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Peer tutoring in first-yearundergraduate mathematicsWarwick Evans a , Jean Flower a & Derek Holtonb

a The Mathematics Centre, University CollegeChichesterb The Department of Mathematics and Statistics,University of Otago, and The MathematicsCentre, University College ChichesterPublished online: 11 Nov 2010.

To cite this article: Warwick Evans , Jean Flower & Derek Holton (2001)Peer tutoring in first-year undergraduate mathematics, International Journalof Mathematical Education in Science and Technology, 32:2, 161-173, DOI:10.1080/002073901300037609

To link to this article: http://dx.doi.org/10.1080/002073901300037609

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Peer tutoring in ®rst-year undergraduate mathematics

WARWICK EVANS, JEAN FLOWER

The Mathematics Centre, University College Chichester

and DEREK HOLTON

The Department of Mathematics and Statistics, University of Otago, andThe Mathematics Centre, University College Chichester

(Received 16 December 1999)

A peer tutoring approach was taken for part of the teaching of mathematicsto two di� erent classes at a tertiary institution, most of whose students werepreparing to be teachers at either the primary or secondary levels. It was hopedthat peer tutoring would increase the learning and understanding of thestudents involved. As many of the students are training to become teachers, itwas hoped that the work on peer tutoring might have particular relevance forthem. From qualitative evidence, the experiment appears to have beensuccessful. The vast majority of the students would like the experience to berepeated during their mathematics course.

1. IntroductionBy peer tutoring we mean that a part of the teaching process involves students

teaching other students. Gri� ths et al. [1], say

Peer tutoring is a structured way of involving students in each other’sacademic and social development. As a learning experience it allows studentsto interact and to develop personal skills of exposition while increasing theirknowledge of speci®c topics.

Peer tutoring has been used quite extensively at the primary and secondary levels[2]. The actual process involved can vary considerably. At the primary level, forexample, older students can act as tutors to younger students in reading, where theolder student listens to the reading of the younger student and makes correctionswhen necessary. At university level, there has been some experimentation withpeer tutoring [3]. It is common, especially in Australia and New Zealand, forsenior students to be involved in tutorial work with junior students. However, herewe will use peer tutoring to mean the assistance given by one student to another inthe learning process, where both students are at a comparable level of developmentand have similar mathematical ability.

The Dearing Report [4, chapter 1] says that

UK higher education must:

. encourage and enable all studentsÐwhether they demonstrate the highestintellectual potential or whether they have struggled to reach the thresholdof higher educationÐto achieve beyond their expectations; . . .

int. j. math. educ. sci. technol., 2001, vol. 32, no. 2, 161±173

International Journal of Mathematical Education in Science and TechnologyISSN 0020±739X print/ISSN 1464±5211 online # 2001 Taylor & Francis Ltd

http://www.tandf.co.uk/journals

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. be at the leading edge of world practice in e� ective learning and teaching;

. . .

. sustain a culture which demands disciplined thinking, encourages curios-ity, challenges existing ideas and generates new ones; . . .

Against this background, we decided to experiment with peer tutoring in an e� ort

to encourage our students both in achievement and in curiosity, and to try an

alternative method of teaching that we felt would be stimulating and have valuableresults for our students.

The main thrust of this paper is a report on the experiment in peer tutoring

that took place at the University College Chichester and involved two di� erent

®rst year classes. The two tutors involved had widely di� erent backgrounds ineducation (one has taught for 26 years in secondary schools and 9 years in Higher

Education, while the other is in the ®fth year of teaching), but their teaching styles

are not wildly di� erent. Both endorse the ethos of their department and typicallycombine lecturing with small group work.

We felt that there were potential bene®ts in peer tutoring where students of

approximately the same ability level taught each other. Among these where that we

expected the students to learn the material better; that it would provide a goodopportunity for them to practise their teaching skills; that it would encourage

students to make up their own mathematics questions; and that it would encourage

informal study groups to establish themselves.

Goldschmid and Goldschmid [5] include a list of potential advantages:

. . . such as ®nding good sources of information, asking appropriate questions,giving pertinent feedback, making contacts, establishing a relationship whichis conducive to learning.

