cross-platform programs with graphical components. |

Matlab/Octave | Numerical computing | Interpreted | Procedural | Weak | Matrices; Prototyping; Visualization; Applications leveraging new algorithms from Applied Math |

Mathematica/Maple | Symbolic and numerical computing | Interpreted | Multi | Weak | Visualization; prototyping; symbolic manipulation; analytical calculations; computer algebra |

R | Statistics | Interpreted | Object-oriented | Weak | Statistical analysis of data |

Julia | Numerical computing | JIT | Multi | Weak | Visualization and prototyping but with high performance; significant interest from Applied Math and CS community. |

- 205.
**python**has become one of the most popular languages for scientific work especially data analysis. Dynamic language features provide ease programming. syntax incorporates indentation to enforce tidy code. Very rich collection of libraries. Environments like Jupyter^{252}provide a free alternative to commercial ones like matlab. High performance implementations like PyPy provide efficient execution https://www.pypy.org (E) - 206.
**c**is a concise language for systems programming that provides excellent performance. Clean and simple syntax has inspired numerous other languages including c++, java, javascript, etc. The standard textbook C Programming Language, 2nd ed. (Pearson,London, 1988),B. W.KernighanandD. M.Ritchieis a classic exemplar of good technical documentation. (A) - 207.
**c++**is an object-oriented extension of c that shares excellent performance. Strongly typed, but templates offer a way to express code that works on more than one type or class without repeating the entire code for each type. Popular for implementing low-level or performance sensitive code. (A) - 208.
**fortran.**One of the oldest languages still in common use (the other is LISP) and historically very important for numerical work for performance reasons. Modern versions incorporate features like OOP. Many legacy codes are in FORTRAN and a good deal of libraries for numerical computing are available. It is not necessary to program in FORTRAN to use these libraries; modern compilers link code across languages quite well. (A) - 209.
**java**. Popularized by the emergence of the web, java provides very convenient cross-platform operation; efficient JITs are available for reasonable performance. Object oriented but easier to program than c++ by avoiding the complexity of pointers. Has rather declined in popularity recently. (I) - 210.
**julia**. A relatively new language that offers the convenience of a dynamic language but with excellent performance comparable to compiled languages. Supports multiple dispatch, whereby multiple copies of a method that act on different data types can be defined; the correct method is selected at runtime. Increasingly popular in the applied math and data science community; one to watch. https://julialang.org (I) - 211.
**matlab.**Proprietary language and environment oriented to numerical work, especially matrix manipulations. Very extensive libraries for numerics and visualization capabilities. Very popular in the academic applied math community and hence a great deal of code available. https://www.mathworks.com/. An annual “Teaching computation with Matlab” conference is sponsored by Mathworks: https://serc.carleton.edu/teaching_computation/index.html (E) - 212.
**octave**.Free and largely compatible alternative to matlab.https://www.gnu.org/software/octave/index (I) - 213.
**mathematica**. Proprietary language and environment that provides a notebook environment for symbolic and numerical computation. Very extensive library of routines across all domains of mathematics. Includes powerful integrated visualization capabilities. https://www.wolfram.com/mathematica/ (E) - 214.
**R**is oriented to statistical computing, data manipulation, calculation and graphical display. https://www.r-project.org. See also RStudio, the recommended development environment.^{256}(E)

#### 2. Typesetting, markup, documentation

- 215.
**html**. The language of the world wide web. The latest version HTML5 incorporates significant features for interactivity. (E) - 216.
**markdown**.Lightweight language for documentation; I recommend following the CommonMark specificationhttps://commonmark.org. See also**reStructuredText**which is popular for python. https://docutils.sourceforge.io/rst.html (E) - 217.
**latex**. Computer typesetting system that is very popular for scientific publishing. See also**Overleaf**, a commercial web based collaborative authoring tool https://www.overleaf.com; LyX, a graphic front-end for writing latex: https://www.lyx.org (I)

