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Procedía Social and Behavioral Sciences 2 (2010) 1469-1476
WCES-2010
Learning Mathematics using a Wiki
Mathias Krebsa *, Matthias Ludwiga, Wolfgang Müllera
aUniversity of Education Weingarten, Kirchplatz 2, 88250 Weingarten, Germany Received October 29, 2009; revised December 7, 2009; accepted January 15, 2010
Abstract
E-learning recently is changing dramatically with the utilization of Web-2.0 technologies in learning processes. The application of Wikis is one example showcasing this trend. Wikis allow for collaborative content creation, in particular writing and refining content iteratively. Recently, there are also several approaches to apply Wikis in such collaborative settings in formal learning contexts. However, collaborative authoring does not happen automatically. In this paper we will present an approach to apply Wikis in Mathematics classes in German Middle schools, to foster collaboration and reflection on mathematical content. Furthermore, we will present and discuss first results of our case study. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Web 2.0; wiki; project based learning; e-learning; mediawiki; collaboration; communication; social software; new learning environments.
. Introduction
In context of lifelong learning students shall be qualified in schools for learning their whole life. This implies that learning is not limited to schools only, but students are taught in different tools and techniques which enable them for learning self-directed and self dependent for their adult life to manage their own educational process. A keystone for this can be seen in knowing different possibilities of using E-Learning. The European Parliament and the Council recommended eight key competences (Official Journal of the European Union L 394 of 30.12.2006). One of them is concerning mathematical competence and basic competences in science and technology. There, it is said: "An individual should have the skills to apply basic mathematical principles and processes in everyday contexts at home and work, and to follow and assess chains of arguments. An individual should be able to reason mathematically, understand mathematical proof and communicate in mathematical language, and to use appropriate aids."
In our opinion Social Software such as Wikis can play a big role, here. Using Social Software allows contacting people with the same interests (Baumgartner, 2006). Due to learning is a social process it is recommended to use Social Software also for learning. Using Social Software, knowledge will be generated in practice and learners have to be responsible for theirs own active learning. Active learning means that content is created in cooperative learning settings. A fairly common usage of Wikis is course-related or group projects (Fountain, 2005). There, Wikis are
* Mathias Krebs. Tel.: +49 (0)751-501-8081 E-mail address: krebs@md-phw.de
1877-0428 © 2010 Published by Elsevier Ltd. doi:10.1016/j.sbspro.2010.03.220
helpful in creating, managing and editing content collaborative. In the following we will describe potentials of Wikis for learning mathematics in a way of project based learning. We will present a new concept how to integrate Wikis in classes for mathematics in school to foster communication and collaboration skills and increase mathematical knowledge.
2. Using a Wiki as a Social Software Tool for Project-based Learning
In the last years, Wikis in different forms have developed to a very important tool for information distribution and exchange in the Word-Wide Web. In addition, Wikis have developed to an important tool in E-Learning scenarios (Byron, 2005; Parker & Chao, 2007). Wikis have a lot of potential like virtual collaboration through to creating and editing content easily, fast publishing, sharing information etc. "The collaborative features (collaborative editing, versioning, discussion next to the content) of Wikis make them particularly well-suited for cooperative learning environments, with no corresponding tool in traditional learning" (Schaffert et al., 2006). Duffy and Bruns (2006) outline different uses of a Wiki. Among others, they mention that a Wiki can be used for acting as ongoing documentation of student's work and to add summaries of their thoughts and building collaborative and annotated content as well as linked network of resources.
There are various reports that virtual or semi-virtual seminars based on internet-based collaboration may provide a number of advantages (e.g., Fischer & Mandl, 2002; Issing & Klimsa, 2002; Ludwig, Müller, & Xu, 2008), for instance
1. A greater extent of engagement due to working time-independently and mobile,
2. A greater relevance of verbalization when content has to be illustrated in written form,
3. An increased quality of developed content due to
4. increased communication between learners and also between learners and lecturers.
In general, Wiki-based learning scenarios carry the potential to lead to similar advantages. However, especially the last two aspects will not be achieved automatically when applying Wikis. Instead, learning processes have to be initiated carefully and following processes need to be supervised actively by teachers to achieve these goals. For a teacher in formal educational institutions this means, they must accept a clearly different role as an initiator and coach in the learning process.
