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PRIMUS: Problems,Resources, and Issues inMathematics UndergraduateStudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20

GROUPS FOR PROOFS:COLLABORATIVE LEARNINGIN A MATHEMATICSREASONING COURSEDaniel S. Alexander a & Luz M. DeAlba MS andPhD aa Department of Mathematics and ComputerScience , Drake University , Des Moines, IA,50311-4505, USA E-mail:Published online: 13 Aug 2007.

To cite this article: Daniel S. Alexander & Luz M. DeAlba MS and PhD (1997)GROUPS FOR PROOFS: COLLABORATIVE LEARNING IN A MATHEMATICS REASONINGCOURSE, PRIMUS: Problems, Resources, and Issues in Mathematics UndergraduateStudies, 7:3, 193-207, DOI: 10.1080/10511979708965860

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Alex ander and DeAlba Groups for Proofs

GROUPS FOR PROOFS:COLLABORATIVE LEARNING INA MATHEMATICS REASONING

COURSE

Daniel S. Alexander and Luz M. DeAlba

ADDRESS: Dep artmen t of Mathematics and Computer Scien ce , DrakeUniversity, Des Moi nes IA 50311-4505 USA. Email : daniel.alex ander@drak e.edu and lu z.d ea lbaOdrake.edu .

ABSTRACT: In t he Mathematics Reas oning co urse a t Drak e Univers itystude nts learn how to cons t ruc t a proof. To this end , t hey studylogic, se t t heory, fun ction s , a nd rela tions. This article describes t hecourse a nd the a ut hors' effor ts to in troduce colla borative, small -group,mixed-ability activit ies in to the classroom. We offer our observationson the case, difficulties , rewards , and demands involved in implement­in g a collaborat ive learning enviro nme nt. 'Ve conclude by sum m a riz­ing stude nt and in struct or a tti tudes towards these collab orative ac­tivities .

KEYvVORDS : Mathematics reasoning, colla borative learning, m athemat­ics worksheets, mixed-ability groups.

INTRODU CTION

In in creasing numbers , under gr aduate ed ucators a re breathing new life intoold material by trying new pedagogical methods. One of t he mo st exc it inginnovations t o emerge from this a t m osp he re of experime ntation is colla bo­rative lea rn ing.

Although t he re a re a variety of s t udies on the ben efits of collabora t ivelearning , t he vast majority of t hese st ud ies focu s on eit he r sec onda ry sc ho olsor undergra dua te offeri ngs below the calculus level. For exam ple, Norwood[5] d escribes her successes with colla borat ive learning with underprepared

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students in a college algebra course one level below pre-calculus. Moreover,she quotes Slavin [7] in support of her observation that there is a "paucity ofstudies examining the effects of coopera t ive learning at the college level" [5,p.245].

We hope to make a small dent in this "paucity" by offering our own,mostly qualitative, account of our experiences with collaborative learningin an upper level mathematics course at Drake University, MathematicsReasoning.

Prior to the fall of 1993, Mathematics Reasoning had been taught usingthe traditional lecture method. But in the fall of 1993 DeAlba decided thatshe would lecture less and instead had the students write proofs duringclasstime. And so the collaborative work began.

Encouraged by DeAlba 's successes with group learning, Alexander de­cided to forego the traditional lecture entirely when he taught the course inthe fall of 1994 and the spring of 1995. In what follows we describe the Math­ematics Reasoning course, the collaborative exercises we developed, and theway we structured our groups. We close with a series of observations andconclusions drawn from our experiences with collaborative learning.

T H E COURSE

The Mathematics Reasoning course is required for mathematics and math­ematics education majors as well as for mathematics minors. However theclass often appeals to students from other disciplines including physics, com­puter science, and actuarial science-we have even had po litical science ma­jors. The course is offered once a semester and while class sizes vary, theytend to remain relatively small. For example, each of the three sections wetaught had about twenty students. The class met for one hour, three timesper week , for fourteen weeks.

