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This article was downloaded by: [University of Western Ontario]On: 14 November 2014, At: 15:16Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK
PRIMUS: Problems, Resources,and Issues in MathematicsUndergraduate StudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20
Using a Wiki to EnhanceCooperative Learning in a RealAnalysis CourseElisha PetersonPublished online: 08 Jan 2009.
To cite this article: Elisha Peterson (2009) Using a Wiki to Enhance CooperativeLearning in a Real Analysis Course, PRIMUS: Problems, Resources, and Issues inMathematics Undergraduate Studies, 19:1, 18-28, DOI: 10.1080/10511970802475132
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PRIMUS, 19(1): 18–28, 2009
Copyright � Taylor & Francis Group, LLC
ISSN: 1051-1970 print / 1935-4053 online
DOI: 10.1080/10511970802475132
Using a Wiki to Enhance Cooperative Learning in aReal Analysis Course
Elisha Peterson
Abstract: This article describes how the author used a wiki-based website in a real
analysis course, and assesses its effectiveness. The wiki was used to post course
materials, maintain a forum, enable students to write collaborative projects, and
enable students to develop a glossary of important terms. The wiki proved to be
very successful; it facilitated student collaboration, exposed students to LaTeX, and
even helped them to study for examinations.
Keywords: Collaboration, cooperative learning, discussion board, forum, Internet,
real analysis, technology, wiki.
1. INTRODUCTION
A wiki is a website which is constructed primarily for the purposes of online
authoring and collaboration [4]. The most famous wiki is Wikipedia [7], the
online encyclopedia constructed with the help of hundreds of thousands of
users. But wikis also exist on much smaller scales; there are thousands of
wiki sites on the web, allowing groups of individuals to collaborate around
shared interests such as an employer, video games, or academic disciplines.
Sometimes wikis are created for use by a single person. (Indeed, this paper
was originally written on the author’s personal wiki.)
There are several features that attract users to wikis:
� Online authoring: a single document may be edited anywhere the Internet
is accessible.
� Collaborative authoring: multiple users may easily edit the same docu-
ment, and user-access may be controlled.
Address correspondence to Elisha Peterson, Department of Mathematical Sciences,
United States Military Academy, 646 Swift Road, West Point NY 10996-1905, USA.
E-mail: [email protected]
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� Streamlined markup language: documents may be edited without needing
to learn more code-intensive languages like HTML.
� Mathematical typesetting: many wikis have the ability to typeset mathematics
using LaTeX.
� Linking structure: wikis are built as collections of pages with an abundance
of links between them; frequently the linking is done automatically.
� Change tracking: wikis allow users to see prior versions of the page, who
edited the page, and what they changed.
Wikis are natural tools for collaboration. In practice, they are frequently
used for collaboratively constructing knowledge, as in the case of Wikipedia.
They are starting to be used more for educational purposes as well, usually
with a similar emphasis on building a collaborative knowledge base [1, 2].
But wikis also have great potential for other kinds of cooperative learning.
Collaborative authoring and mathematical typesetting are two promising
features for mathematics instruction. In addition, change tracking provides
an easy way to monitor and assess student contributions to wikis.
I first used a course wiki to supplement an undergraduate real analysis
course. Given the features described above, I felt that a wiki would be a natural
tool for enhancing cooperative learning, students’ ability to write mathematics,
and instructor-student communication. My specific objectives for the course wiki
were to (i) encourage collaboration among the students; (ii) develop students;
ability to communicate mathematics to others and give them the tools to do so;
(iii) provide students a ‘‘gentle’’ introduction to LaTeX; and (iv) expose students
to collaborative technologies and the movement towards collaborative projects.
