Article Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX
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Helping Students Connect Interdisciplinary Concepts and Skills in Physical Chemistry and Introductory Computing: Solving Schrödinger’s Equation for the Hydrogen Atom Oka Kurniawan,* Li Ling Apple Koh, Jermaine Zhi Min Cheng, and Maggie Pee Singapore University of Technology and Design, 8 Somapah Road, 487372, Singapore
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S Supporting Information *
ABSTRACT: Integrating knowledge across disciplines has been shown to be a challenging, and yet it is a necessary skill that university students need to develop. Students who are able to connect different concepts, perspectives, and angles of a given topic are generally more engaged and have better understanding. However, designing lesson plans or assignments that integrate knowledge across disciplines is a challenge to most instructors. This paper describes our effort to provide an interdisciplinary and independent assignment that couples topics from a Physical Chemistry course with skills taught in an Introductory Computing course. It was shown that such an interdisciplinary assignment, though done independently, can help to boost students’ learning. However, it helps only in the understanding of certain aspects of the topics. We present the design of the assignment and the lessons learned for other instructors willing to adopt this kind of interdisciplinary approach. KEYWORDS: General Public, First-Year Undergraduate/General, Interdisciplinary/Multidisciplinary, Computer-Based Learning, Physical Chemistry, Atomic Properties/Structure
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INTRODUCTION
Molecular Workbench is one successful example in applying computational science to education.8,9 Though computer-based education has some beneficial effect in learning factual knowledge,10−12 studies by Russel and Brattan suggested that there seems to be no significant difference in students’ learning with regards to understanding when compared to other teaching media.13,14 However, these studies focus on using computers as a tool, either to deliver the content or to visualize it. It has been argued that technology should not be used only as a mere conveyor of information, but rather as a way to facilitate the construction of understanding.15 It is thus our aim to provide such construction with this interdisciplinary assignment which involves studying chemistry by writing computer code for a better understanding of chemistry and computing. The introductory computing course in Singapore University of Technology and Design (SUTD) teaches computational thinking to all our first-year students. Wing defines computational thinking as an approach to solving problems, designing systems, and understanding human behavior that draws on concepts fundamental to computing.16 In this sense, it is a kind of analytical thinking that is meant for everyone.17 It is a combination of abstraction and automation principles that can be applied to different disciplines, including the sciences and the humanities.18 Computational thinking has been trans-
Computational thinking has been recognized as one of the necessary 21st century work skills. Many schools have offered computer science courses to expose students at a very early age. Many of these courses, however, offer computing as an isolated course rather than integrated into other courses. It is important for the future workforce to be able to integrate computational thinking with other disciplines. Integrating knowledge across several disciplines is one of the most challenging tasks faced by educators. This is especially true in a higher education institution like a university. The aim of a university is to prepare students for work or further studies. In this context, many of the problems faced in real life require students to apply multidisciplinary science or engineering knowledge.1 Jacobs and Frickel, however, claimed that interdisciplinary studies may not necessarily integrate knowledge and may, in fact, lead to new rounds of differentiation and fragmentation.2 Nevertheless, they do admit that interdisciplinary studies may have transformative potential if they can interpenetrate disciplines and change how we do things.2 Some of these transformative powers can be seen from how people study chemistry using computers.3,4 Gilbert et al. propose that chemistry education should reflect what is happening in chemistry research, in which computational science has become an integral part.5 For example, Gordin and Pea discussed how to use scientific visualization as an educational technology.6 An interactive educational workbench, which uses augmented reality, has been developed.7 © XXXX American Chemical Society and Division of Chemical Education, Inc.