#### 2. Data

- 218.
**xml**. Standardized interchange format emerging from html and electronic publishing. https://www.w3.org/XML/ (I) - 219.
**json**. Lightweight and easy to parse language for interchange of structured data. https://www.json.org/json-en.html (E) - 220.
**sql.**Ubiquitous language for querying databases. https://www.w3schools.com/sql/ (A)

#### 3. Other interesting languages

- 221.
**rust.**New language that provides exceptional performance while eliminating memory management bugs common in other high performance languages. https://www.rust-lang.org (A) - 222.
**Haskell.**Purely functional language with lazy evaluation. Rich tools for formal verification/provably correct programming. https://www.haskell.org (A) - 223.
**Processing**.Easy to learn and use. Facilitates creation of visual and interactive programs. Widely used in education and visual arts. https://processing.org (E) - 224.
**Lisp.**Functional language with minimal syntax; code and data intermix freely. As old as fortran and historically popular in CS and especially artificial intelligence. See also Scheme which is easier to learn and**Clojure**a modern derivative. (A) - 225.
**javascript**is the language of interactivity on the web; ubiquitously available via web browsers. Particularly valuable for deployment of physics education tools. A common transpilation target for other languages. (A) - 226.
**Popular languages beyond physics: Go**is an open source C-family language by Google with memory safety, garbage collection, structural typing, and support for concurrency;**Swift**is Apple's language of macOS and iOS development. (A) - 227.
**Scripting languages: Lua**is a language intended to be embedded in applications to promote extensibility;**Wren**is a recent and elegant alternative.**Ruby**is an object-oriented language designed with an emphasis on programming productivity and simplicity.**Morpho**is an object-oriented language oriented to scientific applications. (A) - 228.
**Influential languages of historical interest: Smalltalk**, important object-oriented language intended for constructionist education (see Sec. V D 1);**Self**, a dynamic prototype-based object-oriented language that popularized many important features especially JITs. (A) - 229.
**Esoteric languages that call into question what constitutes programming: Befunge**arranges instructions in 2D;**Piet**programs appear as abstract art. (I)

#### 4. Parallel computing

A significant trend over the last couple of decades has been the replacement of CPUs with single compute units (“cores”) with multi-core machines. As of writing, 8 or 12 core processors are ubiquitous and as many as 64 core devices emerging. These advances have been driven by the need for higher performance with manufacturers increasingly limited by their ability to reduce the feature size of elements in the processor and also to limit energy consumption: In applications like cellphones, some cores are less capable but consume less power, while others are brought into action when higher performance is needed. To exploit the capabilities of these computers, it is necessary to use parallel programming techniques.

- 230.
**Message Passing Interface**(MPI). Relatively low-level framework for parallel computing. Ubiquitous in high performance computing. (A) - 231.
**OpenMP.**https://www.openmp.org Relatively high-level API that provides a portable implementation of multithreading, supporting shared-memory multiprocessing programming. (A) - 232.
**Pthreads**.Very low-level multithreading API for C. (A) - 233.
**ZeroMQ**.Surprisingly easy to use (in relative terms) low level library to implement distributed applications. https://zeromq.org (A) - 234.
**python libraries for parallel processing**. The above solutions have python wrappers; there are other convenient libraries listed at: https://wiki.python.org/moin/parallelprocessing (A)

### B. Visualization and image/data processing

- 235.
**VTK**.Open source library for manipulating and displaying scientific data including 3D and GUI widgets. https://vtk.org (I) - 236.
**Paraview**.Open-source, multi-platform data analysis and visualization application, particularly for PDE solutions. Suitable for large datasets. https://www.paraview.org (A) - 237.
**Visual Molecular Dynamics (VMD)**.Molecular dynamics visualization program. https://www.ks.uiuc.edu/Research/vmd/ (I) - 238.
**UCSF Chimera**.Interactive visualization and analysis of molecular structures and related data. https://www.cgl.ucsf.edu/chimera/ (I) - 239.
**GNUplot**.Convenient command line plotting and visualization application also used by other projects. http://www.gnuplot.info (I) - 240.
**Gephi**.**Visualization of graphs and networks**. https://gephi.org (I) - 241.
**VisIt**.Scalable visualization and analysis program particularly useful for very large datasets. https://visit-dav.github.io/visit-website/ (A) - 242.
**ImageJ**.Open source image processing package. https://imagej.net A version with many additional plugins and features included by default called**Fiji**is highly recommended: https://imagej.net/software/fiji/ (E) - 243.
**Root**.Open-source data analysis framework for very large datasets used by high energy physics. https://root.cern (A)