There was at least one potential disadvantage to peer tutoring. For instance, we

were concerned that by taking this approach we would spend more time with fewer

topics and so reduce the amount of material that would be covered in a term.Of the two classes that were to be involved in our study, one had 18 students

who were embarking on a four year initial teacher training course for mathematics

specialists leading to a Bachelor of Arts (Quali®ed Teacher Status, QTS) degree.The other class consisted of twelve students of whom ®ve were graduates in a

mathematically related discipline and seven had taken mathematics courses in

Higher Education Institutions. This second class was beginning an accelerated

two-year course giving Quali®ed Teacher Status to the ®ve graduates and a BAdegree with QTS to the other seven.

It was di� cult to quantitatively assess the bene®ts or otherwise of the experi-

ment. In the previous year, `task sheets’ had been given to students each week

which they were expected to complete and hand in for assessment. However, giventhe time taken to complete task sheets and given the College policy limits on the

size of the assessments in ®rst-year modules, it had been decided to ask students

for written work of an essay nature during the experiment. We felt that it was notappropriate or ethical to impose more assessment on the students simply for the

bene®t of our trial. Hence we relied for notes taken during discussions with

students, student questionnaires, written student statements and videotaped

records, on which to base our conclusions.

162 W. Evans et al.

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2. Practice 1

The peer tutoring experiment began in the ®rst of three terms of the two-yearand four-year courses. The ®rst term lasted 10 weeks and included one three-hourmeeting time per week for each class. In the ®rst term, it was decided to try peertutoring twice. The ®rst session spanned weeks 2 and 3 and the second session,weeks 7 and 8. Tutor A taught the two-year class and tutor B taught the four-yearclass. These both took place in the ®nal hour of the available class time of weeks 2and 7. The peer tutoring then occupied the ®rst two hours of class time in weeks 3and 8, respectively. In the ®rst of those hours A’s class taught B’s class and in thesecond hour, B’s class taught A’s. It was decided to take advantage of thepossibilities of small group teaching and the class was split accordingly. In the®rst attempt at peer tutoring, each teaching group consisted of one or two studentsfrom the two-year class and two or three students from the four-year class. Thisimbalance was caused because of the relative sizes of each class.

In the middle of week 2 of the term, the tutors discussed the details of the peertutoring experiment with their classes. The students were told that they would beasked to teach their peers from the other class in small groups and that they wouldhave to prepare some notes, worked examples and references for the students theywere to teach. The students were not given any guidelines as to how to teach butwere given a handout on the topic they were teaching and were expected to write alearning aid for their peer-students.

During this initial discussion, students from each class were interviewed togauge their reactions. They understood what the trial entailed and thought it was a`brilliant’ idea, though some of them were clearly nervous at the thought ofteaching their peers. Their main concern was in regard to the assessment. `Whatwill we put down in 500 words that hasn’t already been said?’ `How can we getmarks for simply repeating what we’ve been given?’

They all had ideas on the form their peer-teaching would take but one studentasked how she could teach the topic better than their tutor. `They manage toproduce new ideas that stick in your mind.’ The students decided that theycouldn’t repeat the teaching methods of their tutors. After a short period wherethey expressed awe at their tutors, the students then started to come up withoriginal ideas for their own teaching. One student, for instance, said `The nationallottery might be a good example. People don’t know the odds against winning.They don’t know that it’s easier to become Prime Minister!’

Tutor A then taught aspects of the Binomial Theorem to the two-yearclass, while tutor B taught arithmetic and geometric progressions to the four-year class.

In the following week, the two-year class ®rst took the role of teachers andtaught their four-year peers. Then in each peer tutoring group the roles werereversed, and the four-year class took the role of teachers. During the two peer-tutoring sessions, tutors A and B circulated among all the groups, observing thembut making very few comments. After the discussion time reported below, thetutors gave out support material in case any student thought that they had missedsome detail when they were being peer-taught. Students were also referred toresources (including videos) relating to the topics that had been covered.