This paper begins with a description of how I implemented a wiki in real
analysis, although all of the ideas presented should naturally extend to other kinds
of courses. During the course, students were surveyed regarding how the wiki was
used in the class. These surveys, together with instructor insight, were used to
assess the efficacy of the course wiki. I will examine each of the ways in which
the wiki was used, and discuss insight gained regarding the difference between a
course wiki and a more traditional course management system such as
Blackboard. I will also discuss my initial concerns with using the wiki, including
lack of participation (a frequent problem with using online course features), the
learning curve required for students to be comfortable using the wiki, and
instructor/student time management. Student feedback and my observations pro-
vide strong evidence that the wiki met these goals, and indicate that wikis could be
a powerful platform for collaborative learning and education in the future.
2. DESCRIPTION OF THE REAL ANALYSIS WIKI
The wiki discussed in this paper was developed for an undergraduate real
analysis course that consisted of nine mathematics majors having a broad
spectrum of GPAs. I used the wiki in four ways:
Wiki, Cooperative Learning, and Real Analysis 19
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� Course materials were posted, including the syllabus, homework assign-
ments, solutions, and supplemental material.
� A forum (discussion board) was maintained.
� Students collaboratively developed a glossary of terms during each block
of material in the course. The resulting document was provided to them
during examinations.
� Students wrote and posted projects.
The wiki was hosted by the free service Wikidot [6] at http://usma387.
wikidot.com. I created the site by inputting some basic information into a web
form at http://www.wikidot.com. When first created, the front page of the wiki
contains information about editing the wiki. I used this front page to post
information about the course, as shown in Figure 1. Other pages on a Wikidot
site have names of the form category:pagename. I used this category structure
to restrict student access on the site, giving them editing privileges in only
Figure 1. Screen shot of course wiki home page.
20 Peterson
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certain categories. I restricted outside access to the website by making the site
private and requiring students to create a user account with Wikidot in order to
access the site.
2.1. Course Materials and Forum
All course materials were maintained at the course wiki, including a syllabus
which was updated over the course of the semester, homework assignments,
and information about upcoming deadlines. I maintained each of these as a
separate page on the site. One of the nicest features of the wiki was the ability
to easily link to other sites, such as Wikipedia [7]. This made it easy to
provide students with links to read about other applications of the subject, or
the historical figures important in its development.
In addition, I created a discussion forum for use with the course. The forum
is automatically generated by Wikidot, when the administrator ‘‘activates’’ it.
Participation in this forum was completely optional, although categories were
made for Reading Questions, Homework Questions, Technology Questions, and
Instructor Notes. I invited students to post questions to the forum, and I posted
homework solutions or hints there.
Posting course materials to the web is nothing new, nor is maintaining a
course forum. What is challenging, however, is getting students to use the
course forum without a ‘‘stick or carrot’’. Although no requirement was made
that students use the forum, many did post questions about specific problems
on the course forum, and I often used it to communicate with the students.
Given that the class was so small, most of the students collaborated on their
own, apart from using the course forum. This is probably mostly due to the
difficulty of the material. One student did report that he would have liked the
forum to be used more, suggesting ‘‘more emphasis on working together
using the wiki’s forums’’ on a feedback survey after the first block of the
course (as a suggestion for improving the course). Given the proper encour-
agement, students most likely would have made more use of this feature.
2.2. Glossaries
During each block of course material, I created a new wiki page at glossary
:chapter-n. I contributed and organized the ‘‘key terms’’ of the block to this
page, using the Wikidot syntax for definition lists [5]. I required students to
contribute two or three entries per block to the glossary, with a small amount
of points assigned to this task. To do this, students clicked an ‘‘edit’’ button on
the glossary page and filled in definitions next to terms. I asked students to use
proper mathematical notation, and provided some basic information on editing
entries and typesetting using LaTeX, as shown in Figure 2.
Wiki, Cooperative Learning, and Real Analysis 21
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The students were highly motivated to contribute to the glossary, since
the final product of each block was made available to them on the exams.
Based on student feedback and my observations, the glossary was the most
successful usage of the wiki. The key benefits were the following:
� students received information about what their instructor considered
important for upcoming exams;
� students had an opportunity to simultaneously review and organize key
terminology prior to the exam; and
� students were able to correct each other’s work.