Received: January 22, 2019 Revised: July 24, 2019
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DOI: 10.1021/acs.jchemed.9b00068 J. Chem. Educ. XXXX, XXX, XXX−XXX
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having more than two persons work on the same set of code simultaneously), especially if this is their first programming course, the grouping was intentionally kept at two to three students to prevent freeloading and/or frustration so as to ensure that all group members achieve a level of concept comprehension in both the chemistry and computing portions. The chemistry and computing portions of the assignment were released together in one document in the second week of the term (each term is 14 weeks long) after the concept of the solutions to the Schrödinger equation for the hydrogen atom was introduced to the students in week 2 of the Physical Chemistry class. The assignment, timeline, and the required deliverables were also introduced in this Physical Chemistry class (Supporting Information). In this class, students were taught how to determine and identify the number and location of radial and angular nodes on the basis of the final wave function expression. The s orbital was used as an in-class example on how to determine the general orbital shape from the number and location of nodes. Students were given 6 weeks (weeks 3−8) to complete both portions of the assignment. In the chemistry portion, the students were tasked to derive the solutions of the Schrödinger equation for the hydrogen atom starting from the time-independent Schrö dinger equation. In the first iteration of this study, each group was assigned to solve a different combination of orbital angular momentum (l) and magnetic (m) quantum numbers in the principal quantum number (n) of 1 or 2. In the second iteration, each group was allowed to choose a principal quantum number of 2, 3, or 4, where the group would then solve for all the related orbital angular momentum (l) and magnetic (m) quantum numbers for the chosen principal quantum number (n). Given that the principal quantum numbers 3 and 4 were more complex, bonus points were awarded to groups who attempted these n numbers if their work, submitted in week 8, was entirely correct. To explain further, a group who attempted n = 2 was graded a maximum of 30 points while a group who attempted n = 3 or n = 4 could potentially obtain 15 or 30 more bonus points, respectively, which indicated a 50% or 100% increase in the grades assigned to this assignment. In this second iteration, there were a total of 221 groups or 463 students attempting this assignment with 5% of the groups choosing n = 2, 33.5% choosing n = 3, and 71.5% choosing n = 4. This equated to 210 or 95% of the groups who were entitled to earning a better grade. However, only 13 or 6.2% of these groups were awarded bonus points as the work from most groups was not entirely correct. In both iterations of the assignment, students were able to derive expressions for both the angular component and the radial component by applying separation of variables after week 3 lessons. Each of these expressions is an ordinary differential equation. Students were then required to solve the equations analytically to find the angular wave function and the radial wave function for the assigned or the selected quantum numbers by week 9. The solution for the overall wave function for that quantum state can then be obtained. The computing portion was broken down into smaller parts and functions to enable the students to learn the importance of modularity in solving big problems. The assignment was designed such that the progress of the computing course in each week was submitted to an online system which checked the correctness of the codes. The learning outcomes of the earlier steps leading to the subsequent steps meant that the
forming the way that traditional disciplines are viewed. Therefore, it is important that the students are able to apply and learn these skills within multiple domains through interdisciplinary activities. The promotion of interdisciplinary activities between chemistry and computational thinking is something we felt worth exploring as it can enable our students to make the leap from learning the subject as an individual silo to making vital connections to see the bigger picture. The use of MATLAB and Mathematica to teach quantum physics have been employed in numerous courses and resources.19−22 However, none has attempted to evaluate the student’s acceptance or the efficacy of such assignments. Many courses or studies employed computers as a tool for understanding or visualizing science concepts and do not develop the computational thinking that is vital to today’s society. Our university presents an exciting opportunity as all students are required to study the basic sciences, such as mathematics, physics, biology, and chemistry, as well as computing. Most of these subjects, however, are designed and conducted separately. In this paper, we will present an integrated interdisciplinary assignment that was jointly developed by the instructors from the Physical Chemistry course and the Introductory Computing course, which requires students to apply the knowledge learned from both courses. Both courses are offered in the last term of the students’ first year at SUTD. The purpose of this assignment is to deepen students’ understanding in a particular area of each subject as well as to help students connect concepts and skills that they would have learned from these two courses to tackle one specific problem. The findings from the student’s assessment of this assignment will be discussed. Here, we will discuss the outline of this interdisciplinary assignment and the findings of this study using the framework proposed by Repko.23