The students used quite a range of teaching styles. When there was more thanone peer-tutor in a group, some had planned which parts of the topic they wouldteach and when they would pass over to their colleagues. Sometimes people just

Peer-tutoring in ®rst-year undergraduate mathematics 163

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joined in when they felt they could provide help. This latter approach often meant

that a peer-tutor could provide individual help to a peer-student who was

experiencing di� culties. As a result, the rest of the peer-students could continue

to follow the main theme.Mostly the groups worked around a table but some groups used an overhead

projector. These latter groups tended to be more directed in their teaching

approach. Some of the peer-tutors provided exercise sheets with gaps for the

peer-students to ®ll in as they went along. Whatever the style though, peer-tutors

tended to be sympathetic and empathetic and tailored the pace to suit their peer-

students.

When both peer tutoring sessions had been completed, there was roughly halfan hour of discussion on the merits (or otherwise) of the experiment. This

discussion, as well as some of the peer tutoring groups, was videotaped. At the

end of the discussion, several students volunteered to write comments on how well

they thought that the experiment had gone. The general feeling was that the

experiment had gone well and the students had enjoyed the experience.

As the result of the discussion and the written work, several themes emerged.

The ®rst theme related to nerves. There was general agreement that all the peer-tutors had been apprehensive and nervous at the start. This was not just because

they had to teach. Some of the two-year peer-tutors felt under greater pressure

because they had to teach by themselves. The concern for many of the four-year

students was that they thought that the Two Year peer-students would know more

mathematics than they did. In addition, both groups were worried that they might

not pitch their lessons at the right `level’ or `get on’ with the students from the

other class. As it turned out this latter fear was not justi®ed. `We appreciated the``audience’’ was enthusiastic and approachable.’ This might have partly been

because they were all facing the same type of problems that morning, and so it

generated a certain camaraderie.

One of the two-year students expressed the general feelings about the pros and

cons of working together. `Personally I was happy to be working alone as I didn’t

have to arrange preparation time with a colleague. Also I didn’t have to worry

about a possible personality clash or disagreements during the teaching session.’But those who worked in groups enjoyed the sense of teamwork and support from

collaborative work. Other students reinforced this latter point. One of them said

`Each group met up privately during the next week, perhaps on several occasions,

to prepare a teaching strategy.’ One student said that `It was easier to work in pairs

because the others thought up ideas that you wouldn’t have thought of by

yourself.’

A third theme seemed to be the di� erence in di� culty between the twotopics. It was generally felt that the Binomial Theorem was more di� cult than

the other topics. Students felt that the work should be balanced more evenly in

future.

There was also apprehension about the assessment. `I think one of the reasons

we were so tense was because we knew that our mark depended on each other.’ But

this was not the only reason that assessment was proving to be a problem. `Anaspect of concern was that the two-year group seemed to have a di� erent under-

standing of the assessment criteria. The person teaching us thought we were

allowed 500 words per person in the group, and couldn’t understand why we had

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not produced more work. Some of the groups of two-year students had produced ateaching plan each, as they expected to get individual marks.’

The last theme of the discussion was the time required by the students to teachthe topics. Generally speaking they took far less time than had been taken by eithertutor. This could have been through omission of content but we believe that thegreater speed was caused by small group size.

One student wrote the following. `It was also worthwhile from the point of viewthat to be able to teach a subject well, you need to have a good in-depth knowledgeand this meant we all went away within our group and researched the subjectsrequired thus acquiring more than just a ``working knowledge’’ passed on fromtutor B. It is also a very good way to check progress.’

These attitudes con®rm an observation by Goodlad and Hirst [2].

Teachers often report that only when they begin teaching do they see how thesubject area `all ®ts together’. Peer tutoring can give anyone who acts in atutorial role this rewarding revelation.

Another student wrote `Overall the students felt they had at least a partial graspof the topics concerned, and would be able to build on this through independentlearning. We had our ®rst experience of issues related to teaching a mixed abilitygroup. Also I learnt how the introduction of formulas often introduces a mentalblock for individuals who prefer ``real’’ examples.’

Overall, students suggested that in future peer tutoring:

. `A choice between our delivery of maths and a lesson plan as the mode ofassessment would be appreciated.’

. `Peer group assessment should be considered.’