Students were asked after the first midterm about the glossary assignment
(Table 1). They were grateful to have the glossaries on the exams, and it allayed
their anxiety somewhat. They liked the collaborative nature of the assignment,
and several students submitted entries beyond the minimum required.
One of the drawbacks of the glossary was that students tended to make
their submissions during the same evening, resulting in some conflicting ‘‘page
Figure 2. Screenshot of wiki developed by the class for the second exam.
22 Peterson
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requests’’. Since only one user may edit a page at a time, this forced some
students to wait before submitting their entries. However, none of the students
reported this as a major problem, probably because the small class size made
this a rare event. For larger classes, it would likely be more of an issue.
2.3. Course Projects
I assigned students two projects over the course of the semester. The first
project required them to read an article in a mathematics journal and write a
summary of the article, focusing especially on its connections with the course
material. They were also challenged to describe a little of the broader context
of the mathematical area addressed by the paper, which can be a challenging
task for an undergraduate. My goal was to provide students practice with
reading mathematics, organizing and condensing thoughts, and writing in the
mathematical language. For the second project, students worked in groups to
write a paper on an advanced topic in real analysis or an application.
Rather than turning in a traditional paper report, students submitted their
projects on the course wiki by directly editing pages. I assigned them specific
page names to use for their projects, and gave them instructions for editing
pages. After the glossary, this was one of the first exposures students had to
Table 1. Student feedback on the glossary
0-min 15-30min 30-45min
45min-
60min
over
60min
How much time did you spend
working on the glossary?
0 3 4 1 0
Strongly
Agree
Agree Neutral Disagree Strongly
Disagree
I liked the collaborative nature
of creating the glossary
2 6 0 0 0
There was enough information
provided about editing pages
and wiki syntax
3 4 0 1 0
I found mathematical typesetting
easier on the course wiki than
in Microsoft’s Equation Editor
1 2 4 0 1
Adding information to the
glossary helped prepare me
for the exam
1 6 1 0 0
Looking at the glossary helped
prepare me for the exam
1 6 1 0 0
I found the glossary useful
during the exam
5 3 0 0 0
Wiki, Cooperative Learning, and Real Analysis 23
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LaTeX. The wiki neatly handled the collaborative aspects of the second
project, with each individual in a group of students able to edit the page.
The key positives of this wiki approach to course projects were the
following:
� all edits to pages were logged, allowing the instructor to see exactly when
the students are working on the project and how long they are taking;
� students had the ability to view each other’s projects, giving them plenty of
good (and some bad) examples to reference; this may have had an impact
on the quality of their work;
� students provided links to the articles they reviewed for the first project
directly on the page;
� some students took advantage of the web format, providing links to
mathematicians referenced in their project as well as to websites that
could offer the reader additional information about the subject; and
� students who were away from campus on the project due date were able to
turn in the assignment as easily as those who remained on campus.
Results of student feedback on the first project are shown in Table 2. Some
students took advantage of the opportunity to view other projects while they
were writing their own, or after they had completed their project. A few students
directly mentioned the ease of LaTeX for mathematical typesetting, the ability
to see others’ work, and the ability to submit the assignment from home. It also
seems the assignment offered the students a change-of-pace from usual projects.
The primary negative of the wiki-based project was the learning curve
associated with wiki notation. While this was a concern, the primary objective
for the first project was reading and writing mathematics; I expected the
students to face the challenge of learning LaTeX. In that sense, the struggles
with formatting mathematics were indicators of success. On the other hand,
there was a time cost due to using a more unfamiliar markup language for
formatting the documents. For the most part, wiki syntax is easy to learn,
and Wikidot provides a toolbar for the more commonly used commands.
However, some students had difficulty with inserting figures into their project
pages, since one must upload the figure to the website rather than copying it
directly onto the page. This process is easily mastered, after it has been
completed once, but it did distract from the main point of the assignment.