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DESIGN OF ASSIGNMENT The topic that was chosen for this independent and interdisciplinary assignment is the solution of the Schrödinger equation for the hydrogen atom. The two reasons for this choice of topic are as follows. First, students find it difficult to visualize the solution and understand the concept due to its probabilistic nature. In addition, visualization of the solution using numerical methods and computing can be an interesting but challenging task. Instructors hope that, by coding the solution, students can clearly understand the nature of separation of variables and how it builds up to its final solutions. At the same time, students will also learn certain aspects of computational science and numerical methods such as the trade-off between accuracy and computational time. Second, through this assignment, instructors hope that students understand that real world problems may not have simple analytical solutions; thus, the need of a multidisciplinary approach is required. In this study, the students, who worked in groups of two or three, were given a semiguided independent assignment that was broken into several steps with the learning outcomes of the earlier steps leading to the subsequent steps. There were two parts of the assignment: the chemistry portion and the computing portion. Although the students were grouped into groups of two or three, they were expected to finish the task in each step independently to ensure that it provided an equal learning opportunity to all students. As coding is a tough to manage assignment when working together in big groups (i.e., B
DOI: 10.1021/acs.jchemed.9b00068 J. Chem. Educ. XXXX, XXX, XXX−XXX
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tasks progressed from simple to more complicated. The first step was for the students to write the Python code for some mathematical functions in week 2. These functions would be used in the subsequent steps (see Supporting Information for weekly deliverables). Step 2 was for the students to use conditionals to choose the Lagrange and the Laquerre functions based on the quantum numbers assigned or selected in the chemistry portion in week 3. Conditionals were taught after mathematical expressions in the Introductory Computing course. Even though this approach can be considered as hard coding the function table, it was adapted so as to ensure that the assignment progressed according to our weekly schedule. One of the deliverables for this second step was that students had to write a recursive function to compute factorials. This factorial function was used in some other functions which when combined in step 3 will create solutions for the angular and radial wave functions. In step 4, the deliverable was for the students to compute the overall solution of the wave function for a given grid requirement. The deliverable in the final step was the plotting of the electron probability in week 9. These plots were submitted to the Physical Chemistry instructors for grading and marked the completion of this assignment. In addition to having the students write the code and visualize the results, they also had to answer some multiple-choice questions related to numerical errors, computational time, and accuracy at the end of the assignment.
iteration), 85 students participated. Overall, we have a total of 186 data points combined in the result of this study.
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RESULTS The results of the survey questions are shown in Figures 1 and 2. Different sections in the bar show the proportion for each
Figure 1. Likert score for survey questions with respect to the chemistry component. The number inside the circle gives the average Likert score for each question. Students were neutral that coding the equation helps them to understand the equation better (Q1). However, visualizing the equation helps to understand the concept better (Q3). Scale has a range of 1−5, where a higher score indicates a more positive response. Total number of responses, N = 186.
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METHODOLOGY In this study, we analyzed students’ perception of the assignment using a survey which was introduced only at the end of the assignment. Although all students taking the course were required to do the assignment, participation in the survey for this study was entirely voluntary and anonymous and had no impact on their grades. Students were asked to rate the following seven statements on a scale of 1 (Strongly Disagree) to 5 (Strongly Agree): 1. Implementing the Schrödinger equation solution in Python helps me to understand the equation better. 2. Solving the Schrödinger equation helps me to understand the probabilistic nature of electron location. 3. Visualizing Schrödinger equation solution helps me to understand the concept better. 4. Coding the Schrödinger equation solutions helps me to see the application of learning computing and programming in Physical Chemistry. 5. Coding the Schrödinger equation solutions helps me to think on the numerical computation aspects of computing, such as rounding error, accuracy, speed considerations, etc. 6. This assignment helps me to see the benefit of learning computer programming. 7. This assignment introduces me to learn Numpy and Python scientific computation libraries. Students were also asked in question 8 how the assignment could be improved and for any other comments they may have. The number of questions was kept to eight because the students may not be particularly excited to volunteer for the survey and having too many questions may dissuade more from participating. This study was done for two years with two different batches of students. In 2016 (first iteration of the assignment), 101 students took part in the study while in 2017 (second
Figure 2. Likert score for survey questions with respect to the computing component. The number inside the circle gives the average Likert score for each question. Survey result is more positive for computing as compared to chemistry. In particular, the assignment helps students see the benefit in learning programming (Q6). Scale has a range of 1−5, where a higher score indicates a more positive response. Total number of responses, N = 186.