. `We should have clearer assessment criteriaÐWhat will you as tutors belooking for in the lesson plan/lesson delivery? What do you want to see?’

. `We should have prior knowledge of the peer group to be taught.’

. `Next time some of those who worked individually hope to work in agroup, and vice versa.’

3. Practice 2

The same time allocation was made for the second trial but, as the result of the®rst experience, several other changes were made. Topics (exponentials andlogarithms) were chosen for weeks 7 and 8 which we hoped would be approxi-mately of equal di� culty and hence might overcome the problem caused by theimbalance in the ®rst practice. Further, the students were shown the mark-schemethat the tutors intended to use for the assessment.

Because of the imbalance in numbers between the two classes, some studentshad had to be sole peer-teachers. What’s more, there had been a concern that itmight be harder for the four-year students to teach the two-year students becauseof the di� erence in their mathematical ability. Consequently it was decided thatthe peer tutoring groups should only contain students from a single year group.This meant that some of tutor A’s students had to attend the session given by tutorB and vice versa.

The tutors again found that the peer-teachers were able to complete thematerial more quickly than the tutors themselves had been able to. Indeed some

Peer-tutoring in ®rst-year undergraduate mathematics 165

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of the peer groups ®nished in half the time that it had taken the tutors to teach the

same topics. However, despite this it was felt that the peer-teaching was more

complete in that they went through algebraic steps more thoroughly compared to

the regular classes when the tutors deliberately left some steps for the students tocomplete by themselves. On re¯ection we believed that the early ®nishers had

indeed completed their task.

Immediately after the second trial, all the students involved, along with the

three authors, took part in an evaluative discussion on the way the experiment had

worked. This discussion lasted a full hour. At the start of the discussion, students

were asked to anonymously ®ll in a questionnaire. They were allowed some time at

the end of the discussion in case they wanted to amend their questionnaireanswers.

On the basis of the discussion and the questionnaires, it seems that almost

all students enjoyed the method of peer tutoring employed in this experiment.

Only two of 24 students did not want the process repeated in the following

term.

Although the topics of exponentials and logarithms were close in di� culty,

they were too interdependent. Students needed some knowledge of one to use intheir teaching of the other. It appears to be a bad idea to separate the teaching of

these topics.

Assessment still seemed to be a problem. Many of the problems of the ®rst trial

had been settled to everyone’s satisfaction in the second trial. However, there were

still a few things to be reconciled. Many students thought that some weight should

be given to their actual teaching performance and that only some of their mark

should be given on the basis of the written work they had to hand in. Some of thestudents felt that it was only the peer-students who could tell how well or poorly

the peer-teachers had performed in their presentation of the material. Hence some

form of peer assessment was proposed and the issue was hotly debated. There was

no unanimity. Those against it thought that it might be capricious, that it might

not be taken seriously, or that it was just too di� cult to do fairly. Those in favour

thought that it could work if su� ciently well structured. The questionnaire shows

that nearly 30% of the class were positively in favour of peer assessment and onlyone student was de®nitely against it.

This also raised a model of peer tutoring that changed the role of the tutor. One

of the tutors suggested that students could be asked to teach a topic without having

been given any prior instruction by the tutor. They would just be given a topic

with perhaps some handouts to show the extent of knowledge required and asked

to undertake the research required to prepare the lesson. There was little

enthusiasm from the students for this approach. It was generally felt that, althoughthey might learn good research skills this way, they could not trust their peer-

teacher. The students were afraid that the peer-teacher might unintentionally

present the material at the wrong level or wander away from the main theme and

explore avenues that were not required by the tutor.

There was su� cient evidence from the two trials for the tutors to use peer

tutoring again in the second term and to include an element of peer assessment inthe process. It is worth noting here though, that when peer assessment was

implemented at Otago, almost all peer-teachers were given full marks for their

teaching. Similar over-marking occurred at Chichester.

166 W. Evans et al.

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4. DiscussionÐthe results of the trials

We have divided the discussion into two parts. In this section, we summarisethe results of this trial. In the next section we look at issues relating to peertutoring in general.