3. ASSESSMENT OF OBJECTIVES AND CONCERNS
3.1. Collaboration and Communication
Student responses indicate that the wiki encouraged them to work more
collaboratively. Students were especially appreciative of the collaborative
24 Peterson
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nature of the glossary, and also liked to see what other students in the class
were doing with their projects. One particularly enthusiastic student devel-
oped his own wiki site for a summer research project.
I did see substantial improvement in the ability of students to commu-
nicate mathematics over the semester, but it is difficult to say how much the
wiki contributed to this objective. Most of the improvement likely came from
more traditional activities such as homework assignments, but the wiki did
provide a good outlet for students to write about mathematics.
3.2. Mathematical Typesetting
One of the goals of the wiki was to help students learn LaTeX. Before the
class, most students had experience with Microsoft’Equation Editor and
Table 2. Results of student feedback on the project (Block II)
Yes No Tried
Did you look at any of the other projects
while you were writing your own?
4 5
Did you look at any of the other projects
after you had finished your own?
4 4 1
Did you look at any of the other projects
at any time?
7 1 1
1. In your opinion, what were some of the advantages of submitting the project to the
Wiki rather than in a more traditional format?
LaTeX formatting power (3)
Unique and different assignment (3)
Ability to submit from home (2)
Could see others’ work (2)
Preview capability
No unnecessary title pages
Later turn-in time
2. In your opinion, what were some of the disadvantages of submitting the project to
the Wiki rather than in a more traditional format?
Learning Curve was steep (5)
Problems with formatting. . . centering,
fonts, pictures, etc. (4)
Problems with math notation (2)
Someone may have lost work
3. Any other comments about the project?
‘‘There is a steep learning curve with the wiki which added a little bit of work time
to the project.’’
‘‘It takes some time to get to know LaTeX, and how to work with wiki, so I spent
more time on that and a little less on the math aspect.’’
Wiki, Cooperative Learning, and Real Analysis 25
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nothing else. All but one student reported that they found it easier to typeset
equations using LaTeX than Equation Editor. I hope that these students will
have an easy transition to full document typesetting with LaTeX in the future.
As a promising indication that this might happen, one of the students began
submitting homework assignments in LaTeX towards the end of the
semester.
3.3. Participation, Learning Curve, and Student/Instructor Time
While students did not take full advantage of some optional wiki features,
such as the course forum, they were quite happy to contribute when grades
or bonus points were attached to participation. Beyond the grades, I did
not need to encourage the students to contribute to the glossaries, for
example.
Most students reported that it was not difficult to learn the wiki
notation and markup, although there were a few exceptions. The process
of adding entries to the glossary took longer than expected, yet was one of
the exercises most valued by the students. In the end, the wiki added to
the students’ overall workload, but by a very small amount; the students
still spent the vast majority of their time in the course on homework
assignments.
From an instructor’s perspective, setting up the wiki required 5–10 hours
of work, and maintaining the site required an hour or two per week. In the
long run, I felt that the wiki actually saved me time, since it makes the
process of posting new material to the web so easy.
3.4. Learning Mathematics
I believe the wiki also contributed to the students’ learning of real analysis in
a few ways. First, I think the streamlined student-instructor communication
helped to keep the students encouraged and engaged in the subject. I was able
to post replies to student questions on the wiki or course forum. My responses
could be seen by all the students, which in some cases allowed students to
continue with a problem where otherwise they may have been stuck. I found
the wiki an ideal mechanism for communicating in mathematics because of
its built-in support for LaTeX. The alternatives, whether communicating
mathematics without special symbols, posting a PDF, or scanning a hand-
written document, would all have been slower and in some cases less precise.
I also used the glossary to convey the fundamental ideas and concepts in each
block of material, allowing students to focus their study. I hope that, by
observing and participating in this process, students will be more prepared to
recognize key ideas and concepts in future courses.
