International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
2
International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
EDITORIAL BOARD
Editor in Chief
Mack SHELLEY - Iowa State University, U.S.A.
Co-Editors
S. Ahmet KIRAY- Necmettin Erbakan University, Turkey
Ismail SAHIN - Necmettin Erbakan University, Turkey
Section Editors
Tolga ERDOGAN - Karadeniz Technical University, Turkey
Hakan AKDAG - Gaziantep University, Turkey
Ozkan AKMAN – Ministry of National Education, Turkey
Editorial Board
Ann D. THOMPSON - Iowa State University, U.S.A
Bill COBERN - Western Michigan University, U.S.A.
Douglas B. CLARK - Vanderbilt University, U.S.A.
Gokhan OZDEMIR - Nigde University, Turkey
Hakan AKCAY - Yildiz Technical University, Turkey
Huseh-Hua CHUANG - National Sun Yat-sen University, Taiwan
Ilhan VARANK - Yildiz Technical University, Turkey
James M. LAFFEY - University of Missouri, U.S.A.
Kamisah OSMAN - National University of Malaysia, Malaysia
Lynne SCHRUM - George Mason University, U.S.A.
Mary B. NAKHLEH - Purdue University, U.S.A.
Mehmet AYDENIZ - University of Tennessee, U.S.A.
Musa DIKMENLI - Necmettin Erbakan University, Turkey
Pasha ANTONENKO - University of Florida, U.S.A.
Pornrat WATTANAKASIWICH - Chiang Mai University, Thailand
Robert E. YAGER - University of Iowa, U.S.A.
Sevilay ATMACA - Cyprus International University, Cyprus
Sinan ERTEN - Hacettepe University, Turkey
Tsung-Hau JEN - National Taiwan Normal University, Taiwan
Yilmaz SAGLAM - Gaziantep University, Turkey
Technical Support
Mustafa Tevfik HEBEBCI & Yasemin AY - Necmettin Erbakan University, Turkey
International Journal of Research in Education and Science (IJRES)
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International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
TABLE OF CONTENTS
TPACK Survey Development Study for Social Sciences Teachers and Teacher Candidates
1
Ozkan Akman, Cemal Guven
The Development of Thai Learners’ Key Competencies by Project-based Learning Using ICT
11
Sasithorn Soparat, Savitree Rochanasmita Arnold, Saowadee Klaysom
An Investigation of Eighth Grade Students’ Problem Posing Skills (Turkey Sample)
23
Hasan Unal, Elif Esra Arikan
A Preservice Mathematics Teacher’s Beliefs about Teaching Mathematics with Technology
31
Shashidhar Belbase
Gender Differences in Mathematics Achievement and Retention Scores: A Case of Problem-
45
Based Learning Method
John T. Ajai, Benjamin I. Imoko
Non-mathematics Students’ Reasoning in Calculus Tasks
51
Ljerka Jukic Matic
Relieving of Misconceptions of Derivative Concept with Derive
64
Abdullah Kaplan, Mesut Ozturk, Mehmet Fatih Ocal
Implementation of Structured Inquiry Based Model Learning Toward Students’ Understanding
75
of Geometry
Kalbin Salim, Dayang Hjh Tiawa
Students’ Understanding of the Definite Integral Concept
84
Derar Serhan
Experimental Studies on Electronic Portfolios in Turkey: A Literature Review
Selahattin Alan, Ali Murat Sunbul
89
International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
TPACK Survey Development Study for Social Sciences Teachers and
Teacher Candidates
Özkan Akman1*, Cemal Güven1
Necmettin Erbakan University, Turkey
1
Abstract
The purpose of this research is to develop a scale for analyzing the technological pedagogical and content
knowledge (TPACK) and self-efficacy perceptions of the social sciences teachers and teacher candidates.
During the development process, an item pool has been generated by evaluating the studies made in the
literature. Also, after opinions have been obtained from the experts in scale development, essential revisions
have been made in accordance with these opinions. The final version of the scale has been applied to 285
teacher candidates for the validity-reliability study. As the result of the confirmatory factor analysis, 7 factors
have been determined related with the scale. The Cronbach Alpha reliability coefficient of the scale has been
found as 0.977. In conclusion, it is determined that the scale is valid and reliable for making the study.
Key words: Technological pedagogical and content knowledge (TPACK); Scale development
Introduction
It is known about the activities which are occasionally made in various countries in the world for providing the
technological integration. For instance, Portuguese has offered laptop computers and the content of all courses
in digital environment to the students with a project which they call MECELLAN. A similar project which is
called FATİH Project in Turkey has been implemented. However, countries like Portuguese and Korea have
provided various courses and seminars with TPACK model for the imperfections of the teachers and teacher
candidates in the education but this system has not been such successful in Turkey. For Turkish context, it is
clear that teachers and teacher candidates shall have TPACK skills. During this implementation, some issues
must be considered. For example, a teacher who has technological knowledge in high degree has problems in
implementing this knowledge by combining it with pedagogy and content knowledge.
It shall be known which subject needs which technology or which teaching strategy is needed to be given with
which technology in the teaching any subject (Gündüz & Odabaşı, 2004; Hew & Brush, 2007; Kurt, 2013;
Mishra & Koehler, 2006). This study is important in terms of realizing this distinction and in determination of
this problem because TPACK model shows distinction from other models which are tried in terms of the
competence of the teachers in integration process. The purpose is to bring the competences of the teacher in the
desired level as soon as possible by considering the studies which are made in this respect.
Theoretical Framework: TPACK
When considering as historical, it was thought that the basis of the teacher’s education was the content
knowledge of the teacher. Technology, pedagogy and content knowledge framework has first occurred as the
pedagogy and content knowledge which was showed by Shulman (1986). Shulman (1986) tried to explain the
education with the complex structure of the pedagogy and content knowledge. He emphasized that the most
important qualification of a good teacher should have not only the content knowledge but also the pedagogical
knowledge at the same time. Shulman (1986) expressed that the teaching knowledge should be given together
with the content knowledge because these are not the concepts which are considered separately from each other
(Angeli & Valanides, 2009; Yiğit, 2014). However, Mishra and Koehler (2006) have made this structure more
complex by adding technology into this structure with the recent development of the knowledge and technology.
Mishra and Koehler (2006) have converted the pedagogy- content knowledge model of Shulman (1986) into a
more complex structure as technology-pedagogy-content knowledge, pedagogy-technologic content- content
pedagogy- technologic pedagogy and content knowledge by adding technology.
*
Corresponding Author: Ozkan Akman, akmanozkan@hotmail.com
2
Akman & Guven
There are many scales which are developed for measuring the technologic pedagogic and content knowledge
self-efficacy of teachers and teacher candidates. These scales are divided into two sections among themselves.