4.1. Student enjoymentAlmost all students seemed to have enjoyed their peer-tutoring experience.

They believed that they had learned how to research a topic and prepare it in aform that they can present to others. They also felt that, certainly in the case of thetopic they peer-taught, that they had learned it to a greater depth than materialpresented in the normal way. It is not clear that the same is true for the materialthat they were peer-taught. This may be the weak side of the peer tutoring process.However, there is no concrete evidence to suggest that being taught by peers ratherthan tutors disadvantaged the peer-taught.

4.2. Constitution of groupsTo ful®l the aim of fostering the development of student study groups, it would

seem necessary to make sure that there were at least two members involved in thepeer-teaching for each group. More than one peer-tutor allows for discussion ofthe material and teaching, which should be of bene®t to all peer-tutors. On theother hand, it seems to be di� cult for many more than two peer-tutors to teachtogether e� ectively. Hence something of the order of four to a group with twoteaching and two learning seems to be about the optimum size. In addition,potential absences need to be catered for.

Not only was the size of group important. In this experiment students wereallowed to self-select their groups. Perhaps we should have noted potentialpersonality clashes [6]:

. . . the learning cell was more e� ective than individual study for extrovertsÐprovided their partners were also extroverts; introverts did well in bothconditions; and mixed pairs of introverts and extroverts were least successful.

However, none of our student responses raised this issue.During the ®rst term, it became apparent that most students had formed study

groups that met outside normal contact time to discuss and follow up the work ofthe formal sessions. This may have happened without the peer-tutoring exercise,and we cannot measure its e� ect in this regard, but we do believe that peer tutoringenabled the groups to establish themselves more readily and earlier.

4.3. Relative weight of tutor/peer tutoringWhile the experiment seems to have been very successful, neither tutors nor

students would want to see it as the only teaching method used. The studentquestionnaire asked them to suggest relative weighting in some hypothetical futurecourse of normal teaching, peer tutoring according to the model that they hadexperienced, and peer tutoring where they had to prepare the material themselves.Based on a 36-hour course, they typically went for approximately 25 hoursteaching the normal way, and 8 hours by the method they had trialled and 3 bythe other peer tutoring method suggested above.

Typically they see the ®rst two methods being combined in one lesson much asit had happened in the trials. The authors would probably agree that this was a

Peer-tutoring in ®rst-year undergraduate mathematics 167

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good combination and a good relative weighting. Because we are not yet convincedthat peer-students learn any better than they do with the usual method of teaching,we would not want to allow peer tutoring to dominate the overall class time.However, the advantages of the system mean that we would want to use it for afraction of the time in future courses. We would need to trial the other peertutoring method before considering its use.

4.4. Time and coverageThe authors were surprised to see tutor groups ®nish more quickly than the

tutors working on the same material. In retrospect, though, maybe we shouldn’thave been. With small groups there is not the variation of student ability that thereis in a larger teaching group. Hence we would expect the same material, even ifbeing delivered by the same person, to be delivered more quickly in the smallergroup. Given that there was usually more than one peer-tutor and that one of thesecould help a slower student, then it is clear that the material as a whole may becovered more quickly in this situation.

There is, of course, the danger that some material was skipped or considered inless depth than was required by the tutor. This could also speed up the peer-tutordelivery. The tutors made spot checks as they went around the peer groups andthere was no direct evidence to support a scanty treatment. However, this is a verydi� cult thing to determine without signi®cant testing. A `content list’ of materialcovered by each tutor was given to all students so that they could convincethemselves their peer-teachers had not omitted any material.

On the matter of coverage of material for the course as a whole, there is noevidence that material had to be omitted. A direct comparison with the previousyear’s content showed that, for the Four Year course, this year one topic had notbeen covered that was covered in the previous year. On the other hand, one extratopic had been covered. This is despite the fact that an hour and a half had beenput aside to discuss the experiment. In the future it will not be necessary to engagein so much discussion and so more time will be available for additional content.

4.5. Choice of topicsFrom what has been said above, it is clear that care has to be taken in choosing

topics for this model of peer-tutoring. Topics covered in pairs like this need to beof approximately the same standard for the sake of fairness to the students. On theother hand, they must also be not too interrelated so that one relies too much onthe other as in our second trial. Choosing topics has to therefore be undertakenwith care.

