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PROMOTING EXCELLENCEIN UNDERGRADUATEMATHEMATICS THROUGHWORKSHOPS BASED ONCOLLABORATIVE LEARNINGA. Bathi Kasturiarachi BSc and PhD aa Department of Mathematics and ComputerScience , Kent State University - Stark Campus ,6000 Frank Ave., NW, Canton, OH, 44720, USAE-mail:Published online: 13 Aug 2007.
To cite this article: A. Bathi Kasturiarachi BSc and PhD (1997) PROMOTINGEXCELLENCE IN UNDERGRADUATE MATHEMATICS THROUGH WORKSHOPSBASED ON COLLABORATIVE LEARNING, PRIMUS: Problems, Resources,and Issues in Mathematics Undergraduate Studies, 7:2, 147-163, DOI:10.1080/10511979708965856
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PROMOTING EXCELLENCEIN UNDERGRADUATE
MATHEMATICS THROUGHWORKSHOPS BASED ON
COLLABORATIVE LEARNING
A. Bathi Kasturiarachi
ADDRESS: Department of Mathematics and Computer Scien ce, KentState University - Stark Campus, 6000 Frank Ave ., NW, Canton OR44720 USA. [email protected].
ABSTRACT: Unde rgradua te mathem a tics plays a cent ral and crucial rolein our sys te m of mathematics educa t ion. The goal in this article isto demonstrate, based on the Academic Mastery Program (AMP) atOccid ental College, how to incorporate colla bo rat ive learning into t hemathem atics curr iculum. AMP is a program th at promotes achievement in mathem a tics by offering freshman undergraduates a collaborative learning enviro nme nt that fost er s success . It help s th em toorganize their lives around th eir acade mic interests. The st ruc t ure ofthe program , we be lieve , minimizes the isolation of wom en , minori ti es ,and b orderline st udents by t ra nsforming each st ude nt's cont ribut ioninto the success of a group. We will describ e in detail the form at ofthe weekly two hour work shops, which are supervised by stude nt facilitators . Issu es such as st ude nt recruitment , hiring facilit a tors, anddesigning work sheets will also be discussed.
KEYWORDS: Collaborative learning, st ude nt facilitators, worksh eets.
Und ergraduate mathematics play s a cent ral and crucial role in our system of mathematics education. Our future enginee rs , scient ists, an d mathematicians begin the most important stage of th eir educa t ion in college. It isfor this reason tha t we hav e to focus our at te nt ion at a very ea rly st age onhighly motivated undergraduate students in mathematics. Ernest L. Boyer,
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of the Carnegie Foundation, alerts us to this need when he writes [2J: "I' mbeginning to believe that the 1990s may well come to be remembered as thedecade of the undergraduate in American higher education." The purposeof this paper is to report on a program that promotes achievement in mathematics by offering undergraduates a collaborative learning environmentthat fosters success. The Academic Mastery Program (AMP) at OccidentalCollege was founded in 1990 by the Dean of Faculty together with Donald Goldberg in Mathematics and Chris Craney in Chemistry. While theformat of AMP is similar to that of P. Uri Treisman's Professional Development Program (P DP) at the University of California - Berkeley, fromthe beginning AMP was designed to fit into the residential liberal arts college environme nt at Occidental Coll ege. AMP, like PDP, is a substitutefor rem edial approaches to mathematics education [5,8,9J. AMP is similar in format to PDP at California State Polytechnic University - Pomona,and the Emerging Scholars Program (ESP) at the University of Texas Austin. It provides a supportive enviro nme nt to undergraduates interestedin mathem ati cs, helping them to organize their lives around their academicinterest s. The AMP is administered by two Directors, one in Ch emistry andthe other in Mathem atics. The Dir ectors manage all aspects of th e programand rep or t to t he Dean of Faculty. The AMP at Occidental College encourages all freshman st udents enrolled in t he ta rgete d classes (usually Calculusand Gen eral Ch emistry) to join the program. On ce t hey join the programt hey a re com mitted to attending a weekly two hour work shop run by student facilitators. The task of each work shop is to master the material bycom plet ing a well design ed work sheet. The weekly workshops are basedon collaborative groups. Key eleme nts of th c program arc active faculty involvem ent, weekly work sh ops based on collabora t ive learning led by st ude ntfacili t a tors, and expectations of a strong commit ment and high GPA fromt he part icipan ts. Key players of t he program are th e parti cipants, undergrad uates who join t he prog ram; facili tat or s, up per class stude nts (two inMathem at ics and four in Ch emistry for each academi c year) who facilitatethe workshops; and facul ty, Dir ectors of AMP and all instructors tc achingt he targetcd courses . This a rt icle will describe the overall st ruct ure of AMPin mathematics and its implem entation at Occid ental College up through1995. A maj or portion of t he article will be devoted to obs ervations andideas developed during t hc acade mic year 1994-95 while the author wasDirector of t hc Academic Maste ry Program in Mathematics at OccidentalCollege. Thro ughout t he a rt icle I will provide information to the readerthat migh t be useful in planni ng to adopt a simila r program at a differentinstit u tion.