First of them is the scales which are developed by researchers. Other one is the adapted forms of these
developed scales. First TPACK scale which is developed for collecting data is the scale which is developed by
Mishra et al., (2006). TPACK which was then developed by Schmith et al., (2009), was actualized with 124
teacher and teacher candidates. This study forms 7 aspects and 47 items. Another similar study is the scale
which was developed by Graham et al., (2010). Şahin (2011) has developed a TPACK scale with 7 dimensions
that involves 47 items for analyzing TPACK levels of the teacher candidates.
There are some scales which adopted the scales which have been developed by other researchers into their
studies. A scale with 31 items which was adopted by Chai et al., (2010), Landry (2009) has been adopted for
mathematics teachers and teacher candidates. A scale with 30 items was adopted for science teachers and
teacher candidates by Graham et al., (2009). A scale with 29 items was adopted for computer teachers by
Doukakis et al., (2010). A scale with 47 items was developed by Öztürk and Horzum (2011) for determining the
TPACK levels of generally all teachers.
In the present study, the TPACK scale which was developed in consideration of all these studies is more
specific and more determinant than the other scales. In literature, there is not a study which measures TPACK
levels of social science teachers and teacher candidates only. This study is important in terms of remedying the
deficiencies and guiding the practitioners in this respect. Based
The final version of the TPACK scale has been originally developed from the scale of Sahin (2011) and
involves 55 items for measuring TPACK levels of social sciences teachers and teacher candidates.
Method
Participants
285 social studies teacher candidates have participated to this study from 4 different universities in Turkey.
While 55% of the participants are women, 45% of those are man.
Data Collection Tool
The writing of the scale materials is formed with the contributions of scale development experts and by detail
scanning of the literature related with the subject. The literature studies related with the subject has been
analyzed in detail from previous years to present. The dimensions of the scales have been formed by the
complete comprehension of the theoretical framework of the subject. Accordingly, seven dimensions have been
determined in the scale which is issued by the researcher. These are dimensions of technological knowledge (T),
pedagogical knowledge (P), of content knowledge (C), of content and pedagogic knowledge (CP), of technology
and pedagogical knowledge (TP), of content and technological knowledge (CT), of technological pedagogy and
content knowledge (TPACK).
An item pool in seven subscales has been formed after examining the relevant literatures. Items have been
formed by benefiting from the expert opinion which was scale development study before formation of this pool.
These items which are formed have been given the last status for being adopted to the scale by benefiting from
Turkish language experts. The scale which is issued has been formed in the type of five points Likert scale. 1-23-4-5 numbers are located across the items of the scales. Respectively, the numbers are given the meaning as; I
do not know, I know in low level, I know in middle level, I know in good level, I know in very good level. Sixty
one items have been formed in the item pool. The developed scale is made on 285 teacher candidates. Validity
and reliability studies are made for the obtained data. According to the reliability studies made, the reliability
coefficient of Alpha Cronbach of the scale has been found as 0.977. Confirmative factor analysis is made on the
obtained data.
Survey Development Process
To increase the content validity of the scale which is developed by the researchers (see: Appendix-1), it is
benefitted from the experts who had scale development studies before. Together with the high level of the Alpha
reliability coefficients, confirmative factor analyses are made. The confirmative factor analyses are analyzed
with AMOS (Analyses of Moment Structures) 16.00 Program. The adaptive index values of factor analysis have
International Journal of Research in Education and Science (IJRES)
3
been found as mentioned in Table 1. There are some index types which are confronted regularly in literature.
These are χ2/df, CFI, RMSEA, GFI, AGFI, NFI, NNFI, SRMR indexes (Karademir & Erten, 2013).
Table 1. Criterion references for fit indices of factor analysis
χ2 / df
RMR
GFI
CFI
RMSEA
1.398
0.038
0 ≤ RMR ≤
0.05
0.853
0.918
0.95≤GFI≤1
0.97≤ CFI ≤ 1
0.050
0≤ RMSEA ≤
0.05
0.90≤GFI≤0.95
or
0.80≤GFI≤0.89
0.90≤CFI≤0.95
or
0.80≤GFI≤0.89
0.05≤ RMSEA
≤ 0.10
Perfect fit
<3
Acceptable fit
<5
0.05 ≤ RMR ≤
0.10
As mentioned in the Table 1, it can be seen that the chi square value is less than 3 and this shows the good
adaptation (Marsh & Hocevar, 1988). It can be seen that the chi- square value (1.398) has a good adaptation.
GFI value changes between 0 and 1. The closer is the value to 1; it means it is appropriate as such (Eroğlu,
2003). In that case, it shows that the GFI value (0.853) has an acceptable adaptation. CFI gives a value between
0 and 1. To become closer to 1, it shows its adaptation. 0.90 value is accepted as the most convenient value
(Eroğlu, 2003). Also, CFI value (0.918) is an acceptable value. It is expected to have RMSEA value close to 0.
The values which are equal to 0.05 or less values are accepted as the adopted values (Karademir, 2013). In this
case, it can be shown that RMSEA value (0.050) has a good adaptation. Generally those values based on the
factor analysis, are good and acceptable.
Results
In this section, the loads and dimensions of the materials which occur before and after the confirmative factor
analysis are shown.
Figure 1. Model before the factor analysis
4
Akman & Guven
As it can be understood from Figure 1, TPACK dimension and their relations before the material factor analysis
have been given. It consists of a total of 61 items. The dimensions and relations after the material factor analysis
are given in Figure 2.
Figure 2. Model after the factor analysis
As understood from Figure 2, 1-2-8-9-10 and 29th items which are not in conformity with the item adaptation
index values are removed after the factor analysis. The item number which was 61 in total has decreased to 55
with the removed items.
As understood from the figure, the values in the relation of the technology dimension with the items change
between β=0.68 and β=0.74. r is found as 0.58 between the technology and pedagogy dimension. The relation
value of the content knowledge dimension with technology has been as r=0.62. The relation level between the
technology and technology pedagogy knowledge has been found as r=0.74. The relation level between the
technology dimension and technology pedagogy and content knowledge dimension has been found as r=0.62.
The relation level of the pedagogy knowledge with materials changes between β=0.64 and β=0.72. The relation
value between pedagogy and technology has been found as r=0.58, with content knowledge r=0.92, pedagogic
content knowledge r=0.90 and technology pedagogy and content knowledge has been found as r=0.85.
The relation level of the content knowledge dimension with materials changes between β= 0.60 and β=0.84. The
relation level between content information with technological knowledge is r= 0. 62, with pedagogic knowledge
r=0.92, with pedagogic content knowledge r=0.85, with technology content r=0.74, technology pedagogy and
content knowledge is found as r=0.77.
The relation level of the technology content knowledge dimension with materials changes between β=0.73 and
β=0.80. The relation level of technology content knowledge and content knowledge is r= 0. 74, relation level
with technology is r=0.66, with pedagogy content r=0.81, between technology pedagogy r=0.92, between
technology pedagogy and content knowledge r=0.93.
