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Capstone: Why it Didn't Workfor UsSandra FillebrownAccepted author version posted online: 12 Mar2013.Published online: 10 May 2013.
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PRIMUS, 23(4): 412–418, 2013Copyright © Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970.2012.716146
Capstone: Why it Didn’t Work for Us
Sandra Fillebrown
Abstract: Saint Joseph’s University, a small comprehensive university, implementeda one-credit capstone project for the mathematics major beginning in 2002. Afterexpansion to three credits in 2007, the capstone, although successful, had unintendedconsequences. In 2010, for both programmatic and practical reasons, the departmentdecided to remove the course from the program. This article describes that process andexplains how the department met its program objectives through other means.
Keywords: Capstone course, problem seminar.
1. BACKGROUND
A little background on the mathematics major at Saint Joseph’s Universityis necessary to understand why we implemented and then abandoned ourcapstone course. Beginning in the 1970’s we were a combined Departmentof Mathematics and Computer Science. As with many schools, there werecomputer science courses required by the mathematics major and mathemat-ics courses required by the computer science major and there was significantoverlap in the two curricula. By the early 2000’s, however, these curriculumsbegan to diverge more and more and fewer and fewer faculty taught courses inboth disciplines. The two departments finally split in 2010.
Also, beginning in the late 1990’s, we saw a significant increase in thenumber of students interested in actuarial science. As a result, we instituteda separate actuarial science major in 2005. This is technically an interdisci-plinary major and not part of the Department of Mathematics; however, theDirector of the Actuarial Science Program has always been a faculty mem-ber in our department. The actuarial science program has been successful atrecruiting students and there are now about a dozen entering freshmen eachyear in this major. As with the mathematics major, there is some attrition over
Address correspondence to Sandra Fillebrown, Department of Mathematics, SaintJoseph’s University, Philadelphia, PA 19131, USA. E-mail: [email protected]
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the 4 years and we have about 8–10 actuarial science students graduating eachyear. Although these students take many of the same mathematics courses asthe math majors during their first 2 years, they now take far fewer upper-levelmathematics courses as they did in the past.
The number of mathematics majors is quite variable, but we graduateapproximately 6–8 each year. About half to two-thirds of these students areinterested in secondary mathematics education. A few complete all of theirrequirements in 4 years and do their student teaching in the Spring semesterof their senior year. The majority, however, are enrolled in a 5-year combinedB.S. and M.S. program with the student teaching in the graduate (fifth) year.The 5-year program gives students more flexibility in their four undergraduateyears to pursue a minor, study abroad for a semester, and take elective courses.
Thus, the students in our upper-division mathematics courses are now pri-marily mathematics majors, many with an interest in secondary education, anda few students who are pursuing a minor in mathematics or who are doublemajors in mathematics and another discipline, such as actuarial science, com-puter science, or economics. Because the number of majors is relatively small,we offer many required courses such as Real Analysis and Abstract Algebraon a 2-year rotation and these classes as well as most electives contain a mixof juniors and seniors and occasionally some sophomores who entered withadvanced placement.
Across the university, there is a general standard that courses need 12 ormore students enrolled or they are cancelled. However, exceptions are made ona case-by-case basis. In particular, if a course is a major requirement and stu-dents need it to graduate on time, a case can usually be made to run the course.If the enrollment is too small – say only one or two students – the departmentis asked to offer it as an independent study, but with three or more students,required courses are usually allowed to run. On the other hand, if departmentsare asking to run many small courses semester after semester, they are askedto find ways to alleviate the need to do so by offering courses every other year,reconsidering their requirements, offering more independent studies, and so on.
Our freshman- and sophomore-level math classes generally have enoughstudents for this not to be a problem since chemistry, physics, and particu-larly actuarial science majors also take most of the mathematics courses at thislevel. Because we have enough actuarial science majors, upper-level mathe-matics courses that these students are required to take, such as probability andstatistics are also sufficiently enrolled. Some electives required for studentsinterested in secondary education are scheduled every other year in the evening(for example, Geometry, History of Mathematics, and Mathematical ProblemSolving) and so in addition to our undergraduate majors, these courses attractstudents who are returning to school to receive mathematics teaching certifi-cation. These, too, are generally popular enough that we do not need specialpermission to run them. However, the rest of our required and elective mathe-matics classes—such as Real Analysis, Abstract Algebra, Complex Analysis,
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Dynamical Systems, Differential Equations, and Topology—frequently havefewer than 12 students, and we must justify the need to run them with lowenrollments.
2. CAPSTONE IDEA #1
Beginning in 2000, we required all mathematics majors to enroll in a one creditcourse called Capstone Seminar Project. From the course catalog:
Each student, under the guidance of a faculty mentor, will undertake an indepen-dent project culminating in a presentation. The topic may be suggested by thestudent, chosen by the mentor, or undertaken as an extension of material coveredin a mathematics course. The venue for the presentation will be chosen jointly bythe student and the mentor. Students should register for this course in the Springsemester of the senior year. Pass–Fail.