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The paper has been organized under the following five general headings:
1. Student recruitment and retention
2. Hiring and training of facilitators
3. Designing worksheets
4. Collaborative learning in a workshop environment
5. Building a successful program
At the end of the paper, I will summarize the main points of the article.
STUDENT RECRUITMENT AND RETENTION
A substantial portion of the AMP resources is devoted to student recruitment at the beginning of each semester (more in the fall). The programtargets certain classes and welcomes all freshman students who are willingto strive for their best and are committed to excellence. A special effortis made to attract minority students (African-American and Latino) to theprogram. In the fall and spring, AMP in mathematics was offered for twolevels and three levels of freshman Calculus respectively. Occidental College has two levels of Calculus I which freshman can enroll based on theirexperience and placement exam score; a normal level and a more advanced(experienced) level. As the students move to Calculus II, they maintain thesame levels. A brochure describing the program is sent to all freshman students before the start of the academic year, and this is usually followed by aletter of invitation which is included in the orientation package. During theorientation week freshman can stop at the AMP· One Stop Shop to inquiremore about the program or to sign up. This same week the Director speaksto the students and answers questions about AMP at special sessions. Thestudents can also sign up during the first week of classes.
Students are well informed about the contracts they have to sign whenthey join AMP. Since no credit is attached for participating in AMP, all students have to sign a contract pledging their commitment and cooperation tothe workshop environment. This extra requirement may not be necessaryfor a program that has credit attached to it. AMP has a very strict attendance requirement: only two excused absences are allowed each semester.A record of the attendance is kept by the facilitators. The students whodo not satisfy the attendance requirement are not eligible to receive theCertificate of Recognition at the end of the semester. This certificate canserve as a resume booster. Occidental College does not require extra qualifications for admittance to any major. However, in institutions which offercompetitive majors, such a certificate can be used as an extra qualification.
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Student retention becomes a significant problem as the semester progresses.This is mainly due to poor time management skills of participants who underestimate the commitment required by AMP. Such occurrences can beminimized by effective advising prior to the signing of contracts. Anotherreason cited frequently by AMP dropouts is their desire to join a differentextra-curricular program that overlaps with AMP workshops. If a sufficientnumber of students show desire to join AMP at a more convenient time, thenan evening workshop should be offered. The Director has to keep a close eyeon attendance and be aware of the participants' needs. I frequently visitedthe workshops unannounced to get an unbiased opinion. Often, a considerable amount of encouragement and advice needs to be given to students whofind it difficult to manage their time. The Director and the facilitators mustbe able to help the students in crisis situations. For instance, if a studenthas a problem with attendance, the facilitators confront the student andseek a resolution without the intervention of the Director. These situationsare always reported at the weekly meetings to the Director, at which stage,the Director may intervene if necessary. The participants can also contactthe Director if they have any concerns regarding the workshops. The Director usually extends office hours for those students who need extra helpoutside the workshops.
During the weekly two hour period the workshop meets, the participants are expected to complete a worksheet in their pre-assigned group. Atthe beginning of the workshop, the facilitators give a brief introduction tothe worksheet and remind them of the key concepts from the past weekof lectures. The students find the weekly worksheets and the mock-testworksheets very helpful. The mock-test worksheets are given before theirmid-term and final exams and are the only instance when they do individualwork in the workshop. However, they are given time to discuss their answersin groups at the end of the workshop. The weeks after exams (and beforethe final) are mainly used to loosen up the stressful atmosphere with anever popular pizza party during the workshop together with a short worksheet. At all times it is necessary to keep up a serious working atmospherein the workshop. The workshops are held in computer classrooms. To prevent unnecessary socializing, the facilitators should never allow outsiders towalk into the room during AMP hours (even if they only want to use thecomputers).