The relation level of the pedagogic content knowledge with material changes between β=0.68 and β=0.75. The
relation level of pedagogy content knowledge with pedagogy is r=0.90, with content knowledge r=0.85, with
technology content knowledge r=0.81, technology pedagogy r=0.74, technology pedagogy and content
knowledge r=0.84. The relation level of technology pedagogy knowledge dimension changes between β=0.67
and β=0.79. The relation level between technology pedagogy technology content knowledge is r=0.74, with
International Journal of Research in Education and Science (IJRES)
5
pedagogy r=0.57, with technology content r=0.92, with pedagogy content r=0.74, between technology pedagogy
and content knowledge r=0.91. The relation level of technology pedagogy and content knowledge dimension
with materials changes between β=0.73 and β=0.80. The relation level between technology pedagogy and
content knowledge technology is r=0.62, pedagogy r=0.85, with content knowledge r=0.77, with technology
content r=0.93, with pedagogy content r=0.84, between technology pedagogy r=0.91.
Discussion and Conclusion
The purpose of this study is to develop scale for understanding the technologic pedagogic and content
knowledge self-efficacy perception level of the social studies teachers and teacher candidates. When the body of
literature is examined, it can be shown many scale development and adaptation studies which are made for
examining the TPACK level of the teachers and teacher candidates. However a part of these studies is
developed for measuring the level of science, mathematics and computer teachers and teacher candidates and a
part of these studies are developed for all the teachers and teacher candidates. This study is developed as more
specific for understanding TPACK level of social studies teacher and teacher candidates with a different point of
view. The scale has been regulated in seven dimensions as in the other studies (Chai et al., 2010; Landry, 2009;
Mishra & Koehler, 2006; Öztürk & Horzum, 2011; Schmith et al., 2009; Şahin, 2011).
The factor loads at the result of the factor analysis are generally between 0.57 and 0.93.These values are
accepted as good levels for scale (Green & Salkind, 2005). Similar results are seen is similar studies (Lux,
2011; Öztürk & Horzum, 2011; Schmith et al., 2009; Şahin, 2010). In the result of the analysis, the adaptation
index values are found as an acceptable value (Byrne, 1998).
For the consistency in the reliability of the scale, Cronbach Alpha internal consistency coefficients are
considered. For the integrity of the scale, Cronbach Alpha value has been found as 0.977. This shows that the
scale has the highest reliability (Büyüköztürk et al., 2010). The reliability values of the scale formed of seven
factors are as follows; 0.887 related with the technology knowledge; 0.916 related with pedagogy knowledge;
0.934 related with content knowledge; 0.925 related with pedagogy content knowledge; 0.846 technology
content knowledge; 0.866 technology pedagogy knowledge and 0.965 technology pedagogy content knowledge.
When compared with the similar studies, it is seen that the values here have higher reliability (Chai et al., 2010;
Graham et al., 2009; Graham et al., 2010; Mishra et al., 2004; Öztürk & Horzum, 2011; Schmith et al., 2009;
Şahin, 2011).
This study is a unique scale development study. As the result of this study, a scale with high validity and
reliability scores is developed. The obtained scale provides opportunity to us for evaluating and understanding
the self- efficacy perceptions of the social studies teachers and teacher candidates regarding their technological,
pedagogical and content knowledge. This scale is especially applicable to measure the TPACK levels of the
teachers and teacher candidates from subject social sciences area. For this reason, it may be adapted into other
subject areas.
Acknowledgements
This paper is resulted from the doctoral study of the first author.
References
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development, and assessment of ICT–TPCK: Advances in technological pedagogical content
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Doukakis, S., Psaltidou, A., Stavraki, A., Adamopoulos, N., Tsiotakis, P., & Stergou, S. (2010). Measuring the
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International Journal of Research in Education and Science (IJRES)
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Appendix 1: The English Version of TPACK Scale
Dear Colleague;
This survey is issued for examining the relation between technology, pedagogy and content knowledge of the
social studies teacher candidates. Your answers in the questionnaire shall be used for research and your
identification and answers shall be definitely kept secret. For this reason, do not hesitate to answer intimately.
The numbers at the right side of the page express these: (1) I do not know, (2) I know in low level, (3) I know in
middle level, (4) I know in good level, (5) I know in very good level
ITEMS OF TPACK SURVEY
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Using Office programs (Like Word, Excel, and PowerPoint) …
Communicating through Internet (E-mail, Skype) …
Using data ( saving to Flash Memory, CD, DVD) …
Using printer, digital camera and Scanner …
Using the programs of concept maps, drawing graphics (Inspiration, Excel etc.) …
Developing daily, annual and unit plan…
Developing classic (multiple choice test, True-False Test, open ended Question
etc) and complementary (Control List, Valuation Scale, Gradational Grading
Key, Self-Efficacy Form, Peer Assessment Form etc.) measurement tools …
Evaluating the performance of the teacher with classic and alternative
(complementary) measuring tools. …
Implementing the different teaching strategies (Presentation Strategy, Invention
Strategy, Research-Analyzing Strategy etc.) …
Implementing different methods (Plain Expression, Case Study, Problem Based
Learning, Project based Learning etc.) …
Implementing different teaching techniques (Brain Storming, Six Thinking Hats,
Demonstration, Metaphor etc.) …
Learning theory and hypothesis (Constructivist Learning, Multiple Intelligence
Theory, Project Based Education etc.)…
How the class management shall be organized and continued in Social Sciences
course …
Content Knowledge related with Individual and Society learning domain…
Content Knowledge related with Culture and Heritage learning domain…
Content Knowledge related with Humans, Places and Environment learning
domain…
Content Knowledge related with Production, Distribution, Consumption learning
domain…
Content Knowledge related with Time, Consistency and Alteration learning
domain…
Content Knowledge related with Science, Technology and Society learning
domain…
Content Knowledge related with Groups, Institutions and Social Organizations
learning domain…
Content Knowledge related with Power, Management and Society learning
domain
Content Knowledge related with Global Connections learning domain
Current releases in Social Sciences field (Releases and books)…
Selecting teaching strategies which are convenient to achievements related with
Social Studies …
Selecting education models which are convenient to achievements related with
Social Studies …
Selecting education techniques which are convenient for teaching achievements
related with Social Studies …
Selecting education methods which are convenient for teaching achievements
related with Social Studies ……
Selecting alternative /complementary and evaluation tools for evaluating
achievements related with Social Studies. …
Preparing daily, annual and unit plan which is convenient to achievements
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
8
Akman & Guven
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
related with Social Studies courses …
Preparing a course plan including the class/ intramural activities for Social Studies
courses …
Technologies which are convenient to teaching approaches/ strategies …
Providing class management while using different education technologies. …
Using technologies which are convenient to different education model and
theories …
Using technologies which are convenient to different education strategies….