#### 1. Tools to support programming

- 244.
**Git.**Open source version control system. https://git-scm.com. Widely used public host servers include**Github**https://github.com and**Bitbucket**https://bitbucket.org/product. These services are generally free for academic open source projects with subscription services for a fee. (A) - 245.
**Slack**is a very widely used platform for group conversations. https://slack.com (E) - 246.
**Documentation**.**Sphinx**assists automatically generating documentation, originally developed for**python**https://www.sphinx-doc.org/en/master/;**Doxygen**is an alternative for C family language https://www.doxygen.nl;**Docutils**uses the reStructuredText format https://docutils.sourceforge.io.**Read the Docs**hosts documentation for open source software. https://readthedocs.org (A) - 247.
**Valgrind**. Powerful profiling tool that helps diagnose and solve memory errors, particularly in c. https://valgrind.org (A)

#### 2. IDEs and programming environments

- 248.
**Visual Studio Code**. Extensible text editor and IDE similar in spirit to Atom. Many languages. https://code.visualstudio.com (E) - 249.
**Textmate**.Highly customizable text editor for macOS. Many languages. https://macromates.com (E) - 250.
**XCode.**Apple's proprietary but extremely usable IDE for C family languages and macOS/iOS: https://developer.apple.com/xcode/ (A) - 251.
**Codelite**.Open source IDE for c, c++, javascript, etc. https://codelite.org (A) - 252.
**Jupyter**.Browser based interactive notebook environment for python.I strongly recommend installing it through the very comprehensive Anaconda collection.^{291}**JupyterLab**is the emerging next-generation version with an interface more like a conventional IDE. See https://jupyter.org (E) - 253.
**Spyder**. Scientific environment for python https://www.spyder-ide.org (A) - 254.
**Colaboratory.**Google's cloud-based python jupyter environment. (E) https://colab.research.google.com - 255.
**Eclipse.**Popular IDE for java. https://www.eclipse.org/eclipseide/ (A) - 256.
**RStudio.**IDE for the R language. https://www.rstudio.com/products/rstudio/ (I)

### C. Libraries

- 257.
**Netlib**. Large collection of collection of mathematical software, papers, and databases. https://www.netlib.org (A) - 258.
**LAPACK**. Dense linear algebra. Legacy fortran API can be challenging; the**LAPACKE**wrapper facilitates calls from c/c++. Part of Ref. 257. (A) - 259.
**Suitesparse**. Sparse matrix library. Also provides some graph algorithms. https://people.engr.tamu.edu/davis/suitesparse.html (A) - 260.
**NAG**. Extensive but proprietary library of mathematical and statistical algorithms. https://www.nag.com (A) - 261.
**NumPy**. Fundamental package for numerical computing in python. https://numpy.org (I) - 262.
**SciPy**. Provides many numerical algorithms for python. https://scipy.org (I) - 263.
**GNU Scientific Library (GSL)**.Wide range of numerical routines for c and c++ programmers. Free software under GNU public license (GPL). https://www.gnu.org/software/gsl/ (A) - 264.
**CGAL.**Algorithms for computational geometry. https://www.cgal.org (A) - 265.
**PetSc**.Portable, extensible toolkit for scientific computation provides parallel linear and nonlinear equation solvers and ODE integrators for use in large-scale application projects. https://petsc.org/release/ (A) - 266.
**SLEPc.**Scalable library for eigenvalue problem computations supports solution of large scale sparse eigenvalue problems on parallel computers. https://slepc.upv.es (A) - 267.
**FFTW**.Compute the discrete fourier transform in one or more dimensions. https://www.fftw.org (A) - 268.
**Gmsh.**Open source 3D finite element mesh generator. https://gmsh.info (A) - 269.
**Fenics**.Open-source computing platform for solving partial differential equations (PDEs).High level interface makes it easy to get started. https://fenicsproject.org The accompanying book is very clear: Automated Solution of Differential Equations by the Finite Element Method, edited byA.Logg,K.-A.Mardal, andG. N.Wells(Springer,Berlin, 2012). (A) - 270.
**Firedrake.**Open source platform for PDEs. Similar in goals to Fenics—they use the same interface—but more application-focused. https://www.firedrakeproject.org (A)