The objectives for the Capstone Seminar Project were twofold: to give ourstudents an experience doing independent research and to provide our studentspractice at giving presentations and communicating mathematics. Both of theseobjectives were stated as goals of the mathematics program and we wanted toensure that all students were meeting these objectives in a more formal wayand not just haphazardly in course work. Because our major already had asignificant number of required and elective courses (12 mathematics coursesbeyond the calculus sequence), we decided to institute this one-credit CapstoneSeminar Project and not add another additional course.
We quickly found that the quality of the projects that students did and theirpresentations on them were extremely variable. At the one end, some studentsdid year-long senior thesis projects and gave presentations in our colloquiumseries or at student mathematics conferences in the area. On the other end,some students simply extended a homework problem from a class and createda poster to display at our annual Math Awareness Day. For example, one stu-dent extended the Gabriel’s Horn problem from Calculus II and explored theconstruction of compact solids of revolution with infinite surface area. Anotherstudent did a statistical analysis of various properties of different brands ofpotato chips. There was no assessment of the projects or presentations; thecourse was Pass–Fail, and to pass a student simply needed to do something.While many students did exceptional work, for others there was no incentiveto work closely with a faculty mentor or to produce quality work.
Another problem was the additional load it put on faculty. Faculty mem-bers were being asked to mentor projects with no compensation and a fewfaculty members were doing the majority of the work. For serious researchprojects this was a welcome task – working closely with motivated students isone of the benefits of teaching in a small university – but for the projects done
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just to fulfill the requirement it was not as satisfying. Although students couldsatisfy this requirement during any semester in their junior or senior year, manywould leave the requirement until their last semester and ask faculty at the lastminute to mentor a project so that they could fulfill the graduation require-ment. Finally, because students could fulfill the requirement in any semesterbut only registered for the Capstone Seminar Project in their second semestersenior year, the bookkeeping related to which student had or had not done theircapstone project became tedious.
3. CAPSTONE IDEA #2
A few years after requiring the Capstone Seminar Project, we were assessingwhether or not it was accomplishing the stated objectives. At the same time,we were also assessing other parts of our curriculum and in particular debatingthe pros and cons of requiring very specific courses versus giving our studentsmore flexibility by reducing the required mathematics courses and substitutinga range of mathematics electives from which they could choose. Historically,we had a fairly rigorous but also rigid set of course requirements. In additionto general education courses, a typical math major took two prescribed mathe-matics courses in each of their first four semesters and one required course andone mathematics elective in each of their final four semesters. However, anystudent interested in either actuarial science or secondary education needed totake a course in Probability and a course in Statistics and took these courses inthe junior year reducing their true electives to two. In addition, students inter-ested in secondary education needed to take a course in Geometry and one inHistory of Mathematics leaving them no mathematics electives.
After much discussion, we decided to significantly revise the curriculum.We took out a sophomore-level course called Problem Solving, no longerrequired Complex Analysis, required only one semester of Abstract Algebrainstead of two, and removed the one credit Pass–Fail Capstone Seminar Project.At the same time, we instituted a regular three credit course called ProblemSolving Capstone. From the course catalog:
This course is intended to provide a capstone experience to senior mathematicsmajors. Students will tackle difficult problems by bringing to bear the knowl-edge and techniques they have gained throughout their major studies. Solutionswill typically require the synthesis of material from two or more courses.Prerequisites: one semester of Abstract Algebra; one semester of Real Analysis.
Problem Solving Capstone was taught for the first time in Spring 2007 andagain in the Spring of 2008 and 2009. A different faculty member taught thecourse each year. Two used Proofs from THE BOOK, by Martin Aigner and
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Günter M. Zeigler [1] and the third used a modified R.L. Moore approach anddid not use a text. Although the course was a general problem-solving course,each instructor chose to spend considerable time delving deeply into one topic.The focus of the course varied with the interests of who was teaching it: oneyear it was topology, another combinatorics, and the third, functional analysis.
Based on student and faculty feedback, the course had some shortcom-ings but was generally successful on a number of fronts. Students worked asa team and the class was essentially a problem seminar. All students in theclass knew each other and had taken many classes together so were comfort-able working together, and the format of the course gave students other thanthe usual “stars,” the ability to occasionally shine. The course did pull togetherideas from several courses and so did meet the objective of integrating at leastsome of the undergraduate curriculum. Although, no new research was actu-ally done, students did get some idea of what mathematical research is likeby tackling problems outside of a textbook setting where you generally knowwhat techniques and theorems are likely to be helpful. Students were requiredto present problem solutions to the class on a regular basis and so practicedcommunicating mathematics.