The social aspect of AMP is very important to the morale of the groups.Most social events are organized jointly by the AMP programs in Chemistryand Mathematics. In the fall, we had a Halloween party and in the springthere was a year-end softball game between faculty and students. These
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events were well advertised, and eac h participant was allowed to bring aguest who wanted to know more about the program. AMP also makes ita point to invite all faculty members who teach the targeted classes for allthe social events.
At the end of the semester the participants are asked to fill out a comprehensive evaluation of the program and its facilita tors. T he purpose of thisevaluation is to get feedback from participants that will be useful in planningworkshops for t he next semester. In particular, evaluations shed light onstudent attendance, quality of worksheets, and rec ruitment practices. In t hepast the facilit a tors, worksheets, collaborative worksh op environment, andthe program in general have been highly rated by t he participants. The top5 items students liked most in AMP in mathematics during 1994-95 were:
1. Ab le to do worksheets /homework wit h someone in own class.2. Extremely helpful facilita tors.3. Group effort and opportunity to ask questions .4. Abi lity to understand t he class work better and the usefulness of mock
tests.
5. P izza parties!
The top 5 items students disliked most in AMP during 1994-95 were:
1. Extra time commitment outside of class (two hours a week).2. Worksheets sometimes do not relate to class work .3. Noise level during group work .4. Need for better hours (night hours to fit schedules).5. Not enough pizza parties!
In Table 1, we have summarized the enrollment data from the AcademicMastery Program in mathematics for fall 1994. Table 2 has cor respondingdata for sp ring 1995. Tables also indicate data on student attendance .
Course Course AMP % attending % attendingenrolled enro lled > 50% > 80%
Basic Calc I 75 9 (12%) 77.8% 66.7%(2 sections)
Basic Calc I 61 16 (26.2%) 93.8% 62.5%experienced (2 sec.)
Multivariable Calc 10 3 (30%) 100% 100%(1 section)
Table 1. Enrollment and Attendance data for AMP-mathem at ics: fall 1994.
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Course Course AMP % attending % attendingenrolled enrolled > 50% > 80%
Basi c Calc II 59 10 (16 .9%) 90% 30%(2 sections)
Basic Calc II 50 6 (12%) 16.7% 0%experienced (2 sec.)
Bas ic Calc I 14 4 (28.6%) 75% 75%(1 section)
Table 2. Enrollment and Attendance data for AMP-mathematics: spring 1995.
The enrollme nt and attendance figur es above showed a marked improvement from corres ponding numbers from th e first three years of AMP (19901993). Despite our best efforts, the spring semester enrollme nt and a ttendan ce was not enco uraging. This t rend occurred only in Ma th em at ics andnot so much in Ch emi stry. On e reason may have been the fact that one ofthe two mathem atics facilitators decided to take leave-of-absence from Occide ntal College for th e spring semester du e to famil y commitme nts . Thisleft th e pro gram with only one facilitator for the first three weeks of t hespring semester . During th ese three weeks th e at te ndance drop ped , and didnot pick up significant ly even afte r a replacem ent was found.
The AMP pr ogram in Math em atics is pr esently improving and expanding under a new Dir ector. AMP is now op en to freshman enrolled in allfirst- year cou rses. The attendan ce has increased (espec ially in t he spring),and a t t rit ion rates have bee n substantially reduced. The success rate of parti cipants of AMP in ma th em atics has not been studied formally. However ,pr eliminary data indicates that three out of the f our students who declaredtheir major as ma th em atics in t he beginn ing of th eir sophomo re year, werefrom AMP 1994-95 class.
HIRING AND TRAINING OF FACILITATORS
The workshop mod el of P. Uri Treisman [8], sugges ts replacing regular calculus hom ework discussion sessions with a workshop, in which students collabora te on non-routine challenging problems . As described by Gillman [51 ,during th e work shops in th e pr ogr am at Berk eley, "[students] begin work ing the probl em s individually, then, when things get tough , in collaborat ionwith one ano t her. These experiences lead to a st rong sense of communityand the forgin g of lasting friendships." The key is to get stude nts involved
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in the learning process in a comfortable environment. The idea of student facilitators arose from the need to promote student involvement inworkshops in place of direct faculty involvement. This involvement certainly increases the comfort level of students attending the workshops andmay also help minimize isolation of minority and borderline students. Theparticipants tend to build friendships with facilitators, and the facilitatorsin turn act as mentors to the participants. A facilitator can be describedas a student leader who runs the weekly workshops. Each academic yeartwo mathematics facilitators are chosen for the AMP program. While someschools (such as University of California - Berkeley and University of TexasAustin) use graduate students as facilitators , Occidental College uses upperlevel undergraduates (juniors and seniors).