## VII. EMERGING TOPICS

I present in this section a number of topics that have emerged in the past decade that physics educators and practitioners working in computation should at least be aware of. Not all of these topics have yet been integrated into even advanced courses, and so they represent rich opportunities for instructional innovation. Many of these will significantly alter how computers are used in physics in the future and how computation in physics is taught.

### A. Diversity, equity, inclusion, and justice (DEIJ)

The continued success of the physics enterprise requires our attention to the equitable participation of various historically marginalized groups. Also referred to as EDI or DEI, there are a number of efforts to improve the climate and participation of these groups. Connecting these efforts to work on computing, which has a specific and different history (though no less problematic) with regard to marginalization than physics, is an important contemporary topic; see 186 for an exemplar earlier effort. Here, I present a few resources for those who wish to educate themselves about these issues.

- 271.
“

Resource Letter: GP-1: Gender and physics,”J.Blue,A.Traxler, andG.Cochran, Am. J. Phys. 87, 616–626(2019). Overview of issues which influence the representation and participation of women in physics. (E) - 272.
“

Resource Letter RP-1: Race and physics,” Katemari Rosa,J.Blue,S.Hyater-Adams,G. L.Cochran, andC.Prescod-Weinstein, Am. J. Phys. 89, 751–768(2021). Introduces theory, frameworks, research, and historical information that connect the experiences of people from ethnic/racial groups that have been minoritized and marginalized in science with a focus on physics. (E) - 273.
LGBT+ Climate in Physics (

American Physical Society,Maryland, 2016),T. J.Atherton,R. S.Barthelemy,W.Deconinck,M. L.Falk,S.Garmon,E.Long,M.Plisch,E. H.Simmons, andK.Reeves. Report prepared for the APS on LGBT+ (Lesbian, Gay, Bisexual, Transgender and other gender and sexual minorities) climate in physics. Includes recommendations directed at the APS. See also “LGBT+ Inclusivity in physics and astronomy: A best practices guide,”N.Ackermanet al which contains further advice: https://arxiv.org/pdf/1804.08406.pdf and Ref. 173 for emerging work in PER. (E) - 274.
“

Sex and gender as non-binary: What does this mean for physics teachers?,” A. Traxler andJ.Blue, Phys. Teach. 58, 395–398(2020). Tractable implications of a sophisticated understanding of gender aimed at physics teachers. (E) - 275.
Coders: The Making of a New Tribe and the Remaking of the World,

C.Thompson(Penguin, New York, 2019). Popular account of the history and culture of programmers with particular attention to the history of gender around programming. (E) - 276.
Hidden Figures,

M. L.Shetterly(William Morrow and Co.,New York, 2016). Popular book documenting the lives and achievements of four Black women who overcame discrimination and racial segregation to become vital parts of mathematics, scientific, and engineering history. A fictionalized movie of the same name based on this book was released in 2016. (E) - 277.
“

Women's history month: Arianna Rosenbluth and the Metropolis Monte Carlo algorithm,”S.Chen, APS News 31 (2022). Article describing the unacknowledged contributions of Arianna Rosenbluth to the creation of Markov Chain Monte Carlo methods. (E) - 278.
Algorithms of Oppression,

S. U.Noble(NYU Press, New York, 2018). How negative biases against women of color are embedded in search engine results and algorithms. (E) - 279.
**APS Inclusion, Diversity, and Equity Alliance (IDEA)**. Community of transformation to empower physics departments, laboratories, and other organizations to identify and enact strategies for improving equity, diversity, and inclusion (EDI). https://www.aps.org/programs/innovation/fund/idea.cfm (E) - 280.
Increase Investment in Accessible Physics Labs: A Call to Action for the Physics Education Community (