4.6. AssessmentThis seems to be the most di� cult aspect of the model of peer tutoring that we

used. It is clear that, as with all assessment, ®rst the tutors have to decide exactlywhat it is that they want to assess. In this instance is it the lesson plan, the way thepeer-teachers understand the material, their delivery or something else?

And then there is the matter of who should be assessed. Perhaps both peer-teacher and peer-student should be assessed.

Next, the tutors have to consider how the assessment should be undertaken. Isa 500-word statement su� cient. Should there be some test of the material? Whatother means should be used? These other means could include peer assessing.

168 W. Evans et al.

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Stefani [6], has found that `student assessment can be as reliable as that oflecturers’ and Cowan [7], claims that the bene®ts of self-assessment are so greatthat they outweigh commonly perceived risks.

4.7. ReplicabilityIt is important to ask whether this peer tutoring experiment could be repeated

elsewhere. There were certainly rare circumstances pertaining in these trials. Notevery tertiary institution has two di� erent teaching groups operating at the same orsimilar times and occasionally covering the same material. When two teachinggroups do cover the same material it is often at quite di� erent levels. The logisticproblems of a peer tutoring approach with a single lecture group, or with a largelecture group might be quite di� erent. In section 6, we discuss other models ofpeer-teaching, which are potentially more widely applicable.

5. DiscussionÐmore detailed analysis

Here we attempt a theoretical discussion of peer-tutoring under the followingfour headings:

content of material covered by the students;understanding gained by students;time taken by students;sta� time.

Initially we will compare two models of teaching; (a) traditional lectures and (b)lectures and peer tutoring (as in this study). The table below indicates thestructures adopted under these two schemes.

Key:`L’ represents a normal lecture.`LL’ represents an accelerated lecture, which is intended to maximize the coverageof material possibly risking a reduction in immediate understanding.`PL’ means that the students are ful®lling the role of `peer-learners’, acting asstudents for their peer-teachers.`PT’ indicates that the students are to be `peer-teachers’.Each session lasts for two and a half hours.

5.1. Content of material covered by the studentsThe peer-tutoring process begins with LL, an accelerated lecture. The tutor is

working under the expectation that all the students will follow up the session inmore detail than usual, with group support, and the lecture is designed with this inmind. The content covered during this session is greater than would be expectedduring a normal lecture.

Peer-tutoring in ®rst-year undergraduate mathematics 169

(a) Lectures (b) Peer-tutoring

Session 1 Session 2 Session 1 Session 2

Group A L L L L Group A L LL PL PTGroup B L L L L Group B L LL PT PL

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The student-learners are in smaller groups during PL than they would havebeen in L. It is easier for the peer-teachers in small groups to focus on speci®cdi� culties that may arise and pace the work tailored for the select audience.Sometimes teaching two-to-two splits up into individual tuition. For this reason,with closer communication and more intensive activity, perhaps small grouptuition is a more e� cient mechanism for conveying content?

The (sta� ) tutors are more experienced at teaching and have a better under-standing of the context of the mathematics. Does this mean that a tutor is bound toconvey more material than peer-teachers in the same time-span can? Is it import-ant for students to witness the interesting `asides’ that can give a lecture itsdistinctive atmosphere and make it more informative than reading a text book.Would peer-teachers be able to provide such mathematical-cultural background?

Any additional content discovered by the peer-teachers as they teach in PT isprobably negligible. However, if we include the preparation tasks for PT into thesymbol `PT’, then anecdotal evidence would suggest that a great deal of content iscovered in research by the peer-teachers between session 1 and session 2. Not allthe researched material will be included in the peer-teaching session but it adds tothe understanding of the peer-teacher. It is not clear whether this additionalmaterial covered is equivalent to additional material that would have been coveredin a normal lecture.

It is possible that some content has been lost in the peer-teaching mode. It isdi� cult to accurately measure a quantity like `mathematical content’. We feel thatif any content has been lost, it is less than one lecture, and we hope that thesigni®cant gains in other aspects of the learning experience will outweigh thispotential loss.