2.1. Problems and experiences by using wikis for formal learning
A considerable amount of application is available on Wikis for learning and teaching (Schwarz et al. 2004; Honegger 2005). Most is concerning to how Wikis can support writing (Rick, Guzdial, Carroll, Holloway-Attaway, & Walker, 2002; Hampel et al., 2005; Bruns & Humphreys, 2005; Forte & Bruckman, 2006; De Pedro et al., 2006). To improve academic writing, Wheeler (S. Wheeler & D. Wheeler, 2007; S. Wheeler & D. Wheeler, 2009) used a Wiki. They come to the conclusion that their "findings indicate that most students raised their skill level in writing as a result of using the Wiki space" (Wheeler, 2007; Wheeler, 2009). But also in other subjects Wikis are used. For example, Notari (2006) used a Wiki in a classroom for learning biology. He highlights the importance of developing a "communication and comment culture" as well as enabling learners "to produce a certain amount of input within a short time".
Himpsl (2007) provides different examples on how to apply Wikis in schools. For example, he used Wikis to collect information and relevant links for a specific learning objective, recording of material in brainstorming sessions, and Wiki-based Web-Quests. For example a Web-Quest was done with the topic "Golden Section". Also small projects were performed with a Wiki.
One other interesting approach is going on at the Regiomontanus Gymnasium Haßfurt (Germany). At this school, students generate content and mathematical quizzes in a Wiki. These tasks are done absolutely voluntary at home by students who do not get any marks for it. In this way, also teachers create learning paths. Learning paths are Wiki-pages (articles) where students can go through2.
See http://wiki.zum.de/Mathematik-digital
Although, as seen, Wikis are used in education experiences for learning Mathematics project-based, fundamental surveys about students reflect on their knowledge is missing.
Before illustrating a concept how to embed a Wiki in schools (see chapter 2.2) we used a Wiki in a seminar at University of Education Weingarten (Germany). Students in Mathematics and Education on project-based Math learning were given the task to use a Wiki for conceptualizing and elaborate own mathematical projects beyond regular lectures (Krebs & Ludwig, 2009). In this seminar, students had twelve weeks in total for realizing their projects. At the end, students had to present their results in front of other students. Some students had to make a written paper in addition to get a certificate. By analyzing the content created by the students as well as evaluating a questionnaire which was filled in by the students after the semester, it was seen that the Wiki was mainly used for presenting the current status of the project and as a kind of a project diary.
Based on the Wiki's page history mechanism, it was possible to trace and analyze the development process of individual pages in some detail. Figure 1 depicts the development and restructuring of articles over time. However, the analysis also revealed that in fact the underlying process of article was rather cooperative than collaborative. Students divided the work into different parts and worked rather independently on their individual tasks. Collaboration (iterative editing and refining of content in the Wiki by different students) took place only seldom. At the figure below the code of the colors indicates that no chapter was edited by more than one user._
1. Objectives 2 Planning
2.1. Thoretlcal background
3, Execution
4, Close-Out/Reflection
5, Project diary
1. Objectives
2. Planning
3. Execution
3.1. Background - physics
3.2. Background - parabula
3.3. Blast pipe in classrooms
3.4. Bridges
3.5. Throws
4. Close-Out/ Reflection
5. Project diary
1. Objectives
2. Planning
3. Execution
3.1. Background - physics
3.2. Background - parabula
3.3. Blast pipe In classrooms
3.4. Bridges/Architecture
3.4.1. Parallelogram of forces
3.4.2. Parabula bridges
3.4.3.Definition throw
3.4.4. Different kinds of throws
3.4.4.1. Throwing vertically
3.4.4.2. Throwing horizontally
3.4.4.3. Throwing oblique
3.4.4.4. Free fall
5. Close-Out Reflection
6. Project diary
1. Objectives
2. Planning
3. Execution
3,1 Background - physics 3.2. Background - parabula 3.3 Bridges/Architecture
3.3.1. What are bridges
3.3.2. Different kind of bridges
3.3.3. Parallelogram of forces
3.3.4. Arched bridges
3.4. Throws
3.4.1. Definition
3.4.2. Different kinds of throws
3.4.2.1. Throwing vertically
3.4.2.2. Throwing horizontally
3.4.2.3. Throwing oblique
3.4.2.4. Free fall
3.4.2.5. Ballistics
3.5. Blast pipe In classrooms
4. Interesting links
5. Close-Out/ Reflection 6 Project diary
Figure 1: Restructuring and elaboration of content at the example "parabola in everyday life" in a student seminar.
Also, content related discussions on the Wiki's discussion pages were quite rare. The difficulty to initiate content-related discussions in a collaborative learning design is also described by Bescherer (Bescherer, Ludwig, Weigand, & Schmidt-Thieme, 2004). One explanation for this is that most students used the Wiki on a rather irregular base. Time to get an answer apparently seemed too long sometimes. Students confirmed this after the project. Further, students expressed in a questionnaire that most problems were discussed in personal communication since they did in fact meet, and other, more accepted communication technologies such as ICQ and StudiVZ were preferred to
perform additional necessary discussions over the Internet. Moreover, some students expressed that using the Wiki was too time-consuming (additional expenditure of time)3.
