In the Classroom
Remedial Mathematics for Quantum Chemistry Lodewijk Koopman,* Natasa Brouwer, and André Heck AMSTEL Institute, University of Amsterdam, Kruislaan 404, Amsterdam,1098 SM, The Netherlands; *
[email protected] Wybren Jan Buma Van ‘t Hoff Institute for Molecular Sciences, University of Amsterdam, Nieuwe Achtergracht 166, Amsterdam, 1018 WV, The Netherlands
In many countries students entering university science courses are confronted with difficulties in applying mathematics. Several reports describe a declining standard of mathematical knowledge and skills of first-year university students and contain recommendations to deal with this problem. Suggestions range from curriculum reforms to remedial help (1–3). Craig (4) and Bressoud (5) summarize the CUPM Curriculum Guide 2004 published by the Mathematics Association of America (6), in which it is stated that the teaching of mathematics within chemistry programs has to be revised to better follow the needs of chemistry students. Zielinski and Schwenz (7) describe necessary changes in the physical chemistry curriculum and pay special attention to mathematics and physics knowledge. This university has chosen a remedial approach to help first-year students with their mathematics deficiencies. A remedial mathematics program was developed as part of two joint projects at several universities: “Brush up Your Maths” (8) and “MathMatch” (9). During Calculus 1, which is a joint course in the first semester for first-year students in mathematics, physics, and chemistry, students are assessed by a diagnostic pretest and a posttest, which help them evaluate their mathematical strengths and weaknesses. Activities to reinforce knowledge and skills are implemented in which computer-algebra-based assessment and practice of mathematics skills play an important role. A similar approach is used at the University of Iowa (10). The pre- and posttest results from students in the 2005– 2006 academic year were thoroughly analyzed (11). Student results from different science disciplines were compared. Chemistry students scored lower than mathematics and physics students. The mathematical skills of all chemistry students improved, but only half of them reached the required level of mathematical knowledge and skills. To some extent this result was not a complete surprise: many chemistry students enter the university after a less mathematics-intensive program at secondary school, because the mathematics entry requirements for chemistry are not as demanding as they are for other science studies. It creates a difficult situation for chemistry students, because they need mathematics throughout their chemistry study. For example, Quantum Chemistry (QC), a compulsory course for chemistry majors in the second semester of the first year, relies on mathematical skills for topics such as principles of quantum mechanics, structure of atoms and molecules, chemical bonds, and molecular orbital theory. Students who attend QC without passing the Calculus 1 course first (officially not an entry requirement for QC) have serious problems learning quantum chemistry. Action was taken and a follow-up to the firstsemester remedial mathematics program was developed for the second semester of the 2005–2006 academic year. Students with weak calculus knowledge were helped to review the required pre-knowledge. Based upon the results of this effort, changes were made to QC and the remedial mathematics activities in the following academic year (2006–2007). This article discusses the
setup and results of the remedial mathematics activities during QC in the 2005–2006 and 2006–2007 academic years. The Mathematics Problem in Quantum Chemistry Exploratory research on the understanding of quantum chemistry showed that students had serious difficulties that seemed to be caused by lack of mathematical skills (12). The problems were so severe that the lecturer could not present an in-depth discussion of quantum chemical concepts. Often the lecturer taught mathematics instead of quantum chemistry. To highlight the seriousness of the mathematics problem, a few examples are given: (i) most students were not able to compute the derivative of eax, or even ex; (ii) students were not able to solve a basic differential equation such as f ″(x) = ‒c 2f (x); (iii) many students found it hard to do calculations with symbols instead of numbers; and (iv) a more abstract exercise such as computing the derivative of g( y) f (x) with respect to x, was an immense obstacle. Tackling the Mathematics Problem The design of the remedial mathematics program in QC, installed in the 2005–2006 academic year, was based on the following considerations: (i) Students are held responsible to successfully review their pre-knowledge. Students should be aware that the review activity is indispensable for QC, but it is not a part of the course. (ii) The lecturer should be able to determine what mathematics pre-knowledge students have or lack in order to optimize his or her teaching. (iii) For motivational reasons, students should immediately see the benefits of practicing mathematics. (iv) The quantity of question items should be large enough so that students can do exercises as often as needed. (v) Students for whom the assignments are not enough are provided extra help. The remedial mathematics program was set up taking these considerations into account. Before the start of QC all students take an online introductory diagnostic test in mathematics. The purpose of this test is to give students an idea of the mathematical entry requirements of the course and to provide the lecturer insight in the mathematics level of the students. For each lecture there is a set of online mathematical assignments. These assignments only treat mathematics needed in the upcoming lecture. Most of the mathematics reappears in the lecture transparencies in the context of quantum chemistry. For example, a pre-lecture exercise composed of the calculations needed in the normalization of a wave function is shown in Figure 1. The explanation is shown after the student has submitted the answer. The lecture transparencies using this mathematic concept is shown in Figure 2. Note that the mathematics exercise in Figure 1 does not contain the context in which it is applied during the lecture. This is done in the belief that context-free practicing of mathematical skills also contributes to the transfer from calculus to science. Having students solve calculations that reappear in the lecture
