Peer Tutoring in Mathematics for University Students
Carmody, Georgina, Wood, Leigh, Mathematics and Computer Education
A peer tutoring service has been implemented with volunteer senior students from third year mathematics and statistics classes. This simultaneously bridges the gap in the university budget and helps both the tutors' education as well as their students' education. In this paper we examine the peer tutoring experience.
There are several (not mutually exclusive) educational and administrative reasons why peer teaching in university mathematics is advocated:
* To improve the supply and quality of tutors for mathematics subj ects 
* To improve the learning of those who are doing the peer teaching 
* To prepare students for teaching situations in the workplace by participation in peer teaching programs
* To provide services that are outside the university budget
Much work has been done on looking at the benefits of peer education as a whole and of different aspects of peer education. Damon and Phelps  distinguish three approaches to peer education namely peer tutoring, cooperative learning, and peer collaboration. Cooperative learning is generally a team-based learning approach where students pool their resources on a particular topic, and in peer collaboration students simultaneously approach broad aspects of a topic, working together. With these two aspects of peer education, the students are generally on an equal level to one another whereas in peer tutoring, one student takes on the role of a tutor and the other(s) take the role of a student. The use of cooperative and collaborative learning in mathematics at university level is addressed by D'Souza and Wood .
Griffin and Griffin  investigated the positive effects of reciprocal peer tutoring at various educational levels. They found that peer tutoring is effective for increasing student achievement for both the tutor and the student, with the tutor often benefiting more than the student. Peer tutoring schemes have been implemented in a variety of subjects and educational levels. Carroll  discusses the effectiveness of senior medical students acting as co-tutors working in tandem with the academic tutor for first year biology students. Bush  describes a peer tutoring program used for introductory accounting courses as a possible suitable substitute to current laboratory classes. In both of these papers the senior students were paid as academic tutors and were able to relieve full-time academic staff while at the same time providing help to first year students.
Both Hopkin  and Houston and Lazenbatt  investigate the use of reciprocal peer tutoring within a class environment in higher education where each student (or group of students) in the class is responsible for a particular topic and then takes on the role of tutor, teaching that particular topic to the other students in the class. This type of peer tutoring fostered independent and responsible learning, and promoted greater levels of communication, student participation and a deeper understanding of the work involved for the tutors [1, 8].
Oates et al.  were unable to find sufficient numbers of suitably qualified tutors for their first year university mathematics subjects and so instituted a second year segment for an undergraduate mathematics degree, where students were taught how to learn and teach mathematics. They observed classes, wrote materials and reflected on their experiences. Students were then paid to conduct tutorials in their third year. The results were positive and the experience also showed that the peer teachers improved their own learning as discussed above.
Preparing students for a career in teaching has some of the same characteristics as preparing them for teaching other students. One essential difference is that the authences in the workplace are more diverse so the students need to be given diverse learning situations that reflect this. The study of the needs of graduates in this area is new and follows from studies of the needs of employers for mathematicians with wide-ranging communication skills . These studies of the needs of graduates provide further impetus for encouraging peer teaching in undergraduate programs.
This paper looks at a peer tutoring program where senior students voluntarily take on the role of an academic tutor for first year mathematics students. Unlike the other papers discussed, mis paper looks at the use of peer tutoring from the view of senior students who are volunteering their time and efforts, not for financial gain or as a means of passing a subject, but rather as a commitment to the further development of mathematics for themselves and others. We discuss the different approaches the tutors took in teaching mathematics, their reasons for volunteering, accomplishments achieved and any correlation between the two, how their views on learning mathematics and mathematics in general have changed since tutoring, and look into the different approaches.
Mathematical and computing support for first year undergraduate students is available in universities through a variety of means. There are mainstream lectures, tutorial and laboratory classes. Some courses implement extra remedial tutorials and students can visit faculty for extra help when needed. At the University of Technology, Sydney (UTS) there is also the Mathematics and Information and Communications Technology Study Center (MICTSC). The operations of the study center include the running of remedial tutorials, workshops, bridging courses as well as a drop-in center where a tutor is available to give support for first year students requiring help in mathematics, statistics, and introductory computing. There are four computers in the drop- in center so students are also able to receive individual assistance in using the mathematical and statistical programs. The drop-in center is a popular environment for mathematics and statistics students from across the university who visit the center for assistance and extra support. Approximately 2000 students use the Center each year.
The transition for students from secondary school to higher education involves major changes not only academically, but also socially and environmentally. The MICTSC aims to create a friendly environment and so is commonly used as a meeting place for collaborative learning where students do group work and receive support from other students. This form of social academic interaction has helped make the transition less stressful for the first year students.