AAPT,Maryland, 2021). Committee on Laboratories Accessible Physics Labs Task Force Report. Report arguing for improved accessibility for physics laboratories; important ramifications for accessibility of computational environments. (E) - 281.
**EQUIP.**Observation tool that teachers can program to collect data particular to their classrooms, e.g., participation of marginalized groups, or how the instructor interacts with different students. https://equip.ninja

### B. GPU computing

Three-dimensional computer graphics calculations ultimately involve the multiplication of many small matrices as a virtual model is placed within a digital world and then projected into two dimensional coordinates for display. These calculations are highly parallel: Objects in the computer are typically represented as polyhedra and each vertex can be transformed independently. To perform these calculations efficiently, most computers are equipped with special graphical processing units (GPUs). Over the last two decades these moved from fixed purpose devices permitting a limited set of operations to becoming very flexible devices that can be independently programmed. Many highly parallel operations including linear algebra, computational geometry, Monte Carlo methods, etc., can exploit these devices to provide excellent performance even on personal computers, and there has been an increasing trend to using GPU computing to accelerate solution of physics problems.

- 282.
**APIs: OpenCL**is an open standard from the Khronos consortium https://www.khronos.org/opencl/;**CUDA**(Compute Unified Device Architecture) is NVidia's proprietary API: https://developer.nvidia.com/cuda-toolkit;**Metal**is Apple's proprietary API: https://developer.apple.com/metal/. (A) - 283.
“

Accelerating molecular dynamics simulation on graphics processing units,”M. S.Friedrichset al, J Comput. Chem. 864–872(2009). Discusses implementation of molecular dynamics on GPUs. (A) - 284.
“

GPU accelerated Monte Carlo simulation of the 2D and 3D Ising model,”T.Preis,P.Virnau,W.Paul, andJ. J.Schneider, J. Comp. Phys. 228, 4468–4477(2009). Highly cited early paper on the Ising model on a GPU. (A) - 285.
“

GPU-computing in econophysics and statistical physics,”T.Preis, Eur. Phys. J. Spec. Top. 194, 87–119(2011) Pedagogical article that provides a detailed account of coding the Ising model for the GPU. (I) - 286.
“

Simulating spin models on GPU,”M.Weigel, Comput. Phys. Commun. 182, 1833 (2011) and “Performance potential for simulating spin models on GPU,” M. Weigel, J. Comput. Phys.**231**, 3064 (2012) also give information about simulating the Ising model on a GPU. (I) - 287.
“

GPU computing,”J. D.Owens,M.Houston,D.Luebke,S.Green,J. E.Stone, andJ. C.Phillips, Proc. IEEE 96, 879–899(2008). Overview of the potential of GPU computing for applications including physics. (I)

### C. Data science, machine learning, and artificial intelligence

A great deal of activity has emerged in exploiting the capabilities of powerful computers with large data sets to advance physics knowledge. A number of (overlapping) terms described these activities: **Data science** has emerged as an umbrella term for interdisciplinary activity using computing for working with data. **machine learning** (ML) refers to algorithms where the computer is not explicitly programmed to perform a task, but “learns” from a given training set of examples. **artificial intelligence** (AI) refers to construction of algorithms to perform tasks typically performed by humans.

Many of the associated methods are older than they appear. As I mention in the Introduction, they are referred to in the first Resource Letter on computational physics 1 and indeed some emerged from physics itself. Nonetheless, there is undoubted contemporary excitement around these elements, they are ever more important to physics practice *and* they are important career destinations for physics students. Tasks such as classification and processing of images are particularly well-suited to ML methods, for example. Unfortunately, efforts to integrate these methods into physics curricula are at a nascent stage. Here, therefore, I present a few selected resources to assist instructors looking to catalyze this integration. The topic certainly deserves a Resource Letter of its own. See also Ref. 36.

