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In the Classroom

In many countries students entering university sciencecourses are conronted with difficulties in applying mathematics.Several reports describe a declining standard o mathematicalknowledge and skills o first-year university students and containrecommendations to deal with this problem. Suggestions rangerom curriculum reorms to remedial help (13).Craig (4) andBressoud (5) summarize the CUPM Curriculum Guide 2004published by the Mathematics Association o America (6),in which it is stated that the teaching o mathematics withinchemistry programs has to be revised to better ollow the needso chemistry students. Zielinski and Schwenz (7) describe nec-essary changes in the physical chemistry curriculum and payspecial attention to mathematics and physics knowledge.

Tis university has chosen a remedial approach to helpfirst-year students with their mathematics deficiencies. A reme-dial mathematics program was developed as part o two jointprojects at several universities: Brush up Your Maths (8) andMathMatch(9).During Calculus 1, which is a joint course inthe first semester or first-year students in mathematics, physics,and chemistry, students are assessed by a diagnostic pretest and aposttest, which help them evaluate their mathematical strengthsand weaknesses. Activities to reinorce knowledge and skills areimplemented in which computer-algebra-based assessment andpractice o mathematics skills play an important role. A similarapproach is used at the University o Iowa (10).

Te pre- and posttest results rom students in the 20052006 academic year were thoroughly analyzed (11). Studentresults rom dierent science disciplines were compared.Chemistry students scored lower than mathematics and phys-ics students. Te mathematical skills o all chemistry studentsimproved, but only hal o them reached the required level omathematical knowledge and skills. o some extent this resultwas not a complete surprise: many chemistry students enterthe university afer a less mathematics-intensive program atsecondary school, because the mathematics entry requirementsor chemistry are not as demanding as they are or other sciencestudies. It creates a difficult situation or chemistry students,because they need mathematics throughout their chemistry

study. For example, Quantum Chemistry (QC), a compulsorycourse or chemistry majors in the second semester o the firstyear, relies on mathematical skills or topics such as principles oquantum mechanics, structure o atoms and molecules, chemicalbonds, and molecular orbital theory. Students who attend QCwithout passing the Calculus 1 course first (officially not anentry requirement or QC) have serious problems learning quan-tum chemistry. Action was taken and a ollow-up to the first-semester remedial mathematics program was developed or thesecond semester o the 20052006 academic year. Students withweak calculus knowledge were helped to review the requiredpre-knowledge. Based upon the results o this effort, changeswere made to QC and the remedial mathematics activities in the

ollowing academic year (20062007). Tis article discusses the

setup and results o the remedial mathematics activities duringQC in the 20052006 and 20062007 academic years.

The Mathematics Problem in Quantum Chemistry

Exploratory research on the understanding o quantumchemistry showed that students had serious difficulties that seemedto be caused by lack o mathematical skills (12).Te problems wereso severe that the lecturer could not present an in-depth discussiono quantum chemical concepts. Ofen the lecturer taught math-ematics instead o quantum chemistry. o highlight the seriousnesso the mathematics problem, a ew examples are given: (i) most

students were not able to compute the derivative o eax

, or evenex; (ii) students were not able to solve a basic differential equationsuch asf (x) = c2f(x); (iii) many students ound it hard to docalculations with symbols instead o numbers; and (iv) a moreabstract exercise such as computing the derivative og(y)f(x) withrespect tox, was an immense obstacle.

Tackling the Mathematics Problem

Te design o the remedial mathematics program in QC,installed in the 20052006 academic year, was based on theollowing considerations: (i) Students are held responsible tosuccessully review their pre-knowledge. Students should beaware that the review activity is indispensable or QC, but it is

not a part o the course. (ii) Te lecturer should be able to de-termine what mathematics pre-knowledge students have or lackin order to optimize his or her teaching. (iii) For motivationalreasons, students should immediately see the benefits o prac-ticing mathematics. (iv) Te quantity o question items shouldbe large enough so that students can do exercises as ofen asneeded. (v) Students or whom the assignments are not enoughare provided extra help.

Te remedial mathematics program was set up taking theseconsiderations into account. Beore the start o QC all studentstake an online introductory diagnostic test in mathematics. Tepurpose o this test is to give students an idea o the mathemati-cal entry requirements o the course and to provide the lecturer

insight in the mathematics level o the students. For each lecturethere is a set o online mathematical assignments. Tese assign-ments only treat mathematics needed in the upcoming lecture.Most o the mathematics reappears in the lecture transparenciesin the context o quantum chemistry. For example, a pre-lectureexercise composed o the calculations needed in the normalizationo a wave unction is shown in Figure 1. Te explanation is shownafer the student has submitted the answer. Te lecture transparen-cies using this mathematic concept is shown in Figure 2.