The prerequisites for Mathematics Reasoning are two semesters of calcu­lus, although most st udents have also taken linear algebra. The Reasoningcourse itself is a prerequisite for almost all the upper division mathematicscourses, including Discrete Mathematics, Modern Geometry, Number The­ory, History of Mathematics, Abstract Algebra, Real Analysis, AdvancedLinear Algebra and Complex Variables.

Although the goal of the course is to teach students to write a solid proof,the course also seeks to help students become conversant in the languageof mathematics and comfortable with the notion that important ideas inmathematics are advanced by rigorous proof.

The topics covered in this course are Logic and Proofs, Set Theory,

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Relations and Functions. The course typically conclude s with a section oncardinality since the differen ce in ca rdinality between the rational numbersand the real numbers shows the students how careful us e of mathematicalreasoning can lead to quite un expected-and ent irely non-intuitive-results.There are many fine textbooks for this course; we used Smith, Eggen andSt. Andre [8]. A course sy llabus, worksh eets and t ests are available on theWorld Wide Web at

http://zelle.drake.edu/dan/MathReason/MathReasonHP.html.

Despite the prer equisite of two se me ste rs of calculus, the mathematicalprepara ti on of students wh o tak e Mathematical Reasoning varies a greatdeal. Because of this , we soug ht a mathematical subject that would se rveas second ary focu s of the co urse, a topic sufficient ly intuitive to be accessibleto our stude nts yet on e not covere d in any of the prerequisite courses.

After looking into the matter quite ca re fully, we chose eleme nt a ry num­ber theory. Students com ing into the course already understand suc h ele­m entary conc ep ts as "divides", "is a factor of', "is even", etc. Moreover,despite the fact that many of the s t ude nts were unfamiliar with the standardnota ti on used in number theory, the notation itself was not a t all cumber­som e a nd, ther efore, did not pr esent a ny real problems (althou gh it shouldbe noted that so me st ude nts initially confused alb with a/ b).

Not su rprising ly, many of the initial collabora t ive clas sroom ac t ivit iesinvol ved wri ting proofs of such pr op osition s as "If alb a nd blc show t hat al e"a nd "Show that ,j2 is irrati on al" . As we had hoped , the fact that t he con ten tof t he ea rly prop osi ti on s was not hard to fathom gene ra lly all owed stude ntsto conc ent ra t e on the m echanics of wri ting a proof. Some students, however,found divisibility proofs suc h as th e firs t one lis ted a bove so "obvious" t hatthey had difficulty wri ting co he re nt proofs. Nonetheless, we found thatthis early em p hasis on number theory enc ouraged st ude nts to in t ernalizethe process of writing a proof and eased t heir transition into the morechalle ng ing to pics of se t theory, fun ction s and rela ti on s , and card inality.

THE MOTIVATION FORCOLLABORATIVE LEARNING

The principal goal of the Mathematics Reasoning course is to t each st ude ntshow to write a clear proof, and collabora t ive, small-gr oup learning offered usa potentially powerful m eans to accom plish this goal. First of all, learninghow to cons t ruct a mathematical proof shares many simila rit ies with learn­ing a new language. Sin ce the best way to learn a language is to practiceit as mu ch as possible , we decid ed t hat st ruc t uring the clas sroom a round

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colla bora t ive, proof-writing activities afforded st udents excellent opportu­nities to both write proofs a nd receive immediate feedback. Moreov er, thena ture of the Mathematics Reasoning cou rse is some wha t unique since thecontent of the course is not a body of theorems or specific skills bu t is in steadthe m astery of a process, namely that of writing a proof. Because of this ,we wanted to give st ude nts as mu ch expe rienc e in const ruc t ing proofs aspossible . So , rather than hope that st ude nts somehow attain a proficien cyin proofs by sim ply watching the in structor write proofs on the board , wemoved the stude nts one step clo ser to mastery of the proof writing pro­cess by letting them spend their class time writing the proofs themselves insmall , mixed-ability groups.

METHODOLOGY

Structure of the Semester

To est a blish the com mon ground of number theory, we spe nt the first weekof the cou rse lecturing on such topics as

• natural numbers, including the Fundamental Theorem of Arithmeticand orde r relat.ions ,

• int.egers , including factors and divisors,

• rational numbers, and

• real number s, including the Law of Trichot.omy a nd the Archimed eanPropert.y.