The recruitment process for facilitators usually begins in the springsemester of the previous academic year. Advertisements describing the position are posted on bull etin boards and copies are distributed to each facultymember and all mathematics and physics majors. Faculty members ar eurged to announce these positions in the upper level classes and if necessary the Dir ector of AMP can visit each class for a brief presentation.The pro spective ca ndidates ar e made aware of the basic qualifications theyshould possess: excellent academic record in mathematics, interpersonalskills, programming knowledge, a positively optimistic attitude, and goodcommunicat ion skills. Faculty members are generally able to identify upperclass st ude nts whom they could recommend as good candida tes. Du e to thehigh level of responsibility attached to th e position of a facilitator, t hey arealso informed of the many benefits, such as high er than minimum wage payand the attrac t ive expe rience that can be included in a resume. The averag esalary of a facilitator in 1994-95 academic yea r was $1020 per semester, andwas based on approximately 12 weekly hours of work. The facilitators areinterviewed and select ed from a rich pool of applicants.
In the 1994-95 academic year workshops met on Monday and Wednesdayevenings for two hours. The participants signed up for only one day, so thatthe total time commit me nt towards AMP was two hours per week. Thework shops on each day were identical except for th e participants. In th efall semes te r, work shops were set up for three tar get ed class es: two levels offirst semester calculus (four sections) and a section of multivariable calculus(Refer to Table 1). Each work shop had three work sheet s, one for each levelof calculus. Participants belonging to each level were divided into groupsduring the work shop. In th e spring semester, th e three targeted classesfor workshops were: two levels of second sem est er calc ulus (four sec t ions)and a section of first sem est er calculus (Refer to Table 2) . The duties of
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the facilitators include: organizing and preparing worksheets; fac ilitatingthe workshops; meeting weekly with the Director; recording attendance,personal information, and exam scores of participants; and attending thetargeted classes. The weekly meetings with the Director will serve as a timeto check on the progress of the workshops, deal with issues that participants and facilitators encounter, discuss worksheets, and talk about futureevents. In addition facilitators are encouraged to meet with the instructorsregularly; these meetings are not only very valuable to the facilitators, butthey also give the instructors insight into the program.
Each year AMP organizes a two day training workshop for the facilitators before the commencement of the fall semester. This is jointly organizedby the Chemistry and Mathematics AMP programs and involves the six facilitators from both programs (2 - Mathematics, 4 - Chemistry). This workshop aims to teach the facilitators workshop scenarios ("this could happento you") , group building activities, ways of dividing into groups, examplesof ice-breakers, role-playing, worksheet development, tips on how to initiategroup work, discussions on process approach, and examples of good worksheets. While the presentations and organization are mainly done by theDirectors, the actual activities are carried out in groups by the facilitators.For example, the trainees were asked to list various characteristics of participants, then as a group we brainstormed about how to get appropriateparticipation in each situation. They were asked to list the top 10 facilitatorRules based on what they learned in the training workshop. They listed thefollowing 10 Rules.
1. Understanding the position of a student
2. Having a positive and const ruct ively critical attitude
3. Emphasizing efficient study habits and test-taking skills
4. Being up-to-date and making worksheets challenging
5. Creating a relaxing environment by maintaining a balance betweenwork and play
6. Incorporating variety into the worksheets by writing problems thathave applications in other disciplines
7. Respecting the opinions and personalities of students and other facilitators
8. Always being prepared
9. Not giving out answers easily; encouraging them by eliciting thinking
10. Having good communication skills
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Among the many handouts given at the training workshop was the veryuseful A Guide for Occidental College's Academic Mastery Program Facilitators. This document was prepared by Grace E. Quimbita, former Directorof AMP, and was based on two documents originating from California StatePolytechnic University - Pomona [1, 4]. The Guide describes the programand carefully explains its goals. In addition, it lists the duties and responsibilities of facilitators, and contains a comprehensive list of problems theymight encounter together with suggestions on how to handle each situation.The effectiveness of the training workshop was evaluated later during thesemester. Overall, the facilitators rated their training very highly.