Note that the mathematics exercise in Figure 1 does notcontain the context in which it is applied during the lecture. Tisis done in the belie that context-ree practicing o mathematicalskills also contributes to the transer rom calculus to science.

Having students solve calculations that reappear in the lecture

Remedial Mathematics for Quantum ChemistryLodewijk Koopman,* Natasa Brouwer, and Andr Heck

AMSTEL Institute, University of Amsterdam, Kruislaan 404, Amsterdam,1098 SM, The Netherlands; *L.Koopman@uva.nl

Wybren Jan Buma

Van t Hoff Institute for Molecular Sciences, University of Amsterdam, Nieuwe Achtergracht 166, Amsterdam, 1018 WV,

The Netherlands

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In the Classroom

in a chemical context, enables them to discover the relevant con-nection between calculus and quantum chemistry. Tis setup isopposite to an integrated approach advocated, or example, byDunn and Barbanel (13).

All exercises are implemented in Maple .A. (14),an onlinecomputer algebra-based testing and assessment system. Te as-signments must be finished beore the lecture. Multiple attemptsare possible, however, with each attempt the assignment appearsin a slightly different orm. Each assignment is expected to takeabout 15 minutes. Te main advantages o using Maple .A. arerandomization o assignments and immediate eedback (15).

Te lecturer monitors the Maple .A. results and commentson the students answers. I many students appear to have a prob-lem with a particular mathematical concept, extra explanationor the whole group is scheduled. Each week a tutor is present orone hour to answer students questions about the assignments.Tis design takes into account the suggestion o Edwards (16)to cope with the mathematics problem on entering university:testing student ability on entry and providing ollow-up supportto those that need it is not sufficient. Careully monitoring thistest-and-support combination to determine its effectiveness is

necessary to adjust the support being offered to students.Based upon the results o the 20052006 academic year, a

ew changes were made to QC and the remedial program theollowing year:

Each set of online assignments was made available intwo versions. Te first was identical to the one used in20052006, while the second version contained ex-amples, explanations, or intermediate steps needed in thecalculation. Students were only allowed to take the second

version i they had attempted the first.

Four interactive introductory lectures were added to QCin which students explored the oundations o quantummechanics.

Students could attain a bonus to their nal grade by hand-ing in answers to sel tests on quantum mechanics giveneach lecture.

In addition, chemistry students received one hour additionaltutorial during Calculus 1 in the first semester. Tey also weregiven problems that would reappear during QC.

Evaluation of the Remedial Program

Te student results o the introductory diagnostic test in20052006 were disappointing. For this reason the lecturerdecided to make the online assignments compulsory. Studentscould only take the two exams (mid-course and end-o-course)

i they attempted all o the online problem sets. Te grades othese online assignments did not count in the final marking othe course.

o measure the effect o this intervention, the lecturesand tutorial sessions were observed. Tis made it possible toqualitatively monitor the effect o the Maple .A. assignmentsdirectly and to compare it with the results obtained in the pre-vious year when no pre-knowledge assignments in mathematicswere given. Te students attempts at taking the online exerciseswere tracked. Tis gave quantitative data on their perormance;we could also see whether students completed the assignments,which questions they ound most difficult, and what progressthey made during the course. Afer completion o the course a

survey was conducted.

In the analysis only first-year chemistry majors were includ-

ed. A test or exam that a student did not participate in is markedzero. Te results or both 20052006 (N= 18) and 20062007(N= 28) are given (able 1). Te lecturer in these years was thesame. It is clear that the 20062007 student population wasbetter than the year beore. In the two years only one studentpassed QC without passing Calculus 1. All other students whopassed QC had also passed Calculus 1.

A majority o the students tried most online assignmentsmore than once, on average twice. Tis shows that they tookresponsibility or the reviewing and indicates a positive effecton participation. It was expected that students would spend 15minutes on the online assignments each lecture. In actuality, theonline assignment took them longer: in 20052006 on average 31

minutes (SD = 13) and in 20062007, 22 minutes (SD = 8).

(a)2

0

0 1

d

(b)0

sin d

(c)1 1 2 dd

x yy y xx

Explanation: e primitive of a constant is equalto the constant times the variable ( in this case).e primitive of sin equals cos . For an integralwith more than one variable, you first have tocompute the integral over one variable (e.g., x). eother variable (in this case y) is considered as aconstant. Next, you compute the integral over theremaining variable ( y).

Compute the following integrals. For more informa-tion, read pages 129131 from the Calculus 1 lecturenotes.