- 288.
“

A high-bias, low-variance introduction to Machine Learning for physicists,”P.Mehta,M.Bukov, C.-H. Wang,A.G.R. Day,C.Richardson,C. K.Fisher,D. J.Schwab, Phys. Rep. 810, 1–124(2019). Very valuable review paper that discusses many methods with physics applications. See also arXiv version https://arxiv.org/abs/1803.08823. python notebooks accompanying the article available at https://github.com/drckf/mlreview_notebooks (I) - 289.
Deep Learning and Physics, 1st ed., A Tanaka,

A.Tomiya, andK.Hashimoto(Springer, Berlin, 2021). Advanced but illuminating book that explores connections among machine learning, information theory, and physics. (A) - 290.
“

Machine-learning-assisted modeling,”E.Weinan,J.Han, andL.Zhang, Phys. Today 74(7), 36–41(2021). Readable article discussing how ML opens up new possible approaches to modeling. (E) - 291.
**Anaconda**. Very comprehensive collection of data science tools, including python, Jupyter, etc., and most commonly used libraries. https://anaconda.org (I) - 292.

https://engage.aps.org/gds/home. A unit of the American Physical Society for those working on or with Data Science methods. (I)**APS Topical Group on Data Science** - 293.
**Data Science Education Community of Practice**(DSECOP) https://www.aps.org/programs/innovation/fund/dsecop/index.cfm. APS-supported community of practice aiming to support instructors in integrating data science into physics classes. See also their Github: https://gds-education-community-of-practice.github.io/ (I) - 294.
The Elements of Statistical Learning, 2nd ed.,

T.Hastie,R.Tibshirani, andJ.Friedman(Springer, Berlin, 2009). Accessible introductory textbook covering basic machine learning methods. (I) - 295.
**Interpretable Machine Learning**,C.Molnar. Digital book that provides a valuable guide to interpreting the output of some machine learning methods. https://christophm.github.io/interpretable-ml-book/ (I)

### D. Quantum computing

Quantum computers differ from their classical counterparts in leveraging the quantum effects of superposition and entanglement to perform calculations that are challenging or intractable for classical computers. Quantum computers use qubits as the fundamental unit of information, which can be in an arbitrary superposition of states $|0\u27e9$ and $|1\u27e9$ that correspond to classical bits. While the idea of simulating quantum systems on other quantum systems is not new, there has been very significant hardware progress in the last decade with current devices having $\u223c100$ qubits. If these machines can be scaled up, they show remarkable promise across numerous computing applications. Here, I collect a few resources for those interested in learning about this exciting new field:

- 296.
“

Resource Letter QI-1: Quantum information,”Frederick W.Strauch, Am. J. Phys. 84, 495–507(2016). Relatively recent resource letter that provides an overview of the field. (E) - 297.
Quantum Information and Quantum Computation,

M. A.NielsenandI. L.Chuang(Cambridge U. P.,Cambridge,2011). A very popular text that starts with overviews of both quantum mechanics and the theory of computation, then builds to advanced topics. Appropriate for advanced undergrads in physics, computer science, or math. (I) - 298.
Quantum Computer Science: An Introduction,

N. D.Mermin(Cambridge U. P.,Cambridge,2007). Focused book on QC that provides a complementary perspective to Nielsen and Chuang. (A) - 299.
Classical and Quantum Computation,

A. Y.Kitaev,A. H.Shen, andM. N.Vyalyi(American Mathematical Society, Rhode Island, 2002). Another good complement to Nielsen and Chuang. (A) - 300.
Quantum Information Theory, 2nd ed.,

M.Wilde(Cambridge U. P.,Cambridge,2017). Approaches QC from an information-theoretic approach that is interesting but more technical than other books. (A) - 301.
Quantum Computing: An Applied Approach, 2nd ed.,

J. D.Hidary(Springer, Berlin, 2021). Very hands-on coding approach to QC. (I) - 302.
**Qubes course**,A.Coladangelo,J.Necaise,K.Setia, andE.Weis.Course aimed at teaching QC to high school students. Available at: https://qbook.qbraid.com/learn/qubes_course. Part of the Qbraid repository which is also valuable https://account.qbraid.com (I) - 303.
“