In a second experiment we changed the setting and set up a collaborative seminar with student groups both from the University of Education in Weingarten (Germany) and students of the East China Normal University (China). Again, students had to utilize a Wiki to develop ideas for project-based learning in a collaborative scenario. In this case, project groups consisted of two German and two Chinese students. Here, the motivation was to foster the utilization of the Wikis also for discussions due to the distributed setting. Unfortunately, due to organizational reasons there was a delay of the project start on the Chinese partner university and Chinese students could only start some weeks after the German students had started. Once more, collaboration and cooperation did not take place in the expected way. In this case, the delayed project start on participating Partner University seems to be responsible for this. Apparently, a parallel project start seems to be essential in Wiki-projects just like in other projects.
2.2. New concept of mathematical project work beyond regular lectures for solving mathematical questions in German schools
The question remains, how is it possible to integrate a Wiki for project based math-learning into schools in such a way that students communicate, discuss and collaborate as a precondition to foster students to describe and reason different mathematical approaches, assess results belonging to the sphere of reception, explain relations and structures and use appropriate representations?
Our approach relates to new approaches for assessment in school. In the last years, different forms than traditional exams have been integrated into different schools in Germany4. In some schools such an evidence of achievement can be e.g. a written homework (in an amount of about 15 pages), an experimental work, an oral presentations or a combination of them. The aim is to enhance competences such as media literacy, self-directed learning, and so called "soft-skills". These evidences of achievement are compulsatory in addition to traditional exams. The assigned marks count like an additional exam.
In our concept, students of grade 8 and grade 9 do an evidence of achievement in the context of E-Learning: students from different schools are using a Wiki for project work in a period of one month. At this, a project group consists of two to four students of disjunctive classes from different schools (see figure 2).
Figure 2: Composition of a project team
The structure of our concept is illustrated in figure 3.
Figure 3: Structure of our concept
On the other hand one project group which had to make a written form for getting a certificate said that using the Wiki was timesaving because most work was already in written form available.
In the federal state Baden-Württemberg (Germany) this is called GFS (abbr. „Gleichwertige Feststellung von Schülerleistungen")
To ease the preparation of the projects the topics of individual projects were predetermined, and students had to choose a project from a given project list. This allowed us to design projects for acceptable group size and to minimize organizational effort in the agreement on topics. Mathematical problems were designed in such a way that they could be extended for those who have finished their work too early within the month. Moreover, the complexity of the projects was chosen in such a way that communication and the exchange of ideas on different strategies would provide a real benefit when performing the project. In our approach, we applied problems related to Fermi problems, such as estimating the necessary number of dentists necessary to provide a sufficient service in Germany.
Face-to-face meetings were planned for kickoff and final project celebration. In the kick-off meeting, students were introduced to the individual projects. We also provided certain guidelines for performing the projects. These guidelines were placed at a wiki page. Project presentation happens in two ways. On one hand, students had to present their results individually at their own school; on the other hand, the whole project team presented its results jointly in a final project meeting. For developing a "communication and comment culture" as well as enabling learners "to produce a certain amount of input within a short time" (Notari 2006) the kickoff meeting is predestined as well as teaching learners how to use the Wiki.
For a Wiki, we utilized a MediaWiki5 set up and administered by a computer expert. Through of this design we expect to solve the disadvantages explained in section 2.1 of this article.
3. Case Study
In a case study we followed the structure sketched in section 2.2. Students of grade 8 and grade 9 do an evidence of achievement in the context of E-Learning. Students from different schools and grades used a Wiki for project work in a period of one month. Among others, following topics related to Fermi problems were given as possible tasks:
5. P1: How many dentists, policemen, math teachers are recommended in Germany?
6. P2: How much money is lost in Germany be dodging the fare?
7. P3: Have bicycle with 21 gears really 21 different gears?
8. P4: How much money is getting lost due to stand-by devices? How much energy could be saved in Germany without stand-by? How large has to be a solar module for illuminating the Allianz-Arena (Footballstadium) in Munich?
9. P5: Is it worth to change from gasoline to natural gas passenger car?
10. P6: Math and football - Why are there ten field players and not more or less? How much time is needed to produce a football?