26 Peterson
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Second, I believe the glossary reinforced the idea that mathematical
language is unforgivingly precise, a key concept in real analysis. For
example, the order of epsilons and deltas in the definition of continuity
cannot be changed. The process of writing out definitions suitable for use
on exams helped students to understand the importance of quantifiers in
analysis.
4. FINAL COMMENTS
The wiki was helpful in achieving several of the learning objectives in the
course. The collaborative nature of wiki technology made it a natural fit for
collaborative student projects. Students writing projects also benefitted from
being able to see each others’ work, and from having an efficient way to
bring additional Internet resources into their projects. Many of the students
were exposed to LaTeX for the first time, and were able to see its many
benefits. The collaborative glossaries were especially beneficial to the students,
and proved to be a useful study tool before examinations. While scaling
a glossary to larger class sizes will likely require some refinement, in the
small-course environment of the real analysis course it was a huge success.
Finally, many students simply enjoyed ‘‘doing something different’’ in the
course.
In future courses, I would like to make more extensive use of the
discussion board and collaborative authoring capabilities of wikis. As sug-
gested by one of the referees, the discussion board could be a great way to
collaboratively explore open-ended questions and explore some of the nuan-
ces that exist in real analysis. I am also intrigued by the possibility of having
students write or correct proofs on the wiki. One student could post a draft of
a proof on the wiki, with another student responsible for editing and polishing
the proof, and correcting it if necessary. Alternately, the instructor could
post an incorrect or poorly-worded proof and require the students to make
the appropriate edits. A third option is to post a sketch of a proof and
require students to fill in the details. Such assignments would give the
students the opportunity to emulate a fundamental part of the mathematical
profession, working together to understand and complete proofs. Using
a wiki’s ability to track changes would permit the instructor to see the
contributions of each student, and thereby have a window into their thought
process.
I’m sure the reader can think of other ways to leverage wikis in real
analysis—or in other mathematics courses. Wiki technology can be both easy
to master and effective as an aid for mathematics instruction. It facilitates
classroom collaborations, without distracting significantly from the main
point of a course. In short, wikis open the door to a whole new way of
looking at cooperative learning in the classroom.
Wiki, Cooperative Learning, and Real Analysis 27
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ACKNOWLEDGMENTS
The author would like to thank the editor and referees for many helpful
comments and passing along some ideas for further ways to use wikis in the
classroom.
REFERENCES
1. Duckworth, W. E. 2008. Student Collaboration using a LaTeX Wiki. MAA
Focus. 28(5): 17–19.
2. Ferris, S. P. and H. Wilder, 2006. Uses and Potentials of Wikis in the
Classroom. Innovate Journal of Online Education. 2 (5). http://www.
innovateonline.info/index.php?view=article&id=258. Accessed 7 July 2008.
3. Tapscott, D. and A. D. Williams, 2006. Wikinomics: How Mass
Collaboration Changes Everything. New York NY: Portfolio.
4. Wiki. http://en.wikipedia.org/wiki/Wiki. Accessed 7 May 2008.
5. Wiki Syntax Description. http://www.wikidot.com/doc:wiki-syntax.
Accessed 7 July 2008.
6. Wikidot. http://www.wikidot.com. Accessed 19 June 2008.
7. Wikipedia. http://en.wikipedia.org. Accessed 19 June 2008.
BIOGRAPHICAL SKETCH
Elisha Peterson received a BS in Mathematics from Harvey Mudd College, and
completed graduate work at Oxford University and the University of
Maryland, College Park, where he was awarded a PhD in 2006. He is a
Davies Fellow and Assistant Professor at the United States Military
Academy (West Point), where he has been since the Fall of 2006. His primary
research interests include diagrammatic algebras and their applications to
geometry, linear algebra, and invariant theory, as well as the mathematics of
cooperation. When possible, Elisha enjoys spending time with his family,
running, and competing in triathlons, as well as implementing and visualizing
mathematics using the Java programming language.
28 Peterson
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