There were, of course, some shortcomings in the Problem SolvingCapstone course. Most students did not remember as much from previoussemesters as we expected and so more time was spent reviewing material thanwe anticipated. Each instructor chose a fairly theoretical field on which to focusand material from more applied courses that students had taken was not inte-grated into the class. Many students did not think of the course as anythingspecial or useful, but rather just as a requirement that they needed to fulfill.And because some students were excused from the capstone course – thosedoing year-long honors or independent research projects and those doing theirstudent teaching, for example – there was not much of a sense of a culminatingexperience for the graduating seniors.
However, overall, the faculty in the department thought the course hadmerit and that it was successful at meeting its stated goals.
4. UNINTENDED CONSEQUENCES
Unfortunately, offering the Problem Solving Capstone for seniors every Springhad an unintended consequence. As noted above, courses with fewer than12 students come under scrutiny by the Dean’s Office and we must justifythe need to offer them. We almost always got permission to run two or three ayear without much difficulty since we also offer many, many general educationcourses that reach their maximum capacity of students.
The Problem Solving Capstone course was only taken by seniors, andnot even by all of the seniors. As noted above, students doing their studentteaching or doing independent research projects were not required to take the
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capstone course. Also, when we first introduced the course, the seniors inter-ested in actuarial science were still mathematics majors and so were requiredto take the course, but, by 2009, seniors interested in actuarial science were nolonger mathematics majors and were not required to take the Problem SolvingCapstone. Thus, the course always had fewer than 12 students and by 2009 farfewer. However, since it was required for graduation, it was allowed to run andwe had to offer it.
This usually meant that all other electives we tried to run that semesterwould also have too few students. In some sense, the seniors were being“siphoned off” into the Problem Solving Capstone course, leaving too fewstudents to populate the other electives we wanted to offer. We would almostalways have to cancel all but one other mathematics elective, essentially turn-ing the “elective” into a requirement. This was exactly the opposite of what wehad intended when we revised the math major curriculum.
In looking more carefully at the Problem Solving Capstone course we real-ized a few things. Since the Problem Solving Capstone course was alwaystaught around a central theme, the material covered would be similar to thematerial covered in an elective course on the same topic. In addition, whilean upper-level course with a pre-requisite of either Real Analysis or AbstractAlgebra would not require students to integrate material from all seven of theirprevious semesters, it would integrate material from several courses at least.Thus, from a content point of view, we did not see a significant advantage ofthe “senior only” Problem Solving Capstone over an elective with appropriatepre-requisites but open to both juniors and seniors.
We also realized that the majority of our majors were already getting aresearch experience and practice communicating mathematics. As mentionedpreviously, we regularly have a few students doing year-long independentresearch projects and presenting their results at professional meetings and indepartment colloquia. All students in our 5-year mathematics education pro-gram do a summer research project at the start of their fifth year as part of therequirements of that program and so have a research experience. In addition,many of our majors now participate in the SJU Summer Scholars program.This program has expanded significantly over the last few years and we havebeen able to fund all of the mathematics majors that apply. Summer Scholarsstudents are paid for 10 weeks in the summer to work with a faculty mentordoing research and there is a weekly seminar where students from all disci-plines present their work, giving our majors an opportunity to practice theircommunication skills. Thus, almost all of our majors are engaged in a researchproject during their time at SJU and give presentations on their results.
In our ongoing assessment of our curriculum, we decided that whilethere were benefits of the Problem Solving Capstone, they were not signifi-cant enough to justify continuing to require the course. Instead, in the Springsemester, we now offer a series of electives on a rotating basis that are acces-sible to both juniors and seniors that have either Real Analysis or Abstract
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Algebra as a pre-requisite (the pre-requisite being whichever course was taughtin the Fall semester). These courses, together with the research projects that anincreasing number of our majors are doing, meet the goals and objectives thatthe Problem Solving Capstone was designed to address. Since the courses areopen to both juniors and seniors, they usually have sufficient numbers to not bein danger of cancellation and so we have the ability to offer other electives thatmight need to run with small numbers of students.
We still occasionally have to cancel courses, but now, since there areno restrictions on which ones need to run since none are required, we havemore flexibility in how this is handled. The net result is that students havesome choice about what they take and the faculty members in the mathemat-ics department have some choice in what is offered. Although we thought theProblem Solving Capstone was a good course that met its stated objectives, webelieve our students are better served in the long run by requiring an additionalmathematics elective open to both juniors and seniors instead of the seniorcapstone.
REFERENCE
1. Aigner, M. and G. Ziegler. 2004, Proofs from THE BOOK. New York:Springer-Verlag.
BIOGRAPHICAL SKETCH
Sandra Fillebrown has been at Saint Joseph’s University since obtaining herPh.D. from Lehigh University in 1986. In addition to mathematics and teachingmathematics, she is passionate about the sport of orienteering.
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