answer, and the number in the circle gives the average Likert score for that particular question. The Likert score ranges from 1 to 5. In general, the students did not respond negatively toward the assignment. In a comparison of Figure 1 with Figure 2, the results seem to suggest that the assignment has a better impact C
DOI: 10.1021/acs.jchemed.9b00068 J. Chem. Educ. XXXX, XXX, XXX−XXX
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The positive result for Q2 means that the assignment helps them to understand the probabilistic nature of electron location. This seems to be correlated to the result from Q3, and it is highly probable that they have a better understanding due to the task of visualizing the equation and in agreement with the many applications of computing as an educational tool to aid visualization in chemistry.6−9 However, the assignment does not appear to be as effective in fostering an understanding of the mathematical equation itself. In order for the assignment to achieve all the learning objectives, the assignment should be modified such that the activity helps them to understand the mathematical equations just as the activity helps them to understand the probabilistic nature. We also noted the limitation of our survey questions, and further studies should be done where the survey questions can be more comprehensive to investigate clearly some of the aspects we wish to study. For example, the survey questions can include aspects investigating whether deep learning actually happens through this particular activity. Moreover, the way the assignment is delivered to facilitate interdisciplinary learning can also be part of the survey questions. Two reasons that may help explain why the students’ perception of the activity was not more positive could be, first, that this was likely their first multidisciplinary assignment that required equal knowledge and application of two very different subjects, and, second, that none of these students taking the courses were science majors. Therefore, the relevance of using computing for a chemistry problem may not be obvious to them. However, the exposure of students to real world problems through a multidisciplinary approach cannot be without merit. In designing this assignment, we focus on integrating the two subjects in such a way that multiple modalities of intelligence can be applied to the learning process. Gardner claims that bringing a broad array of frameworks and methodologies will enhance student’s engagement and thus learning.24 In our assignment, this is especially true as we enable students to apply visual−spatial, logical−mathematical, and interpersonal intelligence. This is affirmed in the positive comments where students find the assignment to be interesting and it boosts their interest to learn more about the subject. One difficulty that students encounter with this assignment is the lack of guidance due to its independent nature. Though we have intended for this assignment to be done independently, we realized that a clear direction is one aspect that motivates students in learning, especially with tasks that are related to computing.25,26 In the present format, the instructions are given through a handout with a brief introduction of the assignment provided by the Physical Chemistry instructors. It seems that more details should be given in terms of helping the students understand the handout better, as well as clarifying the expectation of the instructors. Multiple checkpoints in some of the weeks will also allow students to ask questions related to the assignment and clear their doubts or even difficulties. However, a common problem across both batches of students is that they tend to start and complete the assignment in as short a period of time as possible and only when the deadline approaches. To overcome this, multiple deadlines on some milestones should be set in the future iterations of the assignment so that instructors can frequently check and comment on their work. Moore discusses the three elements of independent learning and teaching, which are the autonomous learner, communica-
on the learning objectives of Introductory Computing’s than Physical Chemistry’s. For example, students seem to have a neutral feeling that implementing the Schrödinger equation solution in Python helps them to understand the equation better (Q1’s score is 2.9 out of 5.0), but they agree more that it does help them to learn certain aspects of numerical computation (Q5’s score is 3.2 out of 5.0). However, for the Physical Chemistry learning objectives, solving the equation using a computer program seems to help students with the understanding of the probabilistic nature of the electron location (Q2) despite the result for Q1. The largest impact on Physical Chemistry’s learning objectives was visualizing the equation (Q3): students claimed that such visualization helps them to understand the concept better. With regards to Computing, students agree positively with all the statements in general. The assignment does help them see the application of computing and programming (Q4). In fact, writing the computer program for this chemistry problem helps students to think and learn some numerical computation aspects such as rounding error, accuracy, and speed (Q5). Thus, the assignment seems to motivate them to learn programming as they see its benefit (Q6). Last, through this independent assignment, they have learned on their own some of the scientific computation libraries in Python (Q7). In the same survey, we also ask students to give some comments on the assignment. A number of students found the assignment to be interesting as shown in the following three quotes: “I found it interesting that the assignment combined the two subjects and we could see how programming can be applied to other fields!” “Thanks profs for this assignment. Makes me want to learn more about the subject.” “It’s a good scientific problem...” At the same time, we realized that quite a number of students found the assignment to be challenging. This can be seen in both the positive and negative comments. The two comments below showed how they found the computing part to be pretty challenging. “Try to include sample codes for certain parts to give people a place to start from.” “I felt that the task difficulty was a little too steep for the content taught in the course.” Some comments also showed that both the computing and chemistry parts were too challenging for them. “Coding for Schrodinger is a bit too far-fetched so I think you should design a new assignment that involves simpler coding but stronger chemistry related concepts. I feel using Schrodinger [...] is not very good as it is not a topic that people (me, at least) do not really appreciate due to its probabilistic nature.” Other comments are related more to the logistics and how the assignment should be delivered. A number of students preferred more guidance in doing the assignment. “Discuss this in [cohort] classes.” “Explain more thoroughly about the assignment before it is given so it is easier to understand.”
