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MATHEMATICS PROBLEM SOLVING USING A WEB-BASEDKNOWLEDGE MAP AND ANALYSIS OF THE PROCESS

Masahiro NAGAI*, Katsuya SHIRAKI**, Hiroaki KOSHIKAWA***,

Kanji AKAHORI*

Tokyo Institute of Technology*, Futtu Junior High School**,

Chiba University***

1. INTRODUCTIONAlthough mathematics education researchers are increasing making use of the Internet in theirresearch, there is little practical research into Web-based collaborative learning whichpromotes interaction and externalizes communication of learners to share knowledge. Theauthors have conducted collaborative learning sessions in real mathematics classes using Webbulletin board, producing positive findings (Nagai, 2000ab). We consider these collaborativeand voluntary learning processes to be Legitimate Peripheral Participation (Lave and Wenger,1991); furthermore, those processes amount to qualitative changes in learning.2. RELATED WORK2.1 Collaborative learning systems and their problemsA number of studies have been conducted on collaborative learning using the Internet. Amongthese, well-known research projects include Web CSILE, from the University of Toronto’sOntario Institute for Studies in Education, and the Collaboratory Notebook used in the CoVisProject of Northwestern University. Common to both of these effective systems isasynchronous learning, in which learning progresses as a result of the discussion which takesplace as learners post their ideas and pose questions about the ideas and opinions of others.However, this may not hold true when we focus on visualization of notes (i.e., externalizationof thought) entered by the learner (Hewitt, 1997 Suthers, 1999).2.2 Previous researchThe authors have previously discussed that mathematical knowledge is built by relating newconcepts with the students previous knowledge and the knowledge of others (Nagai, 1996).We believe that the communication involved in Web-based collaborative learning plays a rolein relational understanding. Hence, after reviewing previous research as outlined above, wedeveloped the NakSun system in 2000 to present this knowledge in a more useful format -using a bulletin board system and knowledge map.

3. THE NAKSUN WEB BULLETIN BOARD3.1 NukSun User InterfaceFigure 1 shows the NakSun knowledge map. Each of the points on the window expresses a

After briefly surveying previous research in collaborative learning systemsincluding NakSun, a bulletin board enhanced knowledge map developed by theauthors, we describe the results of its use at a Junior High school. From ourobservations, the correct answer to mathematical problems was discovered earlythrough collaboration between students. However, in subsequent discussion, wefound that students both submitted useful mathematical views and ideas duringcollaboration and that incentive for learning and discussion were both increased.

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note and a line that links two points shows that they are related. Figure 2 shows some notesthat students have entered. For example, notes can be displayed by: (1) all notes, (2) onlynotes that are linked to a note selected by the learner. In implementing the system, we usedthe Java programming language to create the knowledge map display, and Perl to operate thedatabase and characters.

Figure 1: Knowledge map

Figure 2: Notes entered by students3.2 Displaying linked notesWhen the learner inputs note numbers that are related to a note he or she entered in theentering field, the system draws a knowledge map. With traditional thread-based discussion,only references to messages in one thread can be made. Using NakSun, learners can referenceany notes they recognize as related. A learner may view any note by double-clicking itscorresponding node, and the system can only display the notes linked to this note.

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4. RESEARCH OBJECTIVES AND METHODSAlthough NakSun was used in four collaborative learning studies over March, May, June andOctober of 2001, only the latest study will be discussed. The research objectives were asfollows:4.1 Objectives To discover how learners given mathematical problems which have a single solution

conduct mathematical activities through collaboration. To determine the effectiveness of the NukSun system using a questionnaire.4.2 Implementation in school mathematics classesLearners attempted to collaboratively solve a mathematical problem using the NakSun systemhosted on a Web server at Chiba University and a questionnaire was handed out after the thirdclass. The experiment was implemented as follows: Fourth implementation (October, 2001) Student participants: Nagaura Junior High School, 3rd grade (1 class, 37 persons) Client Operating System: Windows 98 (One student per computer) Lesson schedule: NakSun (3 classes) followed by a questionnaire Mathematics problem: ‘The number of greetings’Mr. A and Mrs. A went to a party where there were another two husbands and wives bringingthe total number of participants to 6 (i.e. multiplying two persons by 3). These people greetedeach other, however each person greeted only one and nobody greeted their own spouse. Afterthe party, Mr. A asked all the other participants. “How many times did you greet?” Everybodyanswered with a different number of times to him.　 -------- Question --------How many times did Mrs. A greet?5. DISCUSSION5.1 Effective links and graph developmentAfter the three classes, 78 noted were entered and we classified these into five groups basedon the number of nodes linked, displayed on the graph whereby the node number representsthe note number (Figure 3). Based on the research of Tarja-Riitta Hurme (2001), we classifiedeach note into nine categories based on their meaning.