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 9 September 2008 • Journal of Chemical Education
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In the Classroom
in a chemical context, enables them to discover the relevant connection between calculus and quantum chemistry. This setup is opposite to an integrated approach advocated, for example, by Dunn and Barbanel (13). All exercises are implemented in Maple T.A. (14), an online computer algebra-based testing and assessment system. The assignments must be finished before the lecture. Multiple attempts are possible, however, with each attempt the assignment appears in a slightly different form. Each assignment is expected to take about 15 minutes. The main advantages of using Maple T.A. are randomization of assignments and immediate feedback (15). The lecturer monitors the Maple T.A. results and comments on the students’ answers. If many students appear to have a problem with a particular mathematical concept, extra explanation for the whole group is scheduled. Each week a tutor is present for one hour to answer students’ questions about the assignments. This design takes into account the suggestion of Edwards (16) to cope with the mathematics problem on entering university: testing student ability on entry and providing follow-up support to those that need it is not sufficient. Carefully monitoring this test-and-support combination to determine its effectiveness is necessary to adjust the support being offered to students. Based upon the results of the 2005–2006 academic year, a few changes were made to QC and the remedial program the following year:
• Each set of online assignments was made available in two versions. The first was identical to the one used in 2005–2006, while the second version contained examples, explanations, or intermediate steps needed in the calculation. Students were only allowed to take the second version if they had attempted the first.
• Four interactive introductory lectures were added to QC in which students explored the foundations of quantum mechanics.
• Students could attain a bonus to their final grade by handing in answers to self tests on quantum mechanics given each lecture.
In addition, chemistry students received one hour additional tutorial during Calculus 1 in the first semester. They also were given problems that would reappear during QC. Evaluation of the Remedial Program The student results of the introductory diagnostic test in 2005–2006 were disappointing. For this reason the lecturer decided to make the online assignments compulsory. Students could only take the two exams (mid-course and end-of-course) if they attempted all of the online problem sets. The grades of these online assignments did not count in the final marking of the course. To measure the effect of this intervention, the lectures and tutorial sessions were observed. This made it possible to qualitatively monitor the effect of the Maple T.A. assignments directly and to compare it with the results obtained in the previous year when no pre-knowledge assignments in mathematics were given. The students’ attempts at taking the online exercises were tracked. This gave quantitative data on their performance; we could also see whether students completed the assignments, which questions they found most difficult, and what progress they made during the course. After completion of the course a survey was conducted. 1234
Compute the following integrals. For more information, read pages 129–131 from the Calculus 1 lecture notes. 2Q
(a)
±0
(b)
±0 sin RdR
(c)
±x 0 ± y –1 x y
dK
Q
1
1
2
dy d x
Explanation: The primitive of a constant is equal to the constant times the variable (K in this case). The primitive of sin R equals cos R. For an integral with more than one variable, you first have to compute the integral over one variable (e.g., x). The other variable (in this case y) is considered as a constant. Next, you compute the integral over the remaining variable ( y). Figure 1. Questions from one of the online assignments.
Figure 2. A lecture transparency (in Dutch), showing as an example the normalization of the function e−r/a0.
Table 1. Average Scores in Calculus 1, General Chemistry, and Quantum Chemistry Average Grades (%) (SD) Course
2005–2006 (N = 18)
2006–2007 (N = 28)
Calculus 1
49 (19)
60 (19)
General Chemistry
51 (15)
64 (17)
Quantum Chemistry
25 (25)
46 (26)
In the analysis only first-year chemistry majors were included. A test or exam that a student did not participate in is marked zero. The results for both 2005–2006 (N = 18) and 2006–2007 (N = 28) are given (Table 1). The lecturer in these years was the same. It is clear that the 2006–2007 student population was better than the year before. In the two years only one student passed QC without passing Calculus 1. All other students who passed QC had also passed Calculus 1. A majority of the students tried most online assignments more than once, on average twice. This shows that they took responsibility for the reviewing and indicates a positive effect on participation. It was expected that students would spend 15 minutes on the online assignments each lecture. In actuality, the online assignment took them longer: in 2005–2006 on average 31 minutes (SD = 13) and in 2006–2007, 22 minutes (SD = 8).
Journal of Chemical Education • Vol. 85 No. 9 September 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
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75
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In the Classroom
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Figure 3. Scatter plots showing the 2005–2006 results for the midcourse (top) and end-of-course exam (bottom) and the average results for the corresponding online assignments (N = 18).
Figure 4. Scatter plots showing the 2006–2007 results for the midcourse (top) and end-of-course exam (bottom) and the average results for the corresponding online assignments (N = 28).