Using technologies convenient to different education methods …
Using technologies convenient to different education techniques …
Using technologies which shall affect the education in positive manner …
Using technologies which are convenient to classic-alternative measurement and
evaluation approaches …
Benefiting from technology by considering the individual differences of the
students …
Preparing daily, annual and unit annual plans in computer …
Evaluating the conformance of a new technology to the education …
Education technologies which are convenient to different learning content of the
social studies courses …
Selecting technologies which are convenient for enriching the content of social
studies course …
Using technologies which are developed by Course Tools Construction Centre
while teaching achievements of Social Studies course …
Technologies which shall provide easier access to the targets/ achievements
mentioned in the social studies course teaching plan …
Using computer aided technologies which are convenient to different learning
content of social studies course …
Using tablet computer and smart board while teaching the different learning
content of social studies courses
Developing projects and class activities including the education technologies in
social studies course …
Integrating the social studies course content with appropriate technology and
formation information …
Selecting appropriate education approaches and contemporary education
technologies which shall provide better teaching of social studies course content
…
Teaching courses by integrating the social studies learning content with my
formation and technological knowledge …
To take the leading to my colleagues about integrating the social studies contend
and formation and technological knowledge …
Teaching a social studies subject by using appropriate technologies according to
different education theories …
To increase the value of the learning of my students through my formation and
technological knowledge while teaching social studies subjects …
To integrate my content, technology and formation knowledge related with social
studies course …
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International Journal of Research in Education and Science (IJRES)
9
Appendix 2: The Turkish Version of TPACK Scale (Original Form)
Sevgili Öğretmen Adayı;
Bu anket Sosyal Bilgiler Öğretmen Adaylarının teknoloji, pedagoji ve alan bilgileri arasındaki ilişkiyi
araştırmak için düzenlenmiştir. Ankette vereceğiniz cevaplar araştırma amaçlı kullanılacak olup kimliğiniz ve
cevaplarınız kesinlikle gizli tutulacaktır. Bu nedenle içtenlikle cevaplamaktan çekinmeyiniz. Sayfanın sağ
tarafındaki rakamlar şunları ifade etmektedir: (1) Hiç Bilmem, (2) Az Düzeyde Bilirim, (3) Orta Düzeyde Bilirim,
(4) İyi Düzeyde Bilirim, (5) Çok İyi Düzeyde Bilirim
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TEKNOLOJİ, PEDAGOJİ VE ALAN BİLGİSİ ÖLÇEĞİ
Ofis programlarını (Word, Excel ve Powerpoint gibi) kullanmayı…
İnternet yoluyla (E-mail, Skype) iletişim kurmayı…
Veri kaydetmeyi (Flash Bellek, CD, DVD’ye kaydetmek gibi) …
Yazıcı, Dijital kamera ve Tarayıcı kullanmayı…
Kavram haritası, grafik çizme (Inspration, Excel vb.) programlarını kullanmayı…
Günlük, yıllık ve ünitelendirilmiş plan geliştirmeyi…
Klasik (Çoktan Seçmeli Test, Doğru-Yanlış Testi, Açık Uçlu Soru vb.) ve tamamlayıcı
(Kontrol Listesi, Dereceleme Ölçeği, Dereceli Puanlama Anahtarı, Öz değerlendirme
Formu, Akran Değerlendirme Formu vb.) ölçme araçlarını geliştirmeyi…
Öğrenci performansını klasik ve alternatif (tamamlayıcı) ölçme araçları ile
değerlendirmeyi…
Farklı öğretme stratejilerini (Sunuş Stratejisi, Buluş Stratejisi, Araştırma-İnceleme
Stratejisi vb.) uygulamayı…
Farklı öğretim yöntemlerini (Düz Anlatım, Örnek Olay, Problem Dayalı Öğrenme, Proje
Tabanlı Öğrenme vb.) uygulamayı…
Farklı öğretim tekniklerini (Beyin fırtınası, Altı Şapkalı Düşünme, Gösteri, Metafor vb.)
uygulamayı…
Öğrenme teori ve kuramlarını (Yapısalcı Öğrenme, Çoklu Zekâ Teorisi, Proje-tabanlı
Öğretim vb.)…
Sosyal Bilgiler dersinde sınıf yönetiminin nasıl organize edeceğini ve sürdürüleceğini…
Birey ve Toplum öğrenme alanıyla ilgili alan bilgisini…
Kültür ve Miras öğrenme alanıyla ilgili alan bilgisini…
İnsanlar, Yerler ve Çevreler öğrenme alanıyla ilgili alan bilgisini…
Üretim, Dağıtım ve Tüketim öğrenme alanıyla ilgili alan bilgisini…
Zaman, Süreklilik ve Değişim öğrenme alanıyla ilgili alan bilgisini…
Bilim, Teknoloji ve Toplum öğrenme alanıyla ilgili alan bilgisini…
Gruplar, Kurumlar ve Sosyal Örgütler öğrenme alanıyla ilgili alan bilgisini…
Güç, Yönetim ve Toplum öğrenme alanıyla ilgili alan bilgisini…
Küresel Bağlantılar öğrenme alanıyla ilgili alan bilgisini…
Sosyal bilgiler alanında çıkan güncel kaynakları (yayın ve kitapları)…
Sosyal Bilgiler dersine ait kazanımlar için uygun öğretme stratejilerini seçmeyi…
Sosyal Bilgiler dersine ait kazanımları öğretmek için uygun öğretim modelleri seçmeyi…
Sosyal Bilgiler dersine ait kazanımları öğretmek için uygun öğretim teknikleri seçmeyi…
Sosyal Bilgiler dersine ait kazanımları öğretmek için uygun öğretim yöntemleri seçmeyi…
Sosyal Bilgiler dersine ait kazanımları değerlendirmek için klasik ve
alternatif/tamamlayıcı ölçme ve değerlendirme araçlarını seçmeyi…
Sosyal Bilgiler dersi kazanımlarına uygun günlük, yıllık ve ünitelendirilmiş yıllık plan
hazırlamayı…
Sosyal Bilgiler dersi için sınıf/okul içi etkinlikleri içeren bir ders planını rahatlıkla
hazırlayabilmeyi…
Öğretme yaklaşımlarına/stratejilerine uygun teknolojileri…
Farklı öğretim teknolojileri kullanırken sınıf yönetimini sağlamayı…
Farklı öğrenme model ve kuramlarına uygun teknolojileri kullanmayı…
Farklı öğretim stratejilerine uygun teknolojileri kullanmayı….