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Figure 3: Graphs classified by the number of nodes5.1.1 Valid links“Valid links” are any two notes which are linked by a meaningful relation as determined bythe experts of mathematics educations. Invalid links are denoted in Figure 3 by a “?”. Thefollowing are the examples of valid links and invalid links. Since notes 34 and 42 are linkedand are meaningfully related, we consider it as a valid link.-------------------------------------------------------------------------------------------------No.34 Eh? This is,,,,,,, The number of references: 4 10/19 Friday 11:49According to the point of the mathematics problem, other people answered different number oftimes, isn’t it? I think a number of times of greetings which are answered by one participantand others must not overlap.-------------------------------------------------------------------------------------------------No.42 Eh! The number of reference: 1 10/19 Friday 11:56But, the range of answer can move from 0 to 4, that is, there are 5 numerals. I think a numberof greetings and others can overlap because participants totaled only 6. But, I am confused.-------------------------------------------------------------------------------------------------Conversely, notes 10 and 23 do not have a valid link since the relation of meaning is not clearfor the learner.-------------------------------------------------------------------------------------------------No.10　ANSWER 　 The number of references: 15 10/16 Tuesday 9:28Assuming that 6 persons are ‘AaBbCc’ (a expresses Mrs. A.),A=Cb, C=AaBb, b=CAa, a=Cb, B=C, c=emptyTherefore, the answer is 2 times.-------------------------------------------------------------------------------------------------No.23　Overlap is OK The number of references: 1　10/19 Friday 11:36Because A asked other 4 persons, I think there is someone whose number of times and A’snumber of greetings overlap. That is, if A greets two times, other 5 persons include someonewho greeted two times.

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-------------------------------------------------------------------------------------------------We found the ratio of valid links to invalid links was 44:49 (44/49=0.90) meaning that linkson the knowledge map are not drawn meaninglessly - 90 percent of links are drawnmeaningfully. It indicates links are made reliably which makes it possible to analyze aknowledge map built by collaborative learning using NakSun.5.1.2 Responses to important notesIn light of mathematics learning, there are important and unimportant categories based oncontent of notes in Figure 3. When comparing important notes and unimportant ones, both thenumber of responses and the number of notes which were responded to differ greatly (Table1).

Name of category Number of responses / Number of notesresponded to/ Total number of notes in thecategory

Unimportant question 1 / 1 / 13Unimportantnote Mere agreement 0 / 0 / 10

Uniqueness of solution 9 / 4 / 13Importantnote Verification of the of

problem12 / 8 / 16

Table 1: Deference of replies between important and unimportant notesSince each note which is linked to another are meaningfully related and there are moreresponses for valuable notes, this indicated that learners could use the knowledge mapeffectively as a method of externalization of thinking about the relationship between notes.5.1.3 Developing the graphLooking at the graph in more detail, we can see there are 30 nodes (Figure 3) and learnersdiscuss whether note 10 is related to the problem. Notes 29, 32, 39, 40, 43, 47 and 54 followon note 23 which is linked to note 10. The contents of note 39 and 47 are displayed below.Note that the note number is based on time and sequence.-------------------------------------------------------------------------------------------------No.39　eh!? The number of references: 6　10/19 Friday 11:55Mr. I. said there was anyone who greeted two times in other 5 people. But, I think it is not truebecause the mathematics problem said there was nobody who answered the same number oftimes.-------------------------------------------------------------------------------------------------No.47　In detail The number of references: 4　10/19 Friday 11:58However the mathematics problem said the number of times must not overlap, Mr. A is thesides with questioner but answerer. That is, the number of greetings of Mr. A is not included inthe answers from other people. So, the answers from other people and A’s number of timescan overlap.-------------------------------------------------------------------------------------------------In the upper section of graph, a discussion of whether the answer provided in note 10 is aunique one begins from note 13. This discussion develops through notes 15, 35, 41, 60, 61and 71. Moreover, notes 66, 67 and 70 are responses to note 59 which is about a differenttopic. These graphs develop in the same way as 5 node (verification of point of problem) and4 node (primitive mathematical thinking) graphs, respectively. Hence, this implies theknowledge map is an effective tool to assist discussion through allowing the relating of notesin a meaningful way.