DESIGNING WORKSHEETS
The success of a workshop depends significantly on the quality of the worksheet , and they need to be carefully planned and designed. Worksheetsmust contain sufficient material from the previous week so that studentsfeel comfortable reviewing them. The first portion of the worksheet maycontain questions similar to routine homework problems. In addition, theworksheets should contain ext ra challenging problems and even problemsdrawn in from future material. A well designed worksheet has a list ofsections covered, with problems arranged in the order of difficulty. Theworkshop usually begins with the facilitators making a roll call and dividing the students into small groups and giving them a brief introductionto the worksheet. The participants begin their work in groups, and thefacilitators offer them help when needed. Once the group completes theworksheet , they call upon the facilitators for a final check, at which pointthey get to see the solutions. The participants take the worksheets andsolutions home. No grades are recorded, since everyone in the group mustget the same answers, and furthermore, these answers have to be correct inorder for the facilitators to decide if the group successfully completed theworksheet. The facilitators can offer help on homework problems only afterthe worksheets are completed . One way to ensure the success of worksheetsis to ensure the presence of facilitators at classes. With this in mind AMPhas a requirement that all facilitators attend three targeted class periodsevery fortnight ; these hours appeared in their stipend as normal workinghours. This feature enables the facilitators to get accustomed to the material as well as relearn the important ideas and concepts for that week. Inaddition the Director always encourages them to meet with the respectiveinstructors when necessary (which they did). In fact, some of the facilitators, whose work-hours were below the maximum allowed by the college,
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even became graders for the targeted classes, a practice they found veryhelpful.
The responsibilities of the Director of AMP include hiring of facilitators ,training them , advertising and promoting the program, recruiting students,meetings with facilitators , organizing social events, and evaluating and improving the program. The workshops are run entirely by the facilitators.In AMP, the facilitators are also responsible for designing the weekly worksheets. While this practice may work well with more mature facilitators (i.e.graduate st udents ), some adjustments need to be made if the facilitators areundergraduates. In fact , in the latter situation, I strongly recommend thatthe Director be involved to some extent in the preparation of the worksheets.For instance, the Director, can suggest interesting problems to the facilitators (however, this must be done in a timely fashion - perhaps at the weeklymeeting, before they meet to discuss the worksheet), and must proofreadthe worksheets and solutions before copies are made. The worksheets aregen erally typed , but the solut ions can be handwritten. The participantsget to see the solutions once th ey complete their group work; th ey couldeven take the solu t ions home for later reviews. I found it very useful tohave a gen eral template in LaTeX whi ch could be used in the pr eparationof worksh eets.
The mathematics courses in which freshman enr oll gen erally have a labcompo ne nt . The worksheets oft en include compute r oriented op en questions reflecting ideas they have encounte red in the regular weekly lab sessions. There is growing consensus among th e mathematics community thattechnology has t o be part of most mathematics courses . As David A. Smith(co-founder of Project CALC a t Duke University ) puts it [7]: "Ma t hema tics courses that ignore technology perpetrate a fraud - we have to educateour studen ts for the world they live in, not the one we grew up in." TheAMP workshops usually meet in a compute rized clas s room (wi th about10 terminals aligned along th e walls) which gives th e facilitators the greatconveni en ce of regrouping th e participants around terminals when th e needarises. The calculus courses at Occidental College use Calculu s in Context Iand II [3] and Calculus (Harvard text) [6] as the course texts. Many of th eroutine drill problem s a t th e beginning of th e work she et follow closely th eform at of the problems taken from the latter text. This gives th e participants a cha nce to pr actice problem s similar to th eir homework. The morechallenging open-ended qu estions were similar to th e problem s found in Calculus in Context I and II; these included vari ations of th e S-I-R epidemicmodel, Newto n's Law of Cooling (as an a pp lica t ion of Euler 's Method) , population growt h, Taylor polyn omial calc ulations (Deriv e was used for graph-
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ics), falling body problems, Inverse Fun ction Theorem (Derive was used forgraphics), contour plots, Newton 's Method (using a True Basic program) ,periodic motion problems, and many other interest ing applications of ratesof change , acc umulat ion, and level curves. Clearly, much of th e content int he work sheets must be determin ed according to the needs of each of t hetargeted classes an d t he goals of each indi vidual program.