Figure 1. Questions from one of the online assignments.

Figure 2. A lecture transparency (in Dutch), showing as an examplethe normalization of the function er/a0.

Table 1. Average Scores in Calculus 1, General Chemistry, andQuantum Chemistry

Course

Average Grades (%) (SD)

20052006(N = 18)

20062007(N = 28)

Calculus 1 49 (19) 60 (19)

General Chemistry 51 (15) 64 (17)

Quantum Chemistry 25 (25) 46 (26)

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In the Classroom

During the lectures, the students were better able to ol-low the mathematical reasoning o the lecturer than the yearspreviously. Te lecturer ofen asked simple questions about themathematics involved to ensure that the students were able tocomprehend the lecture. For instance, he asked or the second de-rivative o ekxwith respect tox. In contrast to previous experience,this posed no problems. Students also said that they recognizedthe mathematics in the lecture rom the assignments that they hadcompleted, which had a motivating effect on the students.

In 20052006 there was a positive correlation (r= 0.73)

between the results o the mid-course exam and the averageresults o the corresponding online assignments (Figure 3, top).Tree scores deviated rom this: these students scored higherthan 70% or the online assignments but scored only 30% orless or the mid-course exam. It is possible that these studentsthought that they would pass the mid-course exam i they gota good grade or the online assignments. wo o these threestudents dropped out o the course. For the rest o the studentsit holds that the higher the score o the mathematics tests, thehigher their grade in the mid-course exam. We can propose theollowing two reasons: (i) these students might have been bettermotivated and thus perormed better or both the mathemat-ics assignments and the mid-course exam or (ii) these students

might have had benefit rom the online activities.

Te results o the end-o-course exam in 20052006 alsoshow a positive correlation (r= 0.88) with the average scoreo the online mathematics assignments (Figure 3, bottom), al-though the evidence is not strong because o the small numbero students participating in the end-o-course exam. Te totaloutcome o the course is disappointing: only 4 out o 18 stu-dents (22%) passed the course with a composite grade o 55%or higher. Te average exam grade was 25% (SD = 25).

In the 20062007 academic year there were more chem-istry majors (able 1). Te 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 performedon the online assignments, the better they performed for QC.Again, there are some students who deviate from this. Tistime some students performed poorly on the online assign-ments, but well on the exams. Some of these students onlymade one attempt at the online assignments. Tis does notmean that they did not learn, only it was not visible to theinstructors from the online assignments. In comparison tothe previous year the final results were more encouraging: 15out of 28 students (54%) passed the course with a compositegrade of 55% or higher. Te average exam grade for QC was46% (SD = 26), nearly twice as high when compared to the

previous year.

Mid-Course Online Assignment Average (%)

Mid-Cou

rseExam

(%)

00

25

50

75

100

25 50 75 100

End-of-Course Online Assignment Average (%)

End-of-CourseExam

(%)

0

25

50

75

100

0 25 50 75 100

Figure 3. Scatter plots showing the 20052006 results for the mid-

course (top) and end-of-course exam (bottom) and the average resultsfor the corresponding online assignments (N= 18).

Figure 4. Scatter plots showing the 20062007 results for the mid-

course (top) and end-of-course exam (bottom) and the average resultsfor the corresponding online assignments (N= 28).

Mid-Cou

rseExam

(%)

50

25

0

75

100

Mid-Course Online Assignment Average (%)

0 25 50 75 100

End-of-CourseExam

(%)

0

25

50

75

100

End-of Course Online Assignment Average (%)

0 25 50 75 100

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In the Classroom

Te students mathematical skills improved as a result othe online assignments. Beore the start o QC students had totake an introductory diagnostic test, which we used as a pretest.A similar posttest was not conducted, but the questions romthe pretest reappeared in the weekly online assignments. Tisenabled us to construct a posttest score. Only students who hadparticipated in the end-o-course exam were included in thisanalysis. Both years show an improvement (able 2).

For both academic years a survey was conducted at the endo the course. In the analysis only responses o students were in-cluded who had at least taken the mid-course exam (or the twoyears under consideration: 12 and 28 respondents, respectively1).

On a five point Likert scale (ranging rom 1 to 5) students in20052006 rated their mathematical skills beore the courseas 2.3; they elt the need to do the remedial exercises. Moststudents said that they benefited rom doing the online assign-ments; they could better understand the lectures. A majority othe students stated that their mathematical skills had improved.Tey ound the online assignments helpul. In 20062007 stu-dents rated their mathematical skills higher (average 3.1) thanthe year beore. Tey were slightly less positive about the effecto the online assignments. Tis might be understandable sincestudents had better Calculus 1 grades.