The quanti®cation of abstract notions continues as we think about the under-standing that students have gained.

5.2. Understanding obtained by studentsLL is an accelerated lecture, designed to cover more material than a normal

lecture. We expect the level of understanding during the lecture to reduce as aresult of the density of material. The tutor is working under the expectation thatthe students will follow up the session more than they do for a normal lecture, withgroup support, and the lecture is designed with this in mind.

The student-learners are in smaller groups during PL than they would havebeen in L. It is easier for the peer-teachers in small groups to focus on speci®cdi� culties that may arise and pace the work tailored for the select audience.Perhaps the understanding gained from being a peer-learner is greater than theunderstanding that would have been gained in a traditional lecture. This gainwould be important if the learner was more comfortable asking their peer-teachersquestions that they might have been embarrassed to ask in large-group lectures. Ifquestions arise during the peer-teaching sessions that the peer-teachers wereunable to address, they would invariably ask other groups, ask a member of sta�present to observe the peer-teaching exercise, or raise the question during thewhole-class session that followed the peer-teaching session. It is important that theopportunity is given for the students to gain access to sta� to resolve trickyquestions.

As the students teach in PT, their understanding is enhanced and con®rmed. Ifwe also include the preparation tasks for PT into the symbol `PT’, then anecdotal

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evidence would suggest that much understanding is gained in research by the peer-teachers between session 1 and session 2. The understanding achieved by PTcertainly exceeds the understanding imparted by a lecture L.

The increase in understanding produced by the activity of peer-teaching willmore than compensate for any loss in understanding by accelerating the lecture Lto LL.

5.3. Time taken by studentsAll activities L, LL, PL and PT took place in a one hour ®fteen minute time

slot. The sessions lasted three hours and there was a break of approximately half anhour mid-way through each session.

The time during these sessions, the contact time, remained unchanged duringthe peer-teaching exercise. On the other hand, the peer-teaching process beganwith an accelerated lecture, then required students to follow up the lecture, toconduct research outside the lectured material, to prepare worked examples,worksheets and handouts, to collaborate with co-teachers, and then spend timeactually delivering the lesson. This all takes considerably more time than would bespent following up two standard lectures.

The time required for PL is the same as the time required for L. If the learnerswant to be sure that they have seen the right material during the peer-teachingsession, then they may spend extra time checking their understanding andcomparing it with the required syllabus.

The students spend more time working for the module under the peer-teachingmodel. Our awareness of these additional demands upon students’ time led us toconduct the peer tutoring exercises early or mid-way through a module, when thepressure for completion of assessments is not so great.

5.4. Sta� timeIf the sta� left the students to peer-teach without supervision, sta� time would

be reduced. However, this year, sta� were present at the peer-teaching sessions tooversee the teaching and to answer any queries that arose from the students.

Perhaps in future, sta� time could be maintained (assuming that the studentshave rights to access sta� for a given number of hours) by expecting the students toconduct the peer-tutoring exercise outside of timetabled slots. We would feeluneasy about this proposal, however. Students with family commitments ®nd itdi� cult to attend additional sessions. The tutors would have no way of overseeingthe peer-tutoring, and some students may not complete their obligations outsideclass.

6. Other models for peer-tutoring

At the College, we were in the fortuitous situation of teaching similar materialto di� erent groups. It would not have been suitable to combine these groupsbecause of the di� erent mathematical background of the students and the di� erentmathematics covered.

Below we present a model for a peer-teaching exercise that involves only onetutor and one large class. The underlying assumptions are that:

the class is split up and part of the class is made expert in one area ofmathematics;

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time is found when experts and novices can meet for peer-teaching;an attempt is made to maximize content covered;blocks of material are chosen that are related but can be learned indepen-dently;a whole-class session(s) exists that builds upon all the material included inthe peer-teaching.

A similar model to this one was employed at the University of Otago to a class

of 120 students. This class had three lectures and one two-hour tutorial each week.For the purposes of Session 1, the tutorials were divided into two, those that were

taught topic X and those that were taught topic Y. Students were assessed on this

work in the same tutorial period.