COLLABORATIVE LEARNING
There has been increasing emphasis on te aching mathematics in a way thatengages students . This is not very surprising, since our audience should notbe a relatively small proportion of st udents bu t the whole class. We needto focus our effort s to maximize the participation at every level of learning. Collaborative learning provides an excellent interactive environment inwhich st udents can learn difficult concepts together. While collabo ration isa social necessity in any workplace, th ere is also evidence suggest ing that iteffects substanti al improvements in tes t scores and peer relationships [10].A few factors such as time const raints and pre-designed syllabi may restrictt he t ime we can allocate for group learning in th e classro om. Successfulgroup learning can , however , be impl emented outside t he classro om oncet he logisti cs of a pr ogram simila r to AMP are set in place.
In mathematics, collabo rative learning can very eas ily be introduced fort he pu rp ose of ret eachin g difficul t concepts. It is importan t to note t hatmost collaborative lea rning meth ods are bas ed on two ass umptions: first ,t hat it may take some st udents several classes to learn a difficult conce pt,and second, st udents general ly learn best from one an other. One of t hebest point s about AMP is t he opportunity t he parti cipants have of meetingst udents from different sections of the same course , since an AMP workshopfor a parti cular course cons ists of st udents from different sections . Theparticipants not only compa re notes but also tend to form st udy groupsoutside of AMP. Indeed , it was observed in 1994-95 that group st udy wasquite commonplace among AMP st udents . In the same yea r the Dir ectorwas abl e to conclude, through informal discussions with minority st udents,that they were more comfort able in classes after th e AMP expe rience.
There is a strong agreement among t he mathematics community aboutour need to focus on problem solving skills of our st udents, as necessitatedby real-life situations. The essence of this can be reached through the introduction of mathematical modeling problems int o the curriculum. On cethe students are trained and exposed to group problem solving skills in theclassroom, it is easier to reiterate the same ideas and methodology in a pro-
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gram similar to AMP. In AMP, it was observed that students not only likedthe challenging modeling problems, but conceded that the cross-disciplinaryproblems in the worksheets were actually helpful to them in other disciplines. I felt that this was a step in the right direction in transformingtheir perception of mathematics as a detached and isolated discipline, toone that has a valued status in all applied sciences. A specific example ofa problem that reaches out to many other disciplines is a growth problem.The exponential growth model used in population growth of a single-species,has its applications in economics and biology. Calculations and comparisonsof the exponen tial growth model and the logistic growth model made up anexcellent worksheet involving substantial group work. This model was latergeneralized to include the more complicated f ermentation process which hasits application in Chemistry.
A few comments are in order concerning how we have gone about addressing the issue of educating the facilitators and faculty on the virtues ofcollaborative learning. Firstly, it is important to have good facilitators whoare not only well versed in mathematics but are also knowledgeable aboutcollaborative learning. However, it is virtually impossible to find manymathematics majors with a strong background in the latter. One way tohandle this sit ua t ion is to do a training workshop (similar to ours) for theselected candidates. Of course, it helps if the facilitators were themselvesat one time participants of AMP. The Director need not be the sole presenter at such a training workshop; in fact the expertise of outside speakerssuch as, educators in Ch emistry or the liberal arts, can greatly enrich thetraining. The Department of Mathematics at Occidental College has for along time embraced an environment in the classroom that promotes collaborative and active learning. After all , th e adaptation of an extra-curricularprogram such as AMP clearly shows that the Department is in the forefrontof curricular and teaching reform. This may not be the case in many otherinstitutions. Sin ce almost all faculty received the majority of their education in traditional lecture courses, getting faculty to buy into the idea ofcollaborative learning might be hard to do. They may show some reluctanceto adopt a new method. Invite them to your class and ask them to observehow occasional group learning works wonders. Accompany them for a visitto a weekly workshop, or invite them to a pizza party where they can talk tothe students. You will be surprised to see how the faculty reacts when theysee their own students doing group work and challenging problems outsideof class. One way to keep a permanent link with faculty is to organize aseminar on innovative teaching methods, where modern developments inteaching can be discussed. Occidental College's Faculty Lunch Discussions,
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organized by the Center for Teaching and Learning (lunch courtesy of theDea n of Faculty), was a forum where pedagogical ideas were presented anddiscussed .
