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<ul><li><p>This article was downloaded by: [Simon Fraser University]On: 12 November 2014, At: 08:03Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK</p><p>PRIMUS: Problems, Resources,and Issues in MathematicsUndergraduate StudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20</p><p>Capstone: Why it Didn't Workfor UsSandra FillebrownAccepted author version posted online: 12 Mar2013.Published online: 10 May 2013.</p><p>To cite this article: Sandra Fillebrown (2013) Capstone: Why it Didn't Work for Us,PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 23:4,412-418, DOI: 10.1080/10511970.2012.716146</p><p>To link to this article: http://dx.doi.org/10.1080/10511970.2012.716146</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the Content) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of theContent.</p><p>http://www.tandfonline.com/loi/upri20http://www.tandfonline.com/action/showCitFormats?doi=10.1080/10511970.2012.716146http://dx.doi.org/10.1080/10511970.2012.716146</p></li><li><p>This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan,sub-licensing, systematic supply, or distribution in any form to anyone isexpressly forbidden. Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Sim</p><p>on F</p><p>rase</p><p>r U</p><p>nive</p><p>rsity</p><p>] at</p><p> 08:</p><p>03 1</p><p>2 N</p><p>ovem</p><p>ber </p><p>2014</p><p>http://www.tandfonline.com/page/terms-and-conditions</p></li><li><p>PRIMUS, 23(4): 412418, 2013Copyright Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970.2012.716146</p><p>Capstone: Why it Didnt Work for Us</p><p>Sandra Fillebrown</p><p>Abstract: Saint Josephs 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.</p><p>Keywords: Capstone course, problem seminar.</p><p>1. BACKGROUND</p><p>A little background on the mathematics major at Saint Josephs Universityis necessary to understand why we implemented and then abandoned ourcapstone course. Beginning in the 1970s 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 2000s, however, these curriculumsbegan to diverge more and more and fewer and fewer faculty taught courses inboth disciplines. The two departments finally split in 2010.</p><p>Also, beginning in the late 1990s, 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</p><p>Address correspondence to Sandra Fillebrown, Department of Mathematics, SaintJosephs University, Philadelphia, PA 19131, USA. E-mail: sandra.fillebrown@sju.edu</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Sim</p><p>on F</p><p>rase</p><p>r U</p><p>nive</p><p>rsity</p><p>] at</p><p> 08:</p><p>03 1</p><p>2 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>Capstone: Didnt Work 413</p><p>the 4 years and we have about 810 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.</p><p>The number of mathematics majors is quite variable, but we graduateapproximately 68 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.</p><p>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.</p><p>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.</p><p>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 classessuch as Real Analysis, Abstract Algebra, Complex Analysis,</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Sim</p><p>on F</p><p>rase</p><p>r U</p><p>nive</p><p>rsity</p><p>] at</p><p> 08:</p><p>03 1</p><p>2 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>414 Fillebrown</p><p>Dynamical Systems, Differential Equations, and Topologyfrequently havefewer than 12 students, and we must justify the need to run them with lowenrollments.</p><p>2. CAPSTONE IDEA #1</p><p>Beginning in 2000, we required all mathematics majors to enroll in a one creditcourse called Capstone Seminar Project. From the course catalog:</p><p>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. PassFail.</p><p>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.</p><p>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 Gabriels 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 PassFail, 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.</p><p>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</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Sim</p><p>on F</p><p>rase</p><p>r U</p><p>nive</p><p>rsity</p><p>] at</p><p> 08:</p><p>03 1</p><p>2 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>Capstone: Didnt Work 415</p><p>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.</p><p>3. CAPSTONE IDEA #2</p><p>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.</p><p>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 PassFail Capstone Seminar Project.At the same time, we instituted a regular three credit course called ProblemSolving Capstone. From the course catalog:</p><p>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.</p><p>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</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Sim</p><p>on F</p><p>rase</p><p>r U</p><p>nive</p><p>rsity</p><p>] at</p><p> 08:</p><p>03 1</p><p>2 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>416 Fillebrown</p><p>Gnter 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.</p><p>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 r...</p></li></ul>