PEER TUTORING PROGRAM
Traditionally the university honors and doctorate students have been paid to act as tutors in the MICTSC. The Center has become increasingly popular and budget restraints meant that demand for services outstripped the supply of assistance. So, in 2005, a volunteer peer tutoring service was incorporated into the drop-in center where third year students joined the tutoring roster. This was initiated by the third year students themselves. They had received support in their first year and wanted to help others. These students were studying for science degrees with majors in mathematics, or for degrees in mathematics and finance.
In the fall of 2005, staffing for the Center consisted of: (paid) one doctorate student, three honors students, one full-time staff member and, (unpaid) 18 third year volunteer students. This combination has been shown to be highly effective for our first year students, the tutors, and the budget, making it a win- win situation for all involved.
The requirement for the volunteer tutors was that they were each to tutor in the drop-in Center for one hour every lecturing week, with their choice of tutoring during the non-lecturing weeks, whereas the paid tutors were in the Center every week. The volunteer tutors could either do this hour on their own or pair up with another volunteer and share a two-hour shift between them. There was no minimum academic achievement result required for the students to volunteer as a tutor as there were always other resources available for the tutors if they did not feel confident in answering a particular student question. Tutors were given a two-hour training session with emphasis being placed on responsibility and communication. A more in-depth tutor education program such as that reported in Oates et al.  was unfortunately not initially possible for this volunteer program but is now under consideration.
Towards the end of semester the tutors were asked to voluntarily complete a survey designed to address the experience gained by the tutors and to gain an insight into peer tutors' different points of views for the learning and teaching of introductory mathematics. In addition to demographic questions, the following questions were asked:
* Why did you originally volunteer or sign on for peer tutoring?
* How do you go about teaching mathematics in the mathematics study center?
* What are your views of mathematics and how have they changed since tutoring?
* What are your views on learning mathematics and how have they changed since tutoring?
* What sense of accomplishment do you feel you have gained through peer tutoring?
Out of the 22 paid and unpaid peer tutors there were 12 responses. Of those students, nine were third year volunteers and three were paid honors students. The authors attribute the lack of response from some tutors to the pressures of final exam week.
METHODS OF TEACHING
The tutors used a wide variety of methods and resources: some of the tutors explained the theory and concepts involved behind the questions, as seen by G and H comments below, while others used examples and explained the steps involved to show how to solve problems in a more systematic manner, see L below. Sometimes these differences in approach were due to the preception of how much students understood or wanted to know, for example J and K.
G. ... I try to give them an overview of the basics to ensure that they aren't just learning how to do the question.
H. I tried to clarify what sort of explanation/help they really require and explain things in the simplest way I can.
L. I attempt the question by myself first and when I get an answer I show the student how I reached that answer. I clearly explain what I did in each step and make sure that they understand the method.
J. ... more times than not it's just that they want help in doing a single problem or it's an assignment problem. It's not often at all that someone will walk up and say 'hey I don't get integration by parts can you help me review it?'.
K. I read the question and ask them what it is they don't understand about it, then I teach them the concept or the topic they are confused with, and after that if they still have trouble doing that particular question, I give them an example similar to that question, and usually at this stage they can do the question on their own.
Some tutors found science and word problems challenging and used a modeling approach, for example C, I and J, below. When difficulties were encountered tutors incorporated the students' textbook or textbooks in the center, the assistance of peers, and sometimes the assistance of a lecturer, see F below. The peer tutors also asked the students requiring assistance what they knew about the topic as both a resource for the tutors, and a solidification of the student's knowledge, as seen by I and J, below. Much of the communication from the students to the tutors seemed just as important to their understanding of the work, J, as from the tutors to the students, helping to solidify the students' knowledge at the same time as having them feel comfortable to ask questions, B.
C. ... it is not very easy to explain mathematics to students. Well I do what I can, drawing a diagram or graphs is often helpful...
I. We may have never seen the application of mathematics with regard to some engineering and finance students, or it may have been years since we had dealt with a topic. So in working with the student we need them to tell us everything they know and maybe to isolate textbook material. This process is often times as important to the solidification of the student's understanding as are the links and explanations that the tutor can offer.
J. I try to think about the best way to explain what it is happening in the question in a pseudo mathematical way (important for word questions). If I don't know I ask to see the textbook and work over that section with them. Or I try to think through the problems from scratch with some help from them ... I then try to walk them through it with them prompting me on what to do or I ask them questions on what should happen next.