Undergraduate computational physics projects on quantum computing,”D.Candela, Am. J. Phys. 83, 688–702(2015). Very useful paper describing quantum computing projects suitable for an undergraduate audience. (E) - 304.
**Qiskit textbook**.Online and continuously updated introduction to the popular Qiskit framework for quantum computing. https://qiskit.org/textbook (I) - 305.
Introduction to Classical and Quantum Computing,

T. G.Wong(Rooted Grove, 2022). Very accessible textbook aimed at undergraduates or for self study. Requires only trigonometry. Particularly aimed at those who lack mathematical background; the only prerequisite is trigonometry. - 306.
**Lecture notes**by John Preskill http://theory.caltech.edu/~preskill/ph229/ and Thomas Vidick http://users.cms.caltech.edu/~vidick/teaching/ provide a good introduction to QC. (A) - 307.
“

Quantum computational chemistry,”S.McArdle,S.Endo,A.Aspuru-Guzik,S. C.Benjamin, andX.YuanRev. Mod. Phys. 92, 015003 (2020). Review paper that focuses on quantum chemistry applications too new to be included in above textbooks. (A)

### E. Augmented and virtual reality

There are quite a few examples of experimental uses of virtual reality (VR) for physics education using custom hardware. However, hardware developments underlying smartphones have facilitated a new generation of and augmented reality (AR) devices that are now very widely available. These have great potential for scientific visualization, education and immersive user interfaces. Present challenges include stratification of platforms and standards as well as the quality (and difficulty) of tools available for creating software with VR support.

- 308.
“

A literature review on immersive virtual reality in education: State of the art and perspectives,”L.Freina, andM.Ott, in Proc. Int. Sci. Conf. Learn. Software Educ. 1, 10 (2015). Broad overview of VR and AR across disciplines. (I) - 309.
“

Learning experience design with immersive virtual reality in physics education,”Y.Georgiou,O.Tsivitanidou, andA.Ioannou, Educ. Tech Res. Dev. 69, 3051–3080(2021). Neat recent example of a VR simulation designed to improve student understanding of relativity. Contains a valuable literature review. (I) - 310.
“

Physics education in virtual reality: An example,”H.KaufmannandB.Meyer, Themes Sci. Technol. Educ. 2, 117–130(2009). Example of a lab-like environment for mechanics. (I) - 311.
“

Non-euclidean virtual reality I: Explorations of$H3$,”V.Hart,A.Hawksley,E. A.Matsumoto, andH.Segerman, arXiv:1702.04004 and “Non-euclidean virtual reality II: Explorations of $H2\xd7E$,” same authors, arXiv:1702.04862. A powerful glimpse into the possibilities of VR where the user can navigate a non-Euclidean space. Usable from a web-browser. (E) - 312.
**OpenXR.**Emerging open standard for AR and VR that aims to improve compatibility between hardware and software. https://www.khronos.org/openxr/ (A)

## VIII. GLOSSARY OF ACRONYMS

- AAPT
American Association of Physics Teachers

- AJP
American Journal of Physics

- API
Application Programmer Interface

- APS
American Physical Society

- CPU
Central Processing Unit

- DEIJ
Diversity, Equity, Inclusion and Justice (Also EDI, DEI)

- GPU
Graphical Processing Unit

- JIT
Just In Time (compiler technology)

- ML
Machine Learning

- MD
Molecular Dynamics

- MPI
Message Passing Interface

- OSP
Open Source Physics

- OOP
Object Oriented Programming

- PER
Physics Education Research

- PERC
Physics Education Research Conference

- QC
Quantum Computing

- VR
Virtual Reality

## ACKNOWLEDGMENTS

The author is deeply grateful to Anna Phillips, Amy Graves, Scott MacLachlan, Will Kirby, Peter Love, William Ratcliffe, Kelly Roos, Tor Odden, Bruce Sherwood, Brandon Lunk, and Karen Camarda for valuable suggestions of resources to include and reading draft versions of the manuscript. This material is based in part upon the work supported by the National Science Foundation under Grant No. DMR-1654283.