During the lectures, the students were better able to follow the mathematical reasoning of the lecturer than the years previously. The lecturer often asked simple questions about the mathematics involved to ensure that the students were able to comprehend the lecture. For instance, he asked for the second derivative of ekx with respect to x. In contrast to previous experience, this posed no problems. Students also said that they recognized the mathematics in the lecture from the assignments that they had completed, which had a motivating effect on the students. In 2005–2006 there was a positive correlation (r = 0.73) between the results of the mid-course exam and the average results of the corresponding online assignments (Figure 3, top). Three scores deviated from this: these students scored higher than 70% for the online assignments but scored only 30% or less for the mid-course exam. It is possible that these students thought that they would pass the mid-course exam if they got a good grade for the online assignments. Two of these three students dropped out of the course. For the rest of the students it holds that the higher the score of the mathematics tests, the higher their grade in the mid-course exam. We can propose the following two reasons: (i) these students might have been better motivated and thus performed better for both the mathematics assignments and the mid-course exam or (ii) these students might have had benefit from the online activities.
The results of the end-of-course exam in 2005–2006 also show a positive correlation (r = 0.88) with the average score of the online mathematics assignments (Figure 3, bottom), although the evidence is not strong because of the small number of students participating in the end-of-course exam. The total outcome of the course is disappointing: only 4 out of 18 students (22%) passed the course with a composite grade of 55% or higher. The average exam grade was 25% (SD = 25). In the 2006–2007 academic year there were more chemistry majors (Table 1). The results show a similar effect as the year before, with a correlation of (r = 0.73) for both exams (Figure 4). Generally speaking, the better students performed on the online assignments, the better they performed for QC. Again, there are some students who deviate from this. This time some students performed poorly on the online assignments, but well on the exams. Some of these students only made one attempt at the online assignments. This does not mean that they did not learn, only it was not visible to the instructors from the online assignments. In comparison to the previous year the final results were more encouraging: 15 out of 28 students (54%) passed the course with a composite grade of 55% or higher. The average exam grade for QC was 46% (SD = 26), nearly twice as high when compared to the previous year.
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 9 September 2008 • Journal of Chemical Education
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In the Classroom Table 2. Average Scores to the Mathematics Pre- and Posttest Average Grades (%) (SD)
Test
2005–2006 (N = 8)
2006–2007 (N = 21)
Pre
61 (12)
58 (12)
Post
71 (10)
76 (14)
Note: Only students that took the end-of-course exam were included.
The students’ mathematical skills improved as a result of the online assignments. Before the start of QC students had to take an introductory diagnostic test, which we used as a pretest. A similar posttest was not conducted, but the questions from the pretest reappeared in the weekly online assignments. This enabled us to construct a posttest score. Only students who had participated in the end-of-course exam were included in this analysis. Both years show an improvement (Table 2). For both academic years a survey was conducted at the end of the course. In the analysis only responses of students were included who had at least taken the mid-course exam (for the two years under consideration: 12 and 28 respondents, respectively1). On a five point Likert scale (ranging from 1 to 5) students in 2005–2006 rated their mathematical skills before the course as 2.3; they felt the need to do the remedial exercises. Most students said that they benefited from doing the online assignments; they could better understand the lectures. A majority of the students stated that their mathematical skills had improved. They found the online assignments helpful. In 2006–2007 students rated their mathematical skills higher (average 3.1) than the year before. They were slightly less positive about the effect of the online assignments. This might be understandable since students had better Calculus 1 grades. Conclusion The objective of this initiative was to improve the mathematical knowledge and skills of students to enable them to successfully complete the QC course. The online remedial exercises helped the students to improve their mathematical skills. In doing so, they were better able to comprehend the lectures and answer elementary questions on the mathematics needed. From the survey it became clear that students were motivated to review their mathematical skills. For both years there was a positive correlation between the exams and the online mathematics assignments. However, in 2005–2006 the course grades were still disappointing, and only 4 students passed the course. In 2006–2007, after the remedial program was improved, the average grade was significantly higher and many more students passed the course: the percentage of students passing the course went from 22% to 54%. This can only partly be explained by the difference in student populations, viz the 2006–2007 population scored higher on relevant first-semester courses. The change in setup of the QC course also had an effect. The exam results show that a student who does not pass Calculus 1 is unlikely to pass QC. We conclude from this that good mathematical skills are necessary, but not sufficient for learning quantum chemistry. In addition to mathematical skills, quantum chemistry requires abstract thinking. From the observations it became apparent that students find this difficult. Mathematical skills might help the students to develop better abstract thinking, but it is not to be expected that by only improving students’ 1236
manipulative mathematical skills that they will be able to apply these skills in abstract courses like QC. Students should be able to understand what the mathematical expressions used in quantum chemistry mean and to apply them. This was one of the objectives of the additional four introductory lectures introduced in 2006–2007. The precise effect of these interventions is a subject of future research. Acknowledgment The authors would like to thank student assistant Koen Peters for implementing the Maple T.A. assignments. Note 1. Because the survey was anonymous it was not possible to exclude students from the survey results that were not first-year chemistry majors (~20% of the population), as was done with the other results.
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