F. If I can answer it straight away, then I do, if not I look for a friend in the Center at that time who could help. I have also called a friend on the phone to help me. If I feel there is no way that I can help, I suggest the student visit their lecturer, or direct them to a lecturer that I know will be able to help.
B. I would also try to develop a friendship between students so as for them to feel comfortable asking questions and being honest with what tìiey do and don't understand.
VIEWS ON MATHEMATICS AND LEARNING MATHEMATICS
Different views of mathematics are apparent through the peer tutors' responses. Reid et al.  identified three levels in which people view mathematics.
* Students who view mathematics as basically a tool for calculating, then approach learning mathematics by rote.
* Students who view mathematics on a mid- and higher-level, usually foster a deeper approach to learning mathematics as they go beyond seeing it as just a tool and consider how it relates to life in a modeling sense, and analytically how it starts to relate to itself, as an abstract language .
* Students who didn't understand the work on a deeper level would be more likely to struggle with following topics, see B1 below.
Most of the peer tutors had a mid- or high-level view of mathematics. A high-level view can be seen in B2 and L, below. In D it is seen in the practical sense, but not very deep in the analytical sense. Most of the tutors viewed mathematics and learning mathematics as challenging, see B2, D, and interesting G.
B1. ... there is a core requirement for understanding mathematics which is basic understanding and this is one thing I discovered whilst tutoring. Students who missed one core fact of a topic would have trouble understanding everything related.
B2. I have always viewed mathematics as a fundamental subject to education which can be very challenging and thus requiring logical and analytical thinking. I believe mathematics exists in all areas of life and even in more subtle situations like decision making which seems mathematics free but also requires analytical thinking.
L. I believe that mathematics is one of the most important subjects. It is incorporated into almost everything around us.
D. ... it is helpful in the real world, but the depths of it are just useful if you are an insightful person or just love mathematics. It is very challenging and logical ...
G. Mathematics is an exciting area of study with a lot of interesting applications. Tutoring in the MSC has made me even more appreciative of these applications and areas of study.
At times, some of the tutors disliked a topic in mathematics or found it boring. It can be seen in the following two quotes that at those times a surface approach to learning was most probably taken; they have not fully understood the ideas and have more than likely made it through those situations by rote learning the components.
A. There are topics I feel are boring and complicated and confusing, these I find hardest to learn because of inability to understand fully. I feel that some of those topics are 'pointless', but still try my best to learn them because I don't want to fail.
D. Mathematics is fun when it is easy, but I dislike it when it gets into physics and very scientific.
The tutors have generally developed a greater understanding of mathematics since they started the peer tutoring program. However a few of the tutors' views of mathematics have not changed since tutoring. By teaching other students, some tutors have developed a greater appreciation of the practicality of mathematics, see D below, a greater understanding of the methodology used in applications, B, and insights in seeing how mathematics plays a fundamental role in real life situations, K. Some tutors, through their teaching, also gained a greater analytical understanding of me underlying theory, see J. Overall, the authors feel that many of the tutors and their students gained a deeper understanding of mathematics.
D. ... since tutoring I noticed how [much] more practical mathematics is.
B. ... through tutoring I have come to understand that more importance should be placed on methodology and understanding than purely repetition.
K. By tutoring other students, you can get the overall picture of the topic in mathematics that you have taught, and clearly you can see what sort of problem you are solving, and [how] they might be used in the workplace.
J. When you first do it, you are more thinking about the mechanics of the problem as in we do this then that when it looks like this. But now when I look back at that work it seems to make more sense.
Damon  observed that students have more to gain in learning from their own age group, as they are more direct in their communication and often at a level more easily understood. A similar view is shared by some of the peer tutors, see by F and I.
F. A lecturer knows their stuff inside out, back to front, and has the answer before you have read the question. A student tutor, however, has a quality that the lecturer has lost and will never regain. The student tutor can teach short cuts to learning and understanding concepts on a much more basic and willingly absorbable level.
I. If you can teach something then it's a sure way to learn it good. I see a lot of students in the study center get onto this fact and jump at the opportunity to get involved in rutoring.
REASONS FOR VOLUNTEERING AND ACCOMPLISHMENTS ACHIEVED
We shall look at the reasons why students volunteered or signed on for peer tutoring in two categories, extrinsic and intrinsic. Extrinsic reasons include the opportunity to help younger students, share knowledge, to take the pressure off other tutors, to put something back into the mathematics department and into the university community, e.g., "When I was in 1st year I noticed the Mathematics Study Center was helpful to me. That is why I wanted to help students wim the extra little bit of help they might need in order to do well in their subject."