Aft.er this init.ial series of lect.ure/discussions , clas ses were gen erally de­voted to small-group , mixed -abilit.y, colla bora t.ive activities (to be describedin more d et.ail below) , although from time t.o t.ime we would occasiona llypresent m ini -lectures t.o introduce new t.opics or clear up difficulties .

A Typical Class

After t.he first. week of the course, students were expected to come to classready to begin group act.ivities . In particular, t.hey were expecte d to haveread in advance the section of the text relevant to the day 's activities andto have begun their homework assignment. Groups were formed during thesecond week of the semester (see the next section for details) , and studentssat with their fellow group m embers as soon as they arrived in the classroom.

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Some groups sat side-by-side while others rearranged their desks so that theyfaced one another.

Unless there was an unfinished worksheet from the previous class meet­ing, the instructor generally began by first answering questions about thetext or homework problems, and then distributing the day's worksheet toeach member of each group. The worksheets and the composition of thegroups are discussed below. Aside from answering questions which mightbe relevant to the class as a whole, the instructor guided the groups in­dividually by offering questions, hints, and suggestions, as depicted in thefollowing recreation of one of DeAlba's classes:

DeAlba arrives in the classroom to find the groups alreadyhuddled over the following worksheet, which was not completedduring the previous class: Prove that if n is a natural number,

1 1then - < ---. Work backward and then forward, and then

9n - 2n + 7prove it again by induction.

After a while some groups finally realize that the instructoris there and begin asking for help. Kim has raised her hand andDeAlba approaches her group.

Kim: What do you mean by "work backward"?DeAlba: Look at the inequality. What other inequality needs

to be true so that it becomes true?Michael: Can I cross-multiply?DeAlba: Kim, What do you think? Can he cross-multiply?Kim : Yes?DeAlba: You're on the right track. Now, just think, if you

do cross-multiply, is there anything that you need to be carefulabout?

Kim: Oh! I see! All the numbers are positive, so when wecross-multiply the inequality is still true.

DeAlba: Correct! The inequality is not reversed, and then?Michael: Well, we get another inequality, 2n + 7 :S 9n.DeAlba: Now what other inequality needs to be true for the

new one to be true?Michael: Oh! I get it. We just keep going until we get

something simple that must be true!DeAlba: Yes! Work on this and I'll be back to check with

you.DeAlba goes over to assist another group that is working on

the same problem but is using induction.

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Scott: We're stuck here! We're trying to prove the inequalityby induction, but we don't know what to do next.

Danielle: Yes, the case for n = 1 is very simple, and we canwrite down the induction hypothesis without any problem, butit's the next step that's a problem.

DeAlba: Maybe it would help if you wrote down exactly whatit is that you want to prove. Also, since you already solved thisproblem using the "backward and forward" method, maybe youcan get some ideas from that proof. Think about this for aminute.

DeAlba goes back to Michael and Kim who seem to be donewith their problem.

Michael: We're done. We get the inequality n ~ 1, so thewhole inequality is true!

DeAlba: So, if you're supposed to work "backward and thenforward," can you tell me from your work, which part is which?

Michael: Mmm. Are you telling us that we're not done? Westill need the "forward" part!

Kim: Oh! Can we just write the steps in the opposite direc­tion?

DeAlba: Correct! You need to write the "forwa rd " phaseneatly and with logical expla na t io ns.

Kim: That 's easy!Michael: I see! The "backward" part is like the scratch work!

DeAlba comes back to Scott and Danielle, who seem to havemade some progress with their proof by induction.

Scott: We have the backwards inequality from the otherproof, 2n + 7 ~ 9n, and this is the one that we want to provenow, 2(n + 1) + 7 ~ 9(n + 1).

DeAlba: O.k. Now how can you get from one to the other?Danielle: Can we just add 2 to one side and 9 to the other?Scott: Oh! 9 is bigger than 2, so if we add the bigger number

to the bigger side, and the smaller number to the smaller side,it works!