Farklı öğretim yöntemlerine uygun teknolojileri kullanmayı…
Farklı öğretim tekniklerine uygun teknoloji kullanmayı…
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Akman & Guven
Öğrenmeyi olumlu yönde etkileyecek teknolojileri kullanmayı…
Klasik - alternatif ölçme ve değerlendirme yaklaşımlarına uygun teknolojileri
kullanmayı…
Öğrencilerin bireysel farklılıklarını dikkate alarak teknolojiden faydalanmayı…
Bilgisayar ortamında günlük, yıllık ve ünitelendirilmiş yıllık plan hazırlamayı…
Yeni bir teknolojinin öğretime uygunluğunu değerlendirmeyi…
Sosyal Bilgiler dersinin farklı öğrenme alanlarına uygun öğretim teknolojilerini…
Sosyal Bilgiler dersinin içeriğini zenginleştirecek uygun teknolojileri seçmeyi…
Sosyal Bilgiler dersinin kazanımlarını öğretirken Ders Aletleri Yapım Merkezi tarafından
geliştirilen teknolojileri kullanmayı…
Sosyal Bilgiler dersi öğretim planındaki belirtilen hedef/kazanımlara daha kolay ulaşmayı
sağlayacak teknolojileri…
Sosyal Bilgiler dersinin farklı öğrenme alanlarına uygun bilgisayar destekli teknolojileri
kullanmayı…
Sosyal Bilgiler dersinin farklı öğrenme alanlarını öğretirken tablet bilgisayar ve akıllı tahta
kullanmayı…
Sosyal Bilgiler dersinde öğretim teknolojileri içeren sınıf etkinlik ve projeleri
geliştirmeyi…
Sosyal Bilgiler ders içeriğini, uygun teknoloji ve formasyon bilgisi ile bütünleştirmeyi…
Sosyal Bilgiler dersi içeriğini daha iyi öğretmemi sağlayacak çağdaş öğretim
teknolojilerini ve uygun öğretim yaklaşımlarını seçmeyi…
Sosyal Bilgiler öğrenme alanlarını, formasyon ve teknoloji bilgim ile bütünleştirerek ders
öğretmeyi…
Meslektaşlarıma Sosyal Bilgiler alanı ile formasyon ve teknoloji bilgisinin
bütünleştirilmesi konusunda liderlik yapabilmeyi…
Farklı öğrenme kuramlarına göre uygun teknolojiler kullanarak bir Sosyal Bilgiler
konusunu öğretmeyi…
Sosyal Bilgiler konularını öğretirken formasyon ve teknoloji bilgim sayesinde
öğrencilerimin öğrenmelerinin değerini artırmayı…
Sosyal Bilgiler dersine ait içerik, teknoloji ve formasyon bilgimi başarılı şekilde
birleştirmeyi…
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International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
The Development of Thai Learners’ Key Competencies by Project-based
Learning Using ICT
1
Sasithorn Soparat 1*, Savitree Rochanasmita Arnold 1, Saowadee Klaysom 2
Phranakhon Rajabhat University, Thailand, 2National Electronics and Computer Technology Center
(NECTEC), Thailand
Abstract
This research aimed to study the use of Project-based Learning using ICT (PBL using ICT) to develop learners’
five key competencies based on Thai Basic Education Curriculum 2008, which consists of 1) communication
capability 2) thinking capability 3) problem solving capability 4) capability in applying life skills and 5)
capability in technological application. The subjects were 212 students from 4 schools in Kanchanaburi, Chiang
Mai, Mae Hong Son, and Si-Saked provinces that were enrolled in the 2010 academic year. The PBL using ICT
approach was used as the research tool. The data collecting instruments were lesson plans, students’ journals,
semi-structured interviews, and social networking where students’ learning artifacts and products were collected
(nectecpbl.ning.com). Content analysis and triangulation methods were used to analyze the data. The results
showed that students were able to perform five key competencies in communication capability, thinking
capability, problem solving capability, capability in applying life skills, and capability in technological
application. Moreover, the students learned the content in the learning areas of Science, Mathematics, Foreign
Languages, Health and Physical Education, and Occupations and Technology. The research findings revealed
that the use of PBL using ICT can help to develop students’ abilities to communicate ideas, problem solving,
life skills, and abilities to use technology, as well as their learning in content of the subject area.
Key words: Project-based learning; Project-based learning using ICT; Learners’ key competencies
Introduction
There is a need to develop learners’ competencies for Thais in order to support the change of global competition
and workplace of technology in the digital world. The learners’ key competencies are analyzed to identify habits
of the learners to become effective as 21st Century Citizens by the Ministry of Education (MoE) of Thailand.
There are five key competencies: 1) communication capability 2) thinking capability 3) problem solving
capability 4) capability in applying life skills and 5) capability in technological application in the Basic
Education Curriculum B.E. 2551 (A.D. 2008) of Thailand. These competencies are the results of the research
findings and the statement on the Tenth National Economic and Social Development (A.D. 2007-2011) which
focuses on learner development for the future (MoE, 2008).To support the learners’ development of key
competencies, learning tools and teaching pedagogy are needed. The Project-based Learning (PBL) using ICT is
an alternative way to support the learners in the digital age which everybody can learn and share anywhere and
anytime.
The Project-based Learning (PBL) is an approach that emphasizes meaningful learning activities that are longterm, interdisciplinary, and student-centered. The learning activities are designed to provide students with real
world relevance, complex tasks, and creative outcomes. The students have chances to learn course content,
master course objectives, choose their own topics, activities, or learning tools (Grant, 2002). In the words of
Papert (2011), “creating is learning” which means the way to use knowledge is the way to get more knowledge.
This can happen by creating projects such as movies, robots, inventions, multimedia, digital storytelling, etc. in
the classroom environment.
According to Redecker (2008), ICT can enhance learning outcome by supporting different senses; supporting
collaborations with new online production, commuting and networking tools; supporting differentiation and
diversity; and empowering learners to personalize their learning process. Therefore, the PBL using ICT should
be introduced to the classroom in order to support students to create their own knowledge. In this case, using
ICT as content based learning is not enough for supporting the learners’ development of key competencies. In
*
Corresponding Author: Sasithorn SOPARAT, sasisopa1979@gmail.com
12
Soparat, Arnold, & Klaysom
the age of global network, computer and internet access should be used more than data searching, emailing, or
chatting. The way to use computers for supporting students’ learning should be focused on creating intelligence
on content knowledge and competencies.
The report of Empower ICT project (Vrasidas, Laohajaratsang & Suwannatachote, 2005) has found that most of
the teachers lacked skills in using ICT. Consequently, they could not design lesson plans by integrating ICT as
learning tools. Therefore, there is a need to instruct teachers to be able to use ICT, and be able to integrate ICT
as learning tools in their classroom. The project of Her Royal Highness Princess Maha Chakri Sirindhorn: Royal
IT Projects (2010) is the project that has developed teachers’ abilities to use ICT in their classroom during 2008
– 2013.The researchers worked with the teachers who have participated in the project of Her Royal Highness
Princess Maha Chakri Sirindhorn and learners in their classrooms. This research aimed to study learners’
development of the five key competencies during the 2010 academic year.
Theoretical Framework
The Development of the Learners’ Key Competencies through PBL using ICT
Project-based learning (PBL) is an instructional method which does not focus on rigid lesson plans. PBL allows
learners to investigate more about the topic through constructing the meaningful artifact (Harris & Katz, 2001).
While creating artifacts such as plays, poems, pie charts, or toothpick bridges, they engage in learning by
sharing and reflecting on their artifacts with others (Grant, 2002).