Different intrinsic reasons include filling time between classes, gaining teaching experience, creating a nice addition to the resume, solidifying knowledge, increasing communication and interpersonal skills, increasing confidence, meeting people, learning from students, and the self-satisfaction of teaching. This last point could possibly be viewed as extrinsic in its origin, as it is likely that the tutor gains enjoyment through making a difference for other students. We shall look at this again through the peer tutors' accomplishments.
After the peer tutoring program, all of the tutors who responded to the survey expressed a sense of accomplishment. Extrinsic accomplishments were all based around being able to help contribute to other students' knowledge and understanding of mathematics and inspire them for future study. Intrinsic accomplishments gained by the tutors included having developed some new friendships, a greater sense of responsibility, increased or grounded their own mathematical knowledge and the development of other skills such as interpersonal skills and the ability to convey knowledge to others. Some also increased their confidence levels in themselves, in tackling unknown problems and rediscovered their overall confidence in mathematics. The most prevalent of the accomplishments is intrinsic in nature with extrinsic origins in that the tutors felt good about themselves by helping make a difference to others. This accomplishment is often a direct link to the reasons people enjoy teaching. It also plays a significant role in the outcome of the other intrinsic achievements, as seen for example in the following two quotes.
F. When I have helped just only a hint, enough for them to recognize the rest of the problem on their own, it gives me such a boost of confidence, motivation and inspiration, that my own study becomes a pleasure.
H. ... the satisfaction of successfully helping others impacted greatly on my selfconfidence.
Only a few of the peer tutors knew that they enjoyed teaching when they originally volunteered in the Center, however, nearly all of the tutors experienced pride and enjoyment from their teaching. This is illustrated by the following quotes.
A. I feel good when I am able to provide help to other students. ... Seeing students who have a better understanding after my help is most rewarding.
C. Well, it always feels good when you find that you've been helpful to some people. And yeah, it kinda gives me pride to hear that students have got good marks thanks to my teaching.
D. It feels great when you successfully help a student with their mathematics. It's the concept of 'giving yourself a pat on the back'.
G. It is great to see that you can make a difference to a student. When they go away with extra knowledge and a smile on their face, you are left with a feeling of achievement.
I. I get a real kick out of helping students out.
L. I've always found it very rewarding to help others in need.
An overall accomplishment brought forth by this program is that a greater sense of community was realised among the peer tutors.
This paper has described a peer tutoring program to support first year students who study mathematics, statistics or mathematical computing in their degree. Research has shown that peer tutoring benefits both the tutors and their students. Volunteer students worked as peer tutors in the Mathematics and Information and Communications Technology Study Center (MICTSC), bridging a financial and educational gap.
It is difficult to provide all the educational and social support required by students when they start at a university, however this program has met the needs of a large group of students (approximtely 2000 students use the center each year). At the same time, peer tutoring has developed skills within the peer tutors. They have gained a greater appreciation of the breadth and depth of their chosen discipline of mathematics. They have gained confidence, self knowledge and developed a more mature approach to life. Research into graduate needs also shows that peer tutoring is an excellent preparation for the workplace where graduates are required to teach their colleagues and present succinct information to their managers.
Future plans include more training for the tutors. An unexpected by-product of the program is an interest in teaching as a profession expressed by some volunteers. This is positive news for the shortage of mathematics and computing teachers in our schools and universities. Even if only a few secondary teachers are developed by this program, it will be a welcome spin-off! There is some evidence that the number of students entering honors (a fourth year of study) has increased as a result of the volunteer program. Some students have become more aware of the possibilities of their discipline and feel part of a community of practice .
The Mathematics and Information and Communications Technology Study Center will continue to incorporate student volunteers into the peer tutoring program and continue to listen to the tutors' suggestions for improvement. With negligible operational costs and positive educational benefits gained by the tutors and their students, this peer tutoring program has created a win-win situation for all.
An early version of this paper was presented at the Adults Learning Mathematics conference in Melbourne, Australia, in July 2005.
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Department of Mathematical Sciences
University of Technology
Sydney, Australia 2007
Division of Economic and Financial Studies
Sydney, Australia 2109
Questia, a part of Gale, Cengage Learning. www.questia.com
Publication information: Article title: Peer Tutoring in Mathematics for University Students. Contributors: Carmody, Georgina - Author, Wood, Leigh - Author. Journal title: Mathematics and Computer Education. Volume: 43. Issue: 1 Publication date: Winter 2009. Page number: 18+. © Mathematics and Computer Education Winter 2009. Provided by ProQuest LLC. All Rights Reserved.
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