B UILDING A SUCCESSFUL PROGRAM
How ca n one build an excellent ext ra curricular program that augments peergroup support in learni ng, increases the number of st udents who are successful in college mathematics, and, more importantly, promotes excellen cein undergraduate mathematics? While it may be impossible to answer thisquestion complete ly, one could perhaps point out some essential ingredi entsnec essary for a good program, based on expe riences from various institutions across the country. As we mentioned earlier, Professional Development Program was developed by P. Uri Treisman for students at Universityof California - Berkeley. This model was later adapted at various other institutions such as UCLA, Cal Poly Pomona, Cal State Fullerton, Universityof Texas - Au stin , Oberlin , and Occid ental. The following points , I believe,should be consid ered at the beginning when designing a program similar toAMP.
1. The administration of the program should be done by the academi cdepartment and not by the Center for Teaching and Learning or asimila r ca m pus body, because it s goal is achieving excellence in aparticular discipline. Keep in mind that the program is not rem edial.Recruiting particip ants, hiring facilitators, and se tting up goals mu stb e done within the department. The Director of the program shouldbe a regular facu lty member whose partial duties will include directingthe program.
2. The department should carefully choose which freshman courses areto be included in the program. It may be necessary to sta r t a pilotprogram first , in order to correct unanticipated problem s particularto the institution that may arise. One such concern is how to assigncredit to participants. If cred it is ass igned , the program should bemandatory for all freshman . If no credit is assigned , there should besome mechanism to commit the participants to join and remain in theprogram.
3. Faculty support is paramount to the success of any program. Theyhave to be involved and info rmed of the major issues that have an effect on their teaching. In his book, Scholarship Reconsidered: Prioriti es of the Professoriat e [2], Ernest L. Boyer , lays out four funct ions of
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the professoriate: th e scholars hip of discovery; the scholarship of integration; the scholarship of application; and the scholarship of teaching.The first two kinds of scholarship are related to the traditional idea of"research" . According to Boyer, when defined as scholarship, teachingshould both educate and entice future scholars. Regarding the thirdfunction of the professoriate Boyer explai ns: "for a service activity tobe considered for scholarship of application, it must be tied directly toon e 's spe cial field of knowledge and relate to, and flow directly out of,this professional activity." The work and effort of running a programof this nature not only falls into th e ca tegory, schola rs hip of teaching ,but clearly into scholars hip of application as well. After all , such aservice activity has it s roots in one 's special field of knowledge, andflows out into th e freshman student body.
4. It is important to recruit women a nd minorities to the program, sincethey are under represented in mathematical sciences. However , ingeneral , every effort should be made to keep enr ollme nt op en to allfreshman .
In AMP, each year th e Dir ector compa res the et hnic and gender distributions of the AMP st ude nt profile to that of th e ent ire st ude ntprofile. If any particular et hnic or gender group is underrepresentedin AMP, th e staff reaches out to th ese groups in th e first week , encouraging them to enr oll.
5. Outsid e funding for an innovative program of this nature can be difficult . It may be possibl e to obtain grant money to get such a progr amoff the ground. This may include a laboratory grant to equip a classroom with computers, which can be used for th e weekly work shops.The overall funding for th e program should com e from the univer sity.After all , ther e will be annual costs for hiring facilitators, organizingsoc ial events, administration, and work shops. If such a mechanism forfunding is not in place when grant money runs out , the pr ogram isbound t o fail.
In order to give th e reader an idea about the cost of running a program simila r to AMP, we will pr esent some figures from th e 1994-95budget. These figures exclude th e cos ts for partial release time forthe faculty directing th e program. The two AMP programs roughlyreceived $18 ,000 for th e academi c yea r, out of whi ch $13 ,000 went forsalaries of facilitator s and $5000 for other expe nses. Two thirds ofthe cos ts wer e for AMP in Ch emistry whil e on e third was for AMP inMathem atics. Mak ing t he assumption that th er e are 6 differ ent workshops (4 in Ch emi stry, 2 in Mathematics) that meet 14 times during
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a semester, each having on the average 15 students, we can come upwith the following numbers for cost-per-participant:
• Cost for each participant per sem est er - $100
• Cost for each participant f or a single two hour workshop - $6.84
6. Broad support will be needed if significant improvements are to bemade in the program. Try to form a network with other successfulprograms and be pr epared to make changes every year. The first signsof succ ess ca n be felt when faculty say that their students ar e learningmore, or when you see a significant interest among freshman wantingto major or minor in mathematics.
