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This article was downloaded by: [University of Western Ontario] On: 14 November 2014, At: 15:16 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/upri20 Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course Elisha Peterson Published online: 08 Jan 2009. To cite this article: Elisha Peterson (2009) Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 19:1, 18-28, DOI: 10.1080/10511970802475132 To link to this article: http://dx.doi.org/10.1080/10511970802475132 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

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Page 1: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

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

To link to this article: http://dx.doi.org/10.1080/10511970802475132

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the “Content”) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of theContent.

Page 2: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan,sub-licensing, systematic supply, or distribution in any form to anyone isexpressly forbidden. Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions

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Page 3: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

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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Page 4: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

� 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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Page 5: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

� 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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Page 6: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

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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Page 7: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

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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Page 8: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

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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Page 9: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

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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Page 10: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

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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Page 11: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

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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Page 12: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

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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Page 13: Using a Wiki to Enhance Cooperative Learning in a Real Analysis Course

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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