PBL is based on constructivism and constructionism. Constructivism explains that learners can construct their
own knowledge through interactions with their environment. The learners are different and each individual
learner can construct new knowledge by building from the current knowledge (Piaget, 1969; Vygotsky, 1978;
Perkins, 1991). Constructionism explains that learners learn when they construct the artifact or project (Harel &
Papert, 1991; Kafai & Resnick, 1996). To create a project, the learners have to start with thinking about
designing their work processes. This includes how the artifacts will look like, what technology, media, and
materials should be used. These will communicate to others how to understand their projects. Therefore, today,
communication is needed more than reading and writing. The more learners have experience to create things
through intermediate media, the more they will understand and insight the important of media analysis from all
around the world (Richards, 2005).
While creating projects, there were many problems. When trying to solve one problem, new problems always
occurred. The learners had to pay attention, think, and work on the problems, use reasoning and systematic
thinking to support their answers in the context of their work. After the students experienced problems and
problem solving, they created more meaningful projects. The learners got some creative answers which derived
from unexpected problems. Therefore, it can be implied that problem-based learning is support creative thinking
which is important for the rapid changes of the world. Moreover, the problem is the important factor for
building interaction between learners and surroundings. This interaction will support the cooperative working,
sharing, and respecting to the ability of others. The learners are not only learning to solve problems, but they are
also prepared to face their new challenge problems. This is a kind of inner motivation that they try to overcome
challenges and frustration facing from creating projects (Richards, 2005; edUTOPIA, 2011).
The project-based learning is an effective way for learners to develop and integrate skills during their small
group cooperation, because they are learning while they are using their knowledge through their work. The more
they use and share in group, the more inspiration for them to know deeper on what they are studying. The
assessment which is designed to measure on the project will give meaningful feedback to the learners for
connecting what they learned and real life (Bautista & Escofet, 2009; Richards, 2005; edUTOPIA, 2011).
The advancement of technology in the present can support the web learning tools for classroom. These learning
tools can be accessed by everybody. Learners can use and share in their real lives. Now a days, the web learning
tools are not distinguished from education anymore. Learners should have chance to learn on the age which is
too different from adult age in the past (Jonassen, Howland, Moore & Marra, 2003; Richards, 2005). Focusing
on the understanding of Project-based learning, there is still different understanding among teachers. Many
teachers are familiar with designing a science project. To do a science project, a student must begin with
questions and processed through the scientific method for seeking the answers which might or might not be
artifacts or products at the end of solution. However, the project-based learning is focused on the “products” as
the aim (Tunhikorn, 2008).
International Journal of Research in Education and Science (IJRES)
13
The Framework for 21st Century Skills
According to the Framework of Education Model in the 21 st century (The Partnership for 21st Century Skills,
2007), there is a different idea of the education model between 20 th and 21st Century. The education model in the
20th century focused on the learning of the core subjects and assessing for learning outcome. While the
education model in the 21st century focuses on learning among the core subjects, 21 st century content, life skills,
learning and thinking skills, and ICT literacy which are known as 21 st Century Skills (The Partnership for 21st
Century Skills , 2007). This idea also covered the focus points of the Basic Education Curriculum B.E. 2551
(2008) of Thailand about content learning standards and the learners’ key competencies. These related points are
shown in Table 1.
Table 1. The relationship between the learners’ key competencies of Thailand in the Basic Education
Curriculum B.E. 2551 (A.D. 2008) and the framework of the education model in the 21st century (The
Partnership for 21st Century Skills, 2007)
The Basic Education Curriculum
The Framework of Education Model in the 21 st Century
B.E. 2551 (A.D. 2008) of Thailand
Outcomes/Skills
Learning Standards:
Core subjects:
Thai Language/ Mathematics/
English, Reading or Language
Science/ Social Studies, Religion
Arts/ World languages/ Arts/
and Culture/ Health and Physical
Mathematics/ Economics/
Education/ Arts/ Occupations and
Science/ Geography/ History/
Technology/ Foreign Languages
Government and Civics
Sub-skills
21st century themes:
Global Awareness/ Finances,
Economic, Business and
Entrepreneurial Literacy/ Civic
Literacy/ Health Literacy/
Environment Literacy
Communication Capacity
Thinking Capacity
Problem-Solving Capacity
Learning and Innovation Skills
Communication and Collaboration
Creativity and Innovation
Critical Thinking and Problem
Solving
Capacity for Technological
Application
Information, Media, and
Technology Skills
Information Literacy
Media Literacy
ICT Literacy
Capacity for Applying Life Skills
Life and Career Skills
Flexibility and Adaptability
Initiative and Self-Direction
Social and Cross-cultural Skills
Productivity and Accountability
Leadership and Responsibility
Many researches explained that learning in the 21st century needs higher order thinking skills related to
sophisticated cognition such as inquiry process, formulating explanations, and communicating understanding.
These skills are difficult to measure with multiple choice or paper-and-pencil tests (Resnick & Resnick, 1992;
Quellmalz & Haertel, 2004; National Research Council, 2006). In order to support Thai new generation citizen
to have the learners’ key competencies which concordant with the skills in 21st century is needed the effective
learning management to support both of learning outcomes and skills. Also the alternative assessment is needed
for both summative and formative assessment to support the new way of learning.
14
Soparat, Arnold, & Klaysom
Method
Subject of the Study
The subjects of this research were 212 students in the classes of eight teachers from four schools. These schools
were located in Karnchanaburi, Cheing-Mai, Mae-Hongsorn, and Sri-Saked provinces. The PBL was introduced
to the classroom by integrating with the lessons taught by eight teachers. These teachers had worked
collaboratively with the researchers to design their own lesson plans by integrating PBL using the ICT
approach.
Research Procedure
This research was based on qualitative research using participatory action research between the researchers and
teachers working cooperatively. The participatory action research attempts to present people as researchers
themselves to the questions of their daily struggle and survival (Tandon, 1998). In this research, teachers were
teamed with the researchers. They understood the research objectives and created the best practice for PBL
using ICT in real classrooms. Moreover, they acted as researchers in collecting data of the learners’ five key
competencies from their students.
The research procedures were conducted in 3 steps: instruction, cooperative working, and follow up. The details
are shown below:
1. Instruction
In this procedure, the 19 teachers had participated in 3 workshops of the project of Her Royal Highness Princess
Maha Chakri Sirindhorn: Royal IT Projects during 2010-2011(the 2010 academic year).The aim of the
workshops was to prepare the teachers to design lesson plans by using PBL using ICT approach.
2. Cooperative working
The teachers and researchers were joined in teams to design the PBL using ICT lesson plans. We connected
through social networking through the website http://nectecpbl.ning.com. Teachers shared their lesson plans for
research teams to critique. They posted and shared questions and interesting topics about PBL and ICT for
others to reflect on. Teachers also wrote their weekly journals on the website. The research team was working
collaboratively to provide feedback on teachers’ lesson plans and data collection tools.
3. Follow up
At the end of the first semester of the 2010 academic year, the researchers followed up on the teachers who were
able to design the PBL using ICT lesson plans for the second semester of 2010 academic year. The researchers
observed each teacher’s classrooms twice, and did group reflection after each class. The learners’ five key
competencies from learning through PBL using ICT were collected during the class and also through the social
network.