SUMMARY
I have set forth in this article, based on th e Academi c Mastery Programat Occid ental Coll ege, to describe how collabora t ive learning ca n be incorpor ated into the mathematics curriculum. The weekly two hour workshopsst ruc t ured a nd supe rvised by stude nts facilitators, ena ble th e freshman participants to attain a higher level of excellence in mathematics. The struc t ureof th e pr ogram , we believe, minimizes the isolation of women and minoriti es , and tran sforms each st ude nt 's cont ribut ion into the success of a group.The nuts a nd bolts of a program as described may vary according to theneeds of each individual institu tion . The needs, once properly understood ,ca n be built into th e program from the outse t . In shaping our stude nts fort heir roles in real life, we must prepare th em to face the growing number ofcom plex intractable pr obl em s which require colla borat ive work. The challen ge is to create a n environme nt in th e classroom and work sh op where t hest ude nts ca n be actively engaged in th e learn ing process.
ACKNOWLEDGEMENTS
The a ut hor wishes to gra tefully acknowl edg e th e contributions made toAMP by Dr. Linda Lasater - Dir ect or , AMP in Ch emistry at Occid entalCollege. Both programs ben efit ed greatly from her supe rb orga niza t ionalskills . The author sin cerely thanks her for the conversa t ions that ass iste din t he prep ar ation of this a r t icle. The a ut hor also wish es to t hank a for mer colleague, Dr. Michael l\lcDonald , and th e present Dir ect or of AMP inMa th em a tics, Lars Kjeseth, for valua ble comme nts made on this ar t icle .
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REFERENCES
Volume VII Number 2
1. A Handbook for Academic Excellence Workshops. 1990. Sponsoredby College of Engineering MEP and College of Science at California StatePolytechnic, Pomona.
2. Boyer, Ernest L. 1990. Scholarship Reconsidered: Priorities of theProfessoriate . The Carnegie Foundation for the Advancement of Teaching.Princeton, N.J.
3. Callahan, J. et al. 1993. Calculus in Context I and II. New York:W. H. Freeman & Co.
4. Chautauqua Short Course. Sponsored by Charles A. Dana Center forMathematics and Science Education, University of California - Berkeley,and California State Polytechnic, Pomona.
5. Gillman, Leonard. 1990. Teaching Programs that Work. Focus: Thenewsletter of the Math ematical Association of America. 10(1): 7-10.
6. Hughes-Hallet , D. et al. 1992. Calculus. New York: John Wiley &Sons Inc.
7. Smith, David A. 1993. Trends in Calculus. Preparing for a NewCalculus. MAA Notes Vol. 36. Ed : Anita Solow . Washington DC: Mathematics Association of America.
8. Treisman, Phillip Michael (Uri). 1985. A study of the mathematics performance of black students at the University of California, B erkeley.Unpublished doctoral dissertation, University of California, Berkeley.
9. Treisman, Uri . 1992. Studying students studying calculus: A look atthe lives of minority mathematics students in college. College Math ematicsJournal. 23(5): 362-372.
10. Wood K. D. , B. Algozzine, and S. Avett. 1994. Using cooperat ivelearning to meet the needs of high-risk learners. in Teaching Reading toHigh Risk Learners: A Unified Perspectiv e. Boston MA: Allyn and Bacon.
BIOGRAPHICAL SKETCH
Bathi Kasturiarachi is an Assistant Professor of Mathematics at Kent StateUniversity - Stark Campus. He holds a BSc (Hons) degree from Universityof Peradeniya, Sri Lanka (1985) and has a PhD degree from University ofNorth Carolina - Chapel Hill (1993). He spent a year at Duke Universityas an Instructor and a Visiting Assistant Professor at University of NorthCarolina - Chapel Hill. He th en spent another year at Occidental Collegeas an Assistant Professor and Director of th e Acad emic Mastery Programin Mathematics, before moving to his curre nt position. His research inter-
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ests are in applied partial differential equations and mathematics education.He lives in North Canton, Ohio, with his wife Sharon and two daughtersBrittany and Naomi.
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