DeAlba: It does, Danielle, doesn't it? Now all you need todo is to write it down clearly with all of the steps explained.

As DeAlba goes over to help yet another group, she seesScott and Danielle leaning over to talk to Kim and Michael.She listens in for a while ... they are trading ideas on the proofby induction .

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The classroom activities continue in this manner. DeAlbamakes contact with all of the groups several times.

Two benefits of this close student-faculty interaction are obvious: notonly can the instructor give immediate and individual feedback to eachgroup, but the instructor comes into much closer contact with his or herstudents than is generally the case in a lecture-based course.

The Composition of the Groups

Both authors used groups of two or three students. The small size of eachgroup not only allowed each group member ample opportunity to contributehis or her own thoughts to the group, but it also made it difficult for anindividual not to participate.

Since the typical class size of Mathematics Reasoning is about twentystudents per section, a group size of two or three meant that there wereabout seven to ten groups per class. This allowed an active and agile in­structor to visit each group several times each class.

We sought a balance of ability and temperament within each group.Examples of the kind of balanced groups for which we searched include

• a quick thinker with poor writing skills and a better writer who is aslower thinker,

• a highly skilled student who has good tutoring skills and a strugglingstudent,

• three highly verbal students,

• an older student who has been away from mathematics for some timeand an undergraduate with a strong mathematics background, and

• three students with different majors.

Aside from the obvious, such as not placing two talkative students whoare good friends in the same group, we found that achieving this sort ofbalance is something that does not happen immediately. Therefore, we ini­tially allowed students to form groups on their own. After taking a weekor so to familiarize ourselves with the abilities and personalities of our stu­dents, we readjusted the composition of the groups in hopes of attaininga better balance within each group. This reshuffling continued throughoutthe semester, with DeAlba readjusting groups weekly while Alexander didso every three to four weeks.

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This frequent readjustment of the groups also widens a student's circleof acquaintances which not only fosters interaction between the groups­something the authors explicit ly en couraged-but it also forestalls a poten­tial drawback to small-group learning, namely the possibility that studentsbecome isolated from those who are not in their group. The authors werequite pleased that the various groups frequently com munica t ed with oneanother, and it was not uncommon for on e group to ask another for help ifthe instructor was engag ed .

The Worksheets

Although ther e was mu ch simila rity between the worksheets we used , eac h ofus d esigned his or her own. It sho uld be noted that since Alexander taughtthe course after DeAlba, he had th e good fortune of using her worksh eets asa model for his own. Every st ude nt was given the same worksheet a nd eac hgroup attempted to com plete the worksh eet colla bora t ively during clas s.

We did not routinely hand out solut ions to the worksh eets or otherwisegive co m plete so lut ions to the class a" a wh ole; instead we left com pletedworksh eets on file at the library. The reason for this was two-fold. Ont he on e hand , t he instructor visited each grou p sufficientl y often so tha tin gen eral eac h group successfully completed their worksh eet. On t he othe rhand , t he instructor placed the responsibility for com plet ion on t he studen ts .If , at the end of a class meeting, a student was un sati sfied with his or herprogress on a particular worksh eet , he or sh e was expected to spend timeouts ide of clas s to co m plete it. For exam ple, the st ude nt could cons ult t heprofesso r or view the work sh eet file a t the library.

A ty pical worksh eet focu sed on on e or more of the following activi ti es:

• making st a te me nts about various mathematical objects, for exam ple,determining wh ether or not t he mapping f(n , m) = 2m

- 1(2n-l) fromN x N to N is a bij ection ,

• reading and in terpreting a n unfamiliar mathematical sta te me nt, forexam ple, introducing the conce pt of a zero divisor, and then findinga ll t he ze ro di visors in Z6,

• co nst ructing example~l(l counte r exam ples, for exam ple, finding a

co unte r exa m ple to U A = U A,A EC A EC

• proving a proposition , often by more than one method, as was don ea bove in the recreation of on e of DeAlba's classes, and

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• determining wh ether a giv en st a tement is true or fal se, and t he n "sa l­vaging" it if it is fal se by modifying it so it becomes true.