Then a week elapsed in which time students were able to work in groups of twoor three and as individuals, to prepare the material that they had to teach. During

the following week, two lecture periods were given over to the peer tutoring. Thiscorresponded to Session 2 in the table.

The material of Session 2 was built upon in subsequent lectures (Session 3).

All students were given an assessment of the material that they had learnt in a

tutorial in the week following the peer tutoring. As a result all students had beenassessed on the material of topics X and Y. (Unfortunately, resources were

unavailable to compare the results of material that was peer-taught versus thematerial that was tutor taught.) In addition, peers assigned a mark up to 5 to their

peer-teachers. This was almost always 5 with only a rare 3 being o� ered.

Student feedback via a questionnaire suggested that a small majority was in

favour of the peer tutoring.One of the major di� culties of peer-tutoring for a large class was the

accommodation available for the peer-teaching sessions. Because of space limita-

tions and because the lecturer wished to be present to provide assistance asnecessary, the peer tutoring had to take place in the regular lecture theatre. This

was a traditional tiered room that was totally unsuitable for groups of four people

to work together in any comfort.

172 W. Evans et al.

Session 1 Session 2 Session 3

Tutor Tutor A teaches B teaches Tutor teaches content Z,teaches teaches content X content Y building from both contentcontent X content Y to to subclass to subclass X and content Y.to subclass B. B. A.subclass A.

Subclass A The tutor The tutor This session can reassureresearches may may students that theirmaterial X choose to choose to knowledge is appropriateand starts be present be present and helps to level out theirpreparation at some or at some or varied experiences asof teaching all of this all of this student learners.materials. session. session.

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7. Conclusions

Generally the students enjoy the peer tutoring experience although neither theynor the sta� involved would want it to be the predominant method of teaching. Itseems to provide an interesting variation to the normal teaching as well as anopportunity to encourage student study groups to be formed. It probably aids instudent learning and encourages communication.

There are some di� culties that need to be taken into consideration whenplanning peer-tutoring. Topics need to be chosen carefully so that they take aboutthe same time to teach, are of roughly the same di� culty, and are independent. Inlarge classes it may be di� cult to ®nd comfortable space for the peer-teachingsessions to take place.

At Otago, the scheme has now run for two years, while at Chichester it hasdeveloped in two ways. First the two-year and four-year groups continued to usepeer tutoring but within their individual year programmes. Second, the studentswho took part in the peer tutoring described above were in the ®rst year of theirdegree course. In the second year, most of them took a level 2 module called`Applications of Calculus’ which was presented using student seminars. This modeof teaching and learning again made use of skills such as the ability to research atopic, to talk about maths in small peer groups, and to take responsibility for theirown learning as well as the learning of their peer group. More details of this can befound in [8]. Consequently, the results that we have had with peer tutoring in bothof the forms above have been su� ciently successful for us to continue with theconcept in other classes.

AcknowledgementParts of this study were presented at the ICTM conference in 1998 [9].

References[1] Griffiths, S., Houston, K., and Lazenbatt, A., 1995, Enhancing Student Learning

through Peer Tutoring in Higher Education (Coleraine: University of Ulster).[2] Goodlad, S., and Hirst, B., 1989, Peer Tutoring: A Guide to Learning by Teaching

(London: Kogan Page).[3] Houston, K., and Lazenbatt, A., 1995, Assessment and Evaluation in Higher Education,

21, 251±266.[4] Dearing, R., 1997, The National Committee of Inquiry into Higher Education: Higher

Education in the Learning Society (London: HMSO).[5] Goldschmid, B., and Goldschmid, M., 1976, Higher Educ., 5, 9±33.[6] Stefani, L., 1994, Studies Higher Educ., 19, 74.[7] Cowan, J., 1988, in D. Boud (ed.) Developing Student Autonomy in Learning (London:

Kogan Page), pp 192±210.[8] Flower, J., Proceedings of the International Commission for the Study and

Improvement of Mathematics Education 1999, to appear.[9] Evans, W., Flower, J., and Holton, D., 1998, Proceedings of the International

Conference on the Teaching of Mathematics, Samos, pp 101±103.

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