Research Tools
The PBL Approach
The PBL approach was used as a tool to determine the development of learners’ five key competencies. The
PBL approach was designed to focus on the roles of learners and teachers in the classroom.
The learners’ roles were to create the artifacts or products in the projects within the social context. The students
in each group mainly rotated their own acts of planning, creating, reflecting, and publishing within the cycle and
could be related to each step as shown in Figure 1.
1) Planning (P); the learners had to think collaboratively to set their group goals, plan and design the
tasks to meet their goals.
2) Creating (C); the learners followed the plan by using appropriate media and technology. When
learners encountered the problems in this step and tried to solve the problems, this led them to
make a new plan.
3) Reflecting (R); this was a very important step that learners used for talking about and critiquing
their own or others’ tasks for improvement.
4) Publishing (P); learners used this step to present their ideas, artifacts, or products.
International Journal of Research in Education and Science (IJRES)
15
Figure 1. The learners’ roles of learning through doing projects within the social context
The teacher’s role included knowing, facilitating, context providing, and assessing as shown in Figure 2.
1) Knowing (K); teachers became informants for the learners. Therefore he/she should have
knowledge not only to provide, but also help learners construct their own knowledge.
2) Facilitating (F); teacher became facilitators and supporters for learners’ needs.
3) Context Providing (C); teachers provided appropriate classroom settings which contained learning
resources, learning tools and media, and social context for learners to work collaboratively.
4) Assessing (A); teachers prepared the assessment tools which reflected for the performance
assessments and followed up on learners’ progress.
Figure 2. The teachers’ roles in PBL
The Data Collecting Tools
Data collecting tools consisted of four tools shown in table 2. These tools were 1) the teachers’ lesson plans, 2)
the learners’ journals, 3) the learner interviews, and 4) the website nectecpbl.ning.com
16
Soparat, Arnold, & Klaysom
Research tools
1) Lesson plans
2) Learners’ Journals
3) Learners’ Interviews
4) Social network
(http://nectecpbl.ning.com)
Table 2. Tools for data collection
Informants
Objectives
Teachers
To study teachers’ and learners’ roles in PBL and level of
integration ICT as the learning tools
Learners
To study the results of PBL using ICT in the classroom (at the
end of every classes)
Learners
To study the results of PBL using ICT in the classroom (at the
end of course)
Learners
To study the learners’ competencies from learning artifacts or
products
The lesson plans that were designed by teachers aimed to study teachers’ and learners’ roles in PBL and level of
integration ICT as the learning tools in the classroom. The data from three other tools was collected from the
learners to provide the learners’ five key competencies resulted from PBL using ICT in the classrooms.
Data Collection and Analysis
The data collection was processed during the 2010 academic year through the four research tools. After
gathering the data, the researchers used content analysis method to find out the learners’ five keys competencies
from learning through PBL using ICT. The data triangulation method was used to confirm the validity of data
analysis by analyzing data among learners within different abilities. The learners’ abilities were justified from
their previous learning outcome; students that scores 80-100 percent were labeled as high ability learners,
students that scores 60-79 percent were labeled as medium ability learners, and students that scores under 60
percent were labeled as lower ability learners.
Results and Discussion
The research was conducted through the participatory action research among the researchers and teachers
working cooperatively. Teachers designed lesson plans and implemented in their classrooms, while the
researchers were facilitators in designing lesson plans and following up. The results of working collaboratively
showed that 8 out of 19 teachers who could design and implement the PBL using ICT in their classrooms within
one year. During implementation in classrooms, the data of learners’ competencies were collected from
students’ journals, interviews, and social networking. The results of using the PBL using ICT presented the
impact on developing both of five key competencies and content learning outcomes. According to teachers’
designed lesson plans, it showed that teachers could integrate PBL using ICT in their lesson plans as shown in
Table 3.
Artifacts/Products
Digital Report
Digial Storytelling
Mind Mapping
Electronic Crafts
- Cards
- Poster
- Slide Show
VDO Role Playing
Website
Table 3. Learners’ artifacts/products
ICT tools
Subject
Microsoft PowerPoint
- Science in grades 9, 11
- Health Education in grade 9
- Occupations and Technology in grade 4
Microsoft Powerpoint
- Math in grade 7
- Science in grade 9
EDraw Mindmap
- Science in grades 11, 12
Glogster
- Science in grades 11, 12
PhotoPeach
- English in grades 11, 12
Tattoons
- Occupations and Technology in grade 4
FlipAlbum
Paintbrush
Windows Movie Maker
- Science in grade 12
Ulead
- English in grade 11
YouTube
Google Sites
- Science in grades 11, 12
- English in grade 12
International Journal of Research in Education and Science (IJRES)
17
The results showed that the learners developed five key competencies which included: 1) Communication
Capacity, 2) Thinking Capacity, 3) Problem-Solving Capacity, 4) Capacity for Applying Life Skills, and 5)
Capacity for Technological Application. The details of their development showed below:
Communication Capacity
The learners developed their communication capacities during the whole process of working; planning, creating,
reflecting, and publishing their artifacts. The results of the study showed that the learners developed
communication capacity in three ways;
1) Perceiving and analyzing the information beyond the classroom
2) Choosing the information and media to fit the tasks
3) Using a variety of ways of communication
The learners learned how to choose, create, manage, and use different types of media such as text, picture,
sound, and movies to create their tasks. In order to communicate their tasks to others, they needed more than
reading and writing skills. Learners had to analyze their final products to determine whether it fit the purpose of
their tasks. It showed that the learners not only created the tasks, but also communicated their tasks
interestingly. From this process learners experienced perceiving and analyzing the information.
Thinking Capacity
The learners developed thinking capacity during the whole process of working; planning, creating, reflecting,
and publishing their tasks. The results of the study showed that the learners developed the higher order thinking
of:
1) Creative thinking
2) Analytical thinking
3) Reasoning and systematic thinking
4) Synthesis thinking
5) Critical and reflective thinking
From learning through PBL using ICT, learners could create creative artifacts. Moreover, the evidence of
learners’ learning showed that doing projects helped them pay more attention to provide reasons and use
systematic thinking. In order to complete the tasks, they had to interact with teachers, peers, and everything
around them. In these situations, it allowed them to think analytically to distinguish all problems and factors
impacts to their work success, to plan and choose media such as texts, pictures, sounds, and movies involving in
content. Moreover, they thought critically and reflectively on the weak points to improve their tasks. When they
tried to explain their tasks, they used synthesis thinking. It helped them connect all working processes to set
their own working system. These direct experiences on designing and creating helped the learners develop a
variety of ways of thinking.
Problem-Solving Capacity
The process of working; planning, creating, reflecting, and publishing their tasks supported the learners to
discover and solve problems from their own context. The results of the study showed that the learners developed
the problem-solving capacity in three ways:
1) Solving problems in a reasonable way
2) Inquiry for solving the problems
3) Effective Decision Making
The learners used analytical and critical thinking to solve many problems while doing projects. Each learner had
a different way of doing projects. Some were well organized and were good planners, and some were not.