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DISCUSSION AND LESSONS LEARNED Our results show that such an interdisciplinary assignment benefits students in some aspects but not in others. This can be seen in the results with respect to the Physical Chemistry part. The score is mainly positive for Q2 and Q3 but not so for Q1. D
DOI: 10.1021/acs.jchemed.9b00068 J. Chem. Educ. XXXX, XXX, XXX−XXX
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tion system, and role of teacher.27 If this assignment is to be successful in enforcing independent learning, we have to create an environment that promotes these three elements. The instructors should design incentives, rewards, and punishment to motivate students to learn independently, which could be in the form of bonus points, penalty for missing deadlines, or other means. Moreover, instructors must set up an easy to use communication system for students to receive instruction and feedback. Such a system should allow students to ask questions and receive immediate feedback either from instructors or peers. This enables the students to seek a clear direction which then enhances their motivation in doing the assignment. Last, a teacher’s role is significant even in such independent assignments. Due to its interdisciplinary nature, all instructors from both courses must be familiar and comfortable with such an interdisciplinary assignment. To ensure this, training and briefing on the assignment for the instructors should be done. In this way, every instructor will be able to provide guidance and clear instructions to the students. In future, Laurillard’s framework can be used to identify gaps in designing computerbased exercises.28 According to constructivism, students learn by constructing knowledge. This learning can be identified as a surface approach, a deep approach, and a strategic approach. As instructors, we need to guide students toward having a deep approach to learning such that students will want to acquire an understanding of the material. Ramsdeen identified several key features that facilitate the deep approach to learning: providing an interesting and relevant assignment, allowing choice, ensuring no excessive load, reducing the anxiety about the exercise, not feeling threatened, involving students actively, interacting with others, and reflecting.29 We have tried to facilitate a deep approach to learning in this assignment. However, an interesting and relevant assignment is not easy to achieve as a particular topic may be interesting to some but not to others. This is reflected in the comments that we received as well. In the second iteration of this assignment, we allowed students to choose the quantum number. This decision is to allow for more choice. To ensure active involvement from all students as well as to allow interaction within and between groups, groups are kept small, either in groups of two or three. The assignment is also spread out through several weeks in order to reduce the workload. However, as mentioned in the previous discussion, students tend to do the assignment near the final deadline if they are left without additional deadlines set as milestones in between. Now, the nature of this assignment will be assessed using the framework suggested by Repko.23 Repko claims that interdisciplinary learning fosters the ability to develop and apply perspective-taking techniques, develop structural knowledge of problems appropriate to interdisciplinary inquiry, integrate conflicting insights, and produce cognitive advancement or interdisciplinary understanding of the problem. Perspective-taking involves understanding alternative viewpoints on a given issue. Some of our students exhibit this as they approach the solutions for the Laguerre polynomial and the Legendre polynomial. From the computing part, such polynomial was chosen using if−else as a kind of a table lookup. This is because this section of the assignment was released on the third week after students learn about conditionals. However, some students commented at the end of the assignment that one should not hard code the solutions but, rather, compute numerically the polynomial by differ-
entiating the equation. This perspective can be gained only if students perform the task required from the chemistry part. In fact, some of these students did exactly this more complicated approach to obtain the solution of the polynomials from differentiation. The second point by Repko focuses on whether interdisciplinary inquiry helps students to develop structural knowledge of problems. Structural knowledge is developed by acquiring declarative knowledge and procedural knowledge that are used in problem-solving or step-by-step task completion. This is well-accomplished in our assignment, as students are required to master Physical Chemistry’s content as well as perform the computation of the orbitals of hydrogen atoms both analytically and numerically. Our students also exhibited the ability to integrate conflicting insights. This was shown in one of the questions posted by students in our online forum. They noted that the plot that was produced by their code was different from the one presented in their chemistry lesson in class. This led to a discussion on complex orbital and real orbital plots. Their original code computes the complex wave function of the hydrogen atom whereas the one presented by the Physical Chemistry instructors during their lesson was the real value plot of the wave function. The real value wave function is a linear combination of two stationary complex wave function solutions. Another example that we found is on the notation for the spherical coordinates. A different convention for the notation of the spherical coordinates results in an opposite sign of the solutions. This too was discovered by the students and discussed in our online forum. One last example of their ability to integrate conflicting insights as well as produce interdisciplinary understanding of a problem is shown when one of the students’ correct code did not show some of the expected lobes in the real value orbital plots. After discussion with instructors, they discovered that the expected lobes were not seen because of the coarse grid spacing used in their numerical computation. As they experiment with a higher number of grid points, they managed to obtain the plots observed in theory. Through this, the students learned the trade-off between accuracy and computational time which is common in many scientific computations.