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5.2 Mathematical activities observed5.2.1 Verification of solutionAlthough the correct answer (No.10) was entered in the first stage of collaborative learning,discussion continued to develop indicating that it can continue after creating the correctanswer. The authors found important mathematical meta-cognitive activities which studentswere verifying the solution in this continued discussion (i.e., the threads beginning from notes23 and 13). This is important for novice learners because they have a tendency to overlookthis process.5.2.2 Discovering rulesThe authors also identified that learners were discovering mathematical rules related to thep r o b l e m , f o r e x a m p l e :-------------------------------------------------------------------------------------------------No.52　Miss G 　 The number of references: 1　10/19 Friday 12:0A husband’s number of times plus wife’s one equals 4. Miss G, is it correct?-------------------------------------------------------------------------------------------------No.72　Marvel The number of references: 0　10/19 Friday 12:11As Mr. S said, I also found that a husband’s number of greetings plus wife’s one wonderfullyequaled 4 when we read students’ answer entered. I think that the hint related to solving thisproblem by the equations is hiding in this phenomenon.-------------------------------------------------------------------------------------------------We can see here that the students discovered the rule that the sum of greetings of any husbandand wife is four. Although this discussion did not continue further due to time constraints, it isstill observable that learners not only found the answer but also attempted to develop othermathematical idea and view.5.2.3 Mathematical expressionAnother interesting mathematical activity was found in student posting. In note 10, using thecapital and lowercase notation the learner expresses the husband and wife relationship,therefore C=AaBb means that Mr. C greets Mr. and Mrs. A and Mr. and Mrs. B. Thisexpression has brevity and rationality which are important mathematical concepts.Subsequently, it was used 5 times derivatively. It appears that learners acquired this method ofmathematical expression through collaborative learning.As mentioned earlier, the beneficial mathematical activities of Verification, Abstraction andExpression were observed in this collaborative learning indicating that by conducting avariety of mathematical activities, collaboration of problem solving developed.5.3 Effectiveness of NakSunThe questionnaire (5 point scale),included the following question andmean response: “Do you think thatstudents discussed actively? 4.0”According to Figure 4, the majority oflearners appreciate “Lots of notes wereentered” or “I was satisfied with thecontent of notes” Moreover, accordingto Burtis (1997), students seemed todiscuss and solve mathematics

34%

9%24%

0%

33%

There are many notesentered

It is easy to see theknowledge map

Learners referred to thenotes many times

Learners were satisfiedwith the content of notes

OthersFigure 4: Effectiveness of NukSun

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problems positively and actively using NakSun, because linked notes were 78%, whereastraditional thread-based one was 48% in our previous research (Nagai, 2002).

６．CONCLUSIONUsing NukSun, the authors have researched Web-based collaborative learning making use ofrelationships and interaction between learners’ notes. We presented them with a single-solution problem. Consequently, although the correct answer to mathematical problems wasdiscovered early stage of collaborative learning, we found that learners continued to discussactively and effectively using a variety of mathematical ideas and views on the knowledgemap. Therefore, this result suggests that collaborative learning is encouraged by theknowledge map.ACKNOWLEDGEMENTSWe express our sincere thanks to Mr. Robert Gravina of the University of Melbourne, studentsand teachers of the Nagaura Junior High School in Japan.REFERENCESBurtis, J.: Sociocognitive Design Issues for Interactive Learning Environments Across

Diverse Knowledge Building Communities. Paper presented at the Annual Meeting of theA m e r i c a n E d u c a t i o n a l A s s o c i a t i o n , C h i c a g o , 1 9 9 7 ,http://csile.oise.utoronto.ca/CSILE_biblio.html.

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Learning through Collaborative Visualization (CoVis) Project, CoVis Homepage,http://www.covis.nwu.edu/.

Nagai, M.: A research on relational understanding in mathematics learning and its buildingprocesses, Master thesis, Chiba Universtiy, 1996.

Nagai, M., Okabe, Y., Nagata, J., Koshikawa, H., Takahashi, T., and Akahori, K.: A Design ofthe Environment Supporting Collaborative Learning on School Mathematics Using theDistribution Network, TSG6 ICME9 at Japan, 2000.

Nagai, M., Okabe, Y., Nagata, J., and Aakahori, K.: A Study on the Effectiveness of Web-based Collaborative Learning System on School Mathematics: Through a Practice of ThreeJunior High Schools, ICCE/ICCAI2000 at Taiwan, Proceedings Volume1 p.279-283, 2000.

Nagai, M., Shiraki, K., Koshikawa, H. and Akahori, K.: Mathematics Problem Solving UsingKnowledge Map on the Web, with an Analysis of the Process, Journal of Science Educationin Japan 2002 Vol.26 No.1, pp.78-90, 2002.

Ontario Institute for Studies in Education of the University of Toronto (OISE/UT), CSILEHomepage, http://csile.oise.on.ca/ .

Suthers, D.: Effects of Alternate Representations of Evidential Relations on CollaborativeLearning Discourse, Computer Supported Collaborative Learning (CSCL'99), p.12-15,1999.

Tarja-Riitta Hurme, S.: Metacognitive processes in problem solving with CSCL inmathematics, Euro-CSCL, http://www.mmi.unimaas.nl/euro-cscl/ presentations.htm, 2001.