Whatever type of planer, they were all involved in solving problems.
Capacity for Applying Life Skills
During the process of working on the project, difficulties and problems were challenges for learners. In this
situation, the learners collaborated and supported each other. They not only shared idea to others, but also
18
Soparat, Arnold, & Klaysom
shared their own potential to make things successful. The results of the study showed that the learners developed
the capacity for applying life skills in five ways:
1) Collaborative Learning
2) Creating potential to overcome difficulties
3) Learning to set the goal for success
4) Creating self-confidence
5) Lifelong learners
These were the intrinsic motivations of learners developed to overcome challenges and disappointment when
they faced difficulties.
Capacity for Technological Application
The learners developed the capacity for technology during the whole process of working; planning, creating,
reflecting, and publishing their tasks. The results of the study showed that the learners developed in three ways:
1) Choose and use technologies appropriately to the task
2) Choose and use technologies as the learning tools
3) Choose and use technologies in a moral way
The learners chose and used many kinds of technology to search for information, communication, design, create
and present their own tasks or projects. Moreover, they used social networking to publish and share their ideas
and works by concerning morality such as using polite words, being aware of and respectful for copy rights, and
thinking critically about reliable information.
Learning Outcomes
The result of the study also showed that learning through the PBL using ICT enhanced the learners to
understand on the contents. Moreover, many learners thought that they enjoyed doing project and it helped them
learned in deeper content. The evidence is shown below.
“I learned on the content about the definition of energy. I can also divide energy into 2 types. The first
is ran out energy and the second is the renewable energy. I could give examples of energy for each type
of those. I knew more about the advantage and disadvantage which it was valuable to use in my
everyday life.”(Learner’s journal writing)
Figure 3. A Learner’s Website about energy in daily life using Google site from Ms. Pan’s classroom in science
grade 11
“This house is designed to use the roof in the dark blue color to prevent heat. The wall is white cream
color, because it’s not an endothermic wall. There are some windows to help air flow…” (Learner’s
journal writing)
International Journal of Research in Education and Science (IJRES)
19
Figure 4. The learner’s artifact “The Energy Saving Home” designed on PowerPoint from Ms. Cha’s classroom
in science grade 9
The researchers found that the PBL using ICT enhanced the learners’ five key competencies and learning
outcomes. The results of the study revealed that the learners were developed in key competencies:
1) Communication Capacity: The learners compiled the substance of the speech, writing, analytical
sharing knowledge and ideas and communicated to others. They had abilities in perceiving and
analyzing information from all around the world, choosing the media fit to the tasks, and creating a
variety of ways of communication.
2) Thinking Capacity: The learners had abilities in analytical thinking, reasoning and systematic thinking,
synthesis thinking, thinking critically and thinking reflectively.
3) Problem-Solving Capacity: The learners had abilities to identify problems and find reasonable
solutions, inquiry for solving the problems, and effective decision making.
4) Capacity for Applying Life Skills: The learners were able to work well with others, and also had social
harmony through the strengthening of good interpersonal relationships. They had abilities in
collaborative learning, creating potential to overcome difficulties, learning to set the goals for success,
creating self-confidence, and lifelong learning.
5) Capacity for Technological Application: The learners chose and used appropriate technologies for the
task, chose and used technologies as learning tools, and chose and used technologies in a moral way.
They used technology to communicate, share and learn, create and publish their knowledge.
As above, the PBL using ICT supported the expectation of Thai new generation to succeed the 21 st Century
skills which corresponded to the aim of the present curriculum “Basic Education Curriculum B.E. 2551 (2008)”
Conclusion
The research results showed that the learners who participated in the PBL using ICT in the different subjects
developed both learning outcomes and key competencies. The learning outcomes were present in Science,
English, Mathematics, Health Education, and Occupations and Technology. The key competencies were
Communication Capacity, Thinking Capacity, Problem-Solving Capacity, Capacity for Applying Life skills, and
Capacity for Technological Application which could be discussed as the following:
Communication capacity of learners was developed by using technological as learning tools. These capacities
are perceiving and analyzing the information beyond the classroom, choosing the media fit to the tasks and
presenting a variety of ways of communication. These tools supported learners to communicate their ideas
through various forms of media from their tasks which was presented in text, sound, picture, movies, etc.
According to Rusk, Resnick & Maloney (2009) the present efficient communication should be more than
reading and writing. When producing tasks, the learners participated in using, managing, and integrating a
variety of media in order to express their creativity.
Thinking capacity of learners was developed in thinking analytically, reasoningly and systematically, synthesis,
critically and reflectively. Creating is at the top of the revised Bloom’s Taxonomy (Anderson & Krathwohl,
2001). These are the higher order thinking which was developed by creating tasks (Rusk, Resnick & Maloney,
2009). During this process, the learners were interacting to their surroundings and facing the problems. These
made them concentrate to complete their tasks.
20
Soparat, Arnold, & Klaysom
Problem-Solving Capacity of learners was developed in solving problems in reasonable ways, inquiry for
solving the problems, and effective decision making. This result is congruent with Rusk, Resnick & Maloney
(2009). They described that to create a project is the way for learners to discover and solve the problems within
the context of meaningfully design. The learner is the thinkerable which they can test and edit their work
throughout the process of creating the project.
Capacity for Applying Life Skills of learners was developed in collaborative learning, creating potential to
overcome difficulties, learning to set goals for success, create self-confidence, and lifelong learning. The
learners have opportunities to work with others such as friends, teachers, and the community. This interaction
allowed them to learn about adaptation and responsibility to work and live with others. It also supported learners
to endeavor for success. This kind of thinking brought their inspiration to overcome all challenges and
disappointment from the process of designing and problem solving (Rusk, Resnick & Maloney, 2009).
Capacity for Technological Application of learners was developed in choosing and using technologies
appropriately for the task, choose and to use technologies as learning tools and choose and use technologies in a
moral way. The learners chose ICT tools for creating, publishing, and sharing ideas. Most of them created
projects by integrating ICT; therefore, it also had the variety of using ICT which supported the learners’ abilities
in ICT. Rusk, Resnick and Maloney (2009) described that using ICT in projects supported the learners to
choose, create, and manage the variety of media which are text, picture, movies, and sound.
According to those results, it could be concluded that the PBL using ICT in the different subjects developed the
learning key competencies. Moreover, it is related to the expected learning outcomes of Thai citizens that are
eager to learn, learning on their own, acquiring knowledge continuously throughout life, the ability to
communicate, analytical thinking, creative thinking, problem solving, having a public mind, having discipline,
regarding to the common good, working in groups with good will, being morally consciousness, having moral
values, and pride in being Thai (Office of the Educational Council ,2009).
Recommendations
Suggestions from the Study
From the study, there were some suggestions as follows:
1) Most of the tasks from students’ projects were small tasks and did not integrate much