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CONCLUSIONS
This article presents the design and evaluation of our interdisciplinary independent assignment between Physical Chemistry and Introductory Computing courses. We have learned that such an assignment requires careful design to meet its objectives. Furthermore, to promote independent learning, careful attention is required in order to create an adequate environment to allow for autonomous learners, prepared instructors fulfilling their role, and a proper communication system between students and instructors. One of the biggest roles of teachers is to provide clear direction that enhances students’ motivation in doing the assignment. Overall, we see that our assignment provides a positive impact on students’ learning in some of its aspects. We have discussed some areas of improvement and assessed its interdisciplinary aspect by reflecting on the four points proposed by Repko. We have observed our students’ comments and their discussion dynamics which show how this interdisciplinary assignment provides unique learning as compared to a discipline-based assignment. E
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(19) Chelikowsky, J. R. Introductory Quantum Mechanics with MATLAB: For Atoms, Molecules, Clusters, and Nanocrystals; John Wiley & Sons, Incorporated: Weinheim, Germany, 2018. (20) Garcia, R.; Zozulya, A.; Stickney, J. MATLAB Codes for Teaching Quantum Physics: Part 1. 2007. https://arxiv.org/abs/0704. 1622 (accessed Jul 19, 2019). (21) Hassani, S. Mathematical Methods Using Mathematica: For Students of Physics and Related Fields; Springer-Verlag: New York, 2003. (22) Trif, D. Matlab Package for the Schrödinger Equation. J. Math. Chem. 2008, 43 (3), 1163−1176. (23) Repko, A. F. Assessing Interdisciplinary Learning Outcomes. Acad. Exch. Q. 2008, 12, 171−178. (24) Gardner, H. Frames of Mind: The Theory of Multiple Intelligences; Wiley Subscription Services, Inc., A Wiley Company: New York, 1983; Vol. 3. DOI: 10.1002/pam.4050030422. (25) Law, K. M. Y. Y.; Lee, V. C. S. S.; Yu, Y. T. T. Learning Motivation in E-Learning Facilitated Computer Programming Courses. Comput. Educ. 2010, 55 (1), 218−228. (26) Yacob, A.; Computer, F.; Terengganu, K. Assessing Level of Motivation in Learning Programming Among Engineering Students. Int. Conf. Informatics Appl. 2012, 425−432. (27) Moore, M. G. Toward a Theory of Independent Learning and Teaching. J. Higher Educ. 1973, 44 (9), 661−679. (28) Laurillard, D. Rethinking University Teaching: A Conversational Framework for the Effective Use of Learning Technologies; Taylor & Francis Group: Hoboken, 2013. (29) Ramsden, P.; Reynolds, C. R.; Brown, R. T. Context and Strategy: Situational Influences on Learning. In Perspectives on Individual Differences; Springer US: Boston, MA, 1988; pp 159−184. DOI: 10.1007/978-1-4899-2118-5_7.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.9b00068.
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Assignment handouts and deliverables, and links to samples of students’ submissions (PDF, DOCX)
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Oka Kurniawan: 0000-0001-9519-0959 Notes
The authors declare no competing financial interest.
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REFERENCES
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