Conclusion

Te objective o this initiative was to improve the math-ematical knowledge and skills o students to enable them tosuccessully complete the QC course. Te online remedial ex-ercises helped the students to improve their mathematical skills.In doing so, they were better able to comprehend the lecturesand answer elementary questions on the mathematics needed.From the survey it became clear that students were motivatedto review their mathematical skills. For both years there was apositive correlation between the exams and the online math-ematics assignments. However, in 20052006 the course gradeswere still disappointing, and only 4 students passed the course.In 20062007, afer the remedial program was improved, theaverage grade was significantly higher and many more students

passed the course: the percentage o students passing the coursewent rom 22% to 54%. Tis can only partly be explained by thedifference in student populations, viz the 20062007 populationscored higher on relevant first-semester courses. Te change insetup o the QC course also had an effect. Te exam results showthat a student who does not pass Calculus 1 is unlikely to passQC. We conclude rom this that good mathematical skills arenecessary, but not sufficient or learning quantum chemistry.

In addition to mathematical skills, quantum chemistryrequires abstract thinking. From the observations it becameapparent that students find this difficult. Mathematical skillsmight help the students to develop better abstract thinking,but it is not to be expected that by only improving students

manipulative mathematical skills that they will be able to ap-ply these skills in abstract courses like QC. Students should beable to understand what the mathematical expressions used inquantum chemistry mean and to apply them. Tis was one othe objectives o the additional our introductory lectures intro-duced in 20062007. Te precise effect o these interventions isa subject o uture research.

Acknowledgment

Te authors would like to thank student assistant KoenPeters or implementing the Maple .A. assignments.

Note

1. Because the survey was anonymous it was not possible to ex-clude students rom the survey results that were not first-year chemistrymajors (~20% o the population), as was done with the other results.

Literature Cited

1. London Mathematical Society. ackling the Mathematics

Problem. http://www.lms.ac.uk/policy/tackling_maths_prob.pd(accessed Apr 2008). 2. Engineering Council. Measuring the Mathematics Problem.

http://www.engc.org.uk/documents/Measuring_the_Maths_Prob-lems.pd(accessed Apr 2008).

3. Mustoe, L.; Lawson, D. Mathematics or the European EngineerACurriculum or the wenty-first Century. http://learn.lboro.ac.uk/mwg/core/latest/sefimarch2002.pd(accessed Apr 2008).

4. Craig, N. C.J. Chem. Educ.2001,78,582586. 5. Bressoud, D. M.J. Chem. Educ.2001,78,578581. 6. Committee on the Undergraduate Program in Mathematics

(CUPM). http://www.maa.org/CUPM(accessed Apr 2008). 7. Zielinski, . J.; Schwenz, R. W. Chem. Educator 2004, 9,

108121. 8. Web-spijkeren (Brush up Your Maths). http://www.web-spijkeren.nl

(accessed Apr 2008). 9. MathMatch. http://www.mathmatch.nl(accessed Apr 2008).10. Pienta, N. J.J. Chem. Educ.2003,80,12441246.11. Heck, A.; van Gastel, L.Int. J. Math. Educ. Sci. Tech.2006,37,

925945.12. Koopman, L.; Ellermeijer, A. L. Understanding Student Difficulties

in First Year Quantum Mechanics Courses.Proceedings o the EPEC-12005 Conerence. Bad Honne, Germany, July 48, 2005,http://www.physik.uni-mainz.de/lehramt/epec/koopman.pd(accessed Apr 2008)

13. Dunn, J. W.; Barbanel, J.Am. J. Phys.2000,68,749757. 14. Maple .A.http://www.maplesof.com/products/mapleta/(accessed

Apr 2008).

15. Heck, A.; van Gastel, L. Diagnostic esting with Maple .A. InProceedings WebALT2006; Eindhoven, Te Netherlands,Jan56, 2006, Seppl, M., Xambo, S., Caprotti, O., Eds.; pp 3751.http://webalt.math.helsinki.i/webalt2006/content/e31/e176/webalt2006.pd(accessed Apr 2008).

16. Edwards, P. Teaching Math. and its Appl.1997,16,118121.

SupportingJCE OnlineMaterialhttp://www.jce.divched.org/Journal/Issues/2008/Sep/abs1233.html

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Table 2. Average Scores to the Mathematics Pre- and Posttest

TestAverage Grades (%) (SD)

20052006 (N = 8) 20062007 (N = 21)

Pre 61 (12) 58 (12)

Post 71 (10) 76 (14)

NOTE: Only students that took the end-of-course exam were included.

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