Studen t s ge ne ra lly found the "s a lvage" worksh eets exe rc ises the mostchal lenging, which of cou rse make t he m quite enjoya ble for the inst ru ctor.Her e 's a recreation of one of Alex ander 's classes showing the st udents atwork on a sa lvage projec t .

The students a re working on the following problem: W ritethe f oll owing in sym bolic terms and either (1) prove, or (2) dis ­prove, sa lvag e and prov e th e sa lva ged version. Let a, b, e EN,and suppose that a lbc. Then a lb or ale .

Alexander is prowling around t he room, looking for studentsin trouble. It doesn 't take long to find some. Leon, who's work ­in g wi th Allan a nd Caro line, a t tracts Alexa nde r's attention:

Leon: Why do you make such a big deal a bout salvaging?The statement 's true isn ' t it.

Alexander: W hy d o you say t hat?Caroline : We t ried several examples a nd t hey were all true,

so we t hink t he statement 's true .Alexander: Give m e a n example.

Leon: ' Veil , a = 2 a nd be = 2 * 3.Caroline: And a = 5 and be = 10 * 10.Alexander: Try some othe r co mbinations whieh equal 100.

Leon: a = 5 and be = 4 * 25.Carolin e: Wait! The statement isn ' t t ru e! T ry a = 50 and

be = 4 * 25 .Leo n: Oh! I see! It doesn't work if a is greater t han both of

t he two nu mbers!Alexander: You got your co unter example! But is t hat the

strong est condition you can find?All an : What do you m ean by "st ronge r" ?Alexander: Good question . I m ean one t hat covers t he case

you just m en ti on ed as well as lo ts of othe rs.Allan: I guess I see wha t yo u mean by "stro nger" now , but

I'm clueless about t he salvage. Can yo u give m e a hint?Alexander: We ll, t hi nk abou t t he kind of numbers you are

us ing . Oops! Gotta run . Ano t he r group need s so me help . Thinka bout it - I'm sure yo u' ll figure it ou t. I'll come back a nd cheekon you in a wh ile.

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Alexander hurries to another group who also initially believethe statement is true as stated. Then he hustles over to a groupwho have found a counter example, but are wrestling with a sal­vage. And so the period goes until, with only about ten minutesremaining, Alexander finally checks back with the first group,who are writing up their results:

Alexander: Did you come up with a salvage?

Leon: Yeah, we think so. If a is prime then the statement istrue. 'Ve know it's true, but we can't seem to prove it.

Alexander: Always the hard part. I just visited a couple ofother groups who are having the same problem. Hmm. There'snot much time left. Maybe I should talk to the entire class.

Alexander (Addressing the class): O.k., everyone. (Mosteveryone listens, but a few students keep on working, which isfine with Alexander. Collaborative work has accustomed him tostudents so intent in their own work that they politely ignore himwhen he addresses the class if they don 't need help.) Everyone'sgot the salvage, but how many people are having problem withthe proof? (Several hands shoot up.) How many have triedan indirect proof. (Fewer hands go up .) I'll get you started.Assume that a does not divide b. What would you need toshow?

Rebecca: That a doesn't divide c?

Alexander: Right. And what don't you want to forget abouta?

Tom: That it's prime?

Alexander: You got it.

The remainder of the class meeting passes with Alexanderassisting various groups with the finer points of their proofs. Noteveryone is finished by the time he reminds everyone that theclass is over. Many students pack up quickly and leave. Otherskeep working. Alexander again reminds those who remain thatanother class (taught be a professor who's always impatient tobegin his class) needs the room. He smiles. He's used to this,too.

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Alexander and DeAlba

Grading

ASSESSMENT

Groups for Proofs

The worksheets were not graded. Grades for the course were based onindividual activities such as homework, quizzes and exams. Nonetheless,the collaborative activities afforded us an excellent opportunity to see thestudents' thought processes firsthand, which in turn influenced our grading.Several students whose letter grades were borderline received the highergrade simply because their work on the classroom exercises indicated thatthcy understood the material better than their scores on homeworks or testssuggestcd. However, our close interaction with students cut both ways: Aborderline B/C student would receive a C if he or she did not demonstratea strong grasp of the material during group activities.