In our survey every project team consisted of three students, each from different classes and schools. All in all six schools participated. At the kickoff meeting 10 of 17 participating students attended. For scheduling the meeting the web-software doodle6 was used. The kickoff meeting lasted two hours and outlined the question on how the Wiki could be utilized at best to support project work. In addition, the individual groups individually developed guidelines and rules for their individual project in a group work phase. Examples for rules developed in this phase are
11. How regularly a group wanted to work with the Wiki and provide new material (daily, every second day, etc.)
12. How to handle questions (e.g., every time a question was asked, others must write to it even he does not know the answer)
13. Working hours (e.g., one week ahead in table form)
14. Hierarchy in groups (e.g., does a formal group leader exist?
15. How often interim feedback was wanted on the part of lecturers?
In the kickoff meeting, students also had to perform three smaller tasks on the computer. These included changing password for accessing the Wiki, creating an individual user page, which included uploading and embedding an image, and writing a favourite mathematical formula on a different user page. Moreover, for homework students had to summarize their results of their group work on project guidelines in the Wiki.
http://en.wikipedia.org/wiki/MediaWiki http://doodle.com
As already mentioned, not all students were able to attend to the kickoff meeting. At this meeting, only two groups were complete; for one group one project member were missing, for two groups two project members were missing, and for one group all team members were missing. For those groups where members were missing in the kickoff meeting, the corresponding team members were informed via an email on the outcome of the meeting one day after the kickoff meeting happened.
4. First Results & Conclusions
At the current state, we can only provide fist preliminary results from our study based on the first two weeks of project work. In figure 4 the cumulative increased amount of pages and edits are illustrated. The pages include articles as well as discussion pages and user pages.
11 ■' 1 Î''2DD9 nyi&2D09 V.'-VMXH
Figure 4: Increased amount of pages end edits during the first two weeks, cumulative
During these two weeks we expected that students were finalizing group rules, establishing project questions and beginning to think about strategies how to solve the project questions. Answers concerning reflection on mathematical content cannot be given at this state of the project yet. In figure 5 all six projects (P1, P2, P3, P4, P5, P6) are illustrated. Tasks and remarkable interventions are pointed out.
P2 P3 P4 P5 P6
1 2 3 4 5 e 7 8 9 10 11 12 13 14
| H | M m □ n n
Legend: □
□ □
No Task
Task performed concerning group rules (group allocation) Task performed to substantiate project question Task performed concerning strategy Help/Tip/lntervention from Lecturer via Wiki
Figure 5: Increased amount of pages end edits during the first two weeks
Students of P1 needed some days to figure out the project questions (to calculate the recommended amount of dentists, math teacher or others). Finally, they agreed on calculating the recommended amount of math teachers in Germany. In which way the increased communication at the beginning of the project has affects on collaboration later on, cannot be said at this stage of the project.
At P2, all group members had joined the kickoff meeting. The project rules were figured out very fast and first strategies and calculations were done. Also some questions were asked by students due to misunderstandings of calculations.
At P5 most page edits happened. All in all 25 edits at the project page and the corresponding discussion page happened. In opposite of them, only 3 page edits happened at P6.
Through to using an internet platform, external and unexpected and unplanned influences happen in the first project phase. At the project group who handled the question "gasoline or natural gas passenger car?" (P5) one student wrote that his father's friend owns a gasoline car and he wanted to interview him. Also asking car dealerships was an idea. On top of that a hyperlink with interesting information was embedded from someone else. Here it can be seen that some first beginnings of using appropriate aids were used. An other example of unplanned influences is concerning to the question "Why are there ten field players and not more or less?". One team member found an extract of an e-Book in which this question was answered. In this case the project team has to understand the content, formulate it in own words and discuss questions and certain aspects.
An explanation for the slow progress of project P6 is that only one student attended the kickoff meeting and the other two students were informed via an email while one of them was informed 4 days after the project had started due to no valid e-mail address. At day ten an intervention was done by the Coach. Although, this intervention included a precise task and a deadline at which the task had to be done, nothing was done by the students. Again, a delayed project start on participating students seemed to be responsible for this. Further, we figured out that one student of P6 who had not attended the kickoff meeting, had problems in understanding how to use the Wiki. Also missing group rules seems responsible for this.
For gain more information about characteristics in collaboration, the progress of the project have to be more proceeded. Further state of the project is needed to get more information in which way Wikis can help students getting competences in networking with learners of the same interest and can help to fulfil aspects like reason mathematically like it is described by the European Parliament and the Council.
At this state of the project, it seems that treating such kind of "real world tasks" using a Wiki can help generating knowledge in practice. Due to the complexity of the tasks, communication with others about mathematical content is essential for good success in our cooperative learning settings.
Acknowledgement
The research is part of a comprehensive post-grade colloquium on "E-Learning in schools as headstone for lifelong learning" and is promoted by the federal state Baden-Württemberg (Germany).
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