The Students' Assessment of the Course

At thc end of the semester students completed

• an 18-question Special Questionnaire taken in its entirety from [3,pp. 62-64] to assess their collaborative experience, and

• a Department Instructor Evaluation designed by the Department ofMathematics and Computer Science.

Student Responses to the Special Questionnaire

We gave the Special Questionnairc to students at the end of the follow­ing semesters: Fall 1993 (DeAlba), Fall 1994 (Alexander) and Spring 1995(Alexander). A total of 47 students responded to the questionnaire. Not allof the students were present when the questionnaire was administered, andthe percentages refer to the percent of the respondents.

• For 57% of the students, Mathematics Reasoning was their "first ex­pcrience in a discussion group."

• Only 17% of the students had previously taken a mathematics coursewhich used "learning through group discussion."

• Thc initial reaction of thc students to "learning through group dis­cussion" was as follows: 13% of the students were "enthusiastic" andanothcr 28% thought it "sounds o.k.," so 41% of the students had apositive initial reaction to the prospect of small-group collaborativelearning.

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• 81% of the students found that "learning through group discussion[had] been useful in the mastery of the material that otherwise wouldhave been difficult."

• 43% of the students felt that their "ability to think critically" had been"enhanced by learning through group discussion." An additional 52%believed that their critical abilities had been enhanced "somewhat."

• The "group sessions" made "the learning task more enjoyable" for85% of the students.

• Over 87% of the students felt that the "subject matter in the bookwas adequately and efficiently covered."

• 85% of the students said that "if the occasion should arise" that theywere in a professional capacity to teach, they would "use learningthrough group discussion" as an educational tool. (Only 27 studentsresponded to this question since it was inadvertently omitted on oneset of questionnaires)

The following comments were III response to discussion questions Ill ­

eluded in the questionnaire:

• Asked to list "the major advantages of learning through group discus­sion as compared to the lecture method," one student responded that"I remember the techniques better when 1 try to work through themon my own first. Also, everyone develops their own special variationon the techniques; theirs is not simply a copy of the instructor's wayof doing things . .. "

• "T he student gets supreme satisfaction from solving a problem .. . sat­isfaction not always present in a lecture."

• "It made me feel that 1 was getting something out of class. 1 couldwork at my own pace .. . it [group work] didn't insult my intelligenceby being talked at."

• "I got a lot more practice doing proofs [as compared to the lecturemethod] ... 1 felt really great when we figured out something on ourown, and the ideas we didn't figure out kept going through my minduntil 1 understood or had to ask the professor."

• "Students pay attention because they are doing the thinking. Withlecture, students daydream."

• "It enables you to really see whether you grasp the material beforeyou leave the classroom."

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Student Comments from the Department Instructor Evaluation

W e administered the Department Instructor Evaluation at the same timeas the Special Questionnaire. Below are sel ected excerp t s .

• "At firs t I thought [the in structor] needed to lecture more, but thenthe small groups sta r ted working. I'm glad this is how it was done.I really think I understood it better by being able to discuss it withpartners."

• "T he group learning slowed it down. That's bad, but I learned morewith that. Kind of a double edged sword."

• "I really did enjoy t he format [the in structor] crea ted . Stick with it! "

• "M ay be a little more lecture, but I really did enjoy working in groups."About a third of the students in the Fall 94 and Spring 95 sec t ionsechoed this desire for a "lit t le" more lecturing on eit her the SpecialQuestionnaire or the Department Evaluation. One s t ude nt expre sseda desire for a 50-50 mix of lecture and group activity. No stud entsugges ted that the group activities b e dropped altogether.

CONCLUSIONS AND OBSERVATIONS

• Preparing a worksheet is roughly the same amount of work for theinstructor as preparing a lecture.

• The collabora t ive act ivit ies offered frequent pleasant su rp rises. Forexa m ple, se ve ral times eac h semester we ran into unanticipated solu­tions to problems posed on a worksh eet.

• Students who arrived in class ea rly often worked on the previous ma­t erial b efore the instructor arrived. We do not remember ever havingto chase st udent s out of a room after a lecture but several times eachof us had to remind students that class time was over.

• The make-up of the individuals in a given class is far more evidentduring a group activity than it is during lecture. For example, itis easy to determine wh ether each group member is on task, and togently correct the sit ua t ion if this is not the case. The instructor alsogains a much b etter sense of who is grasping the material and whois struggling. Subtler things become evident , too, such as a s t udentwho understands the material but has difficulty getting thoughts ontopaper.

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iI=tililID September 1997 Volume VII Number 3

• Collaborative activities are more fun. Classes are more lively sincestudents are more vocal about the material than they are during alecture. We also took great delight in seeing students' thought pro­cesses on display.

• The collaborative activities used in Mathematics Reasoning had atrickle down effect on our other courses since we each use collaborative,small-group activities to some degree in every course we teach.

• Norwood [5, p. 234] observed that while several recent studies (forexample, [4, 6, 7]) indicate that team rewards are "essential" ingredi­ents for successful collaborative learning, Davidson in [1] argued thatat the college level team inducements were by no means necessary.We concur with Davidson's position.

REFERENCES

1. Davidson, N. 1985. Small-group learning and teaching in mathemat­ics: A selective review of the research. In et. al R. E . Slavin, ed it or , Learningto Cooperate, Cooperating to Learn. New York: Plenum.

2. Hagelgans, N. 1. et. al. 1995. A Pra cti cal Guide to CooperativeLearning in Coll egiate Mathematics , MAA Notes, 37. Washington DC:Mathematical Association of America.

3. W. F. Hill. 1989. Learning Through Discussion, S econd Edition.Newbury Park CA: Sage Publications, second edition.

4. Newmann, F. N. and J. Thompson. 1987. Effects of CooperativeLearning on Achievement in Secondary Schools: A Summary of Research.Madison WI: University of Wisconsin, National Center on Effective Sec­ondary Schools.

5 . Norwood, K. 1995. The effects of the use of problem solving and coop­erative learning on the mathematics achievement of underprepared coll egefreshmen. PRIMUS. 5(3) :229-252.

6. Slavin, R. E. 1993. Cooperative Learning. New York: Longman.

7. Slavin, R. E . 1989-90. Research on cooperative learning: Consensusand controversy. Educational Lead ership, 35(4):52-54.

8. Smith, D., M. Eggen, and R. St. Andre. 1990. A Transition toAdvanced Mathematics, Third Edition. Pacific Grove CA: Brooks/Cole.

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Alexander a nd DeAlba Groups for Proofs

BIOGRAPHICAL SKETCHES

Dan Alexander is an Assistant Professor of Mathematics at Drake Universityin Des Mo in es , IA with a doctorate in mathematics from Boston Univer­sity. He t each es a wide variety of courses and uses colla bora t ive activitiesin all his classes. Although he is becoming more and more involved withp edagogical issues, his primary research interest is in the history of mod­ern mathematics , in particular, complex analysis. His book, A Hist ory ofComplex Dynamics was published by Vieweg in 1993. When not involved inmathema ti cs or t eaching, his favori t e recr eations are hiking, cross-countryskiing and canoeing .

Luz M aria DeAlba is Professor of Mathematics a nd Computer Scien ce a tDrake University wher e she has taught since 1984. Sh e earned both MSand PhD degrees in Mathematics from Iowa State University. Her researchinter ests lie in t he a reas of Operator Theory, and Matrix Theory. DeAlbahas in cr eas ed her enjoyment of t ea ching by using nontraditional methodsin the clas sroom. These include: colla bora t ive learning ac t iv it ies in Math­em a tics R eas oning, Calculus, a nd Lin ear Alg ebra ; portfolios for assessmentin Introduct ory Linear Alg ebra ; and calcula tor proj ects in all of t he co u rsest hat she teaches . To relieve stress she swims laps a t t he Drake pool or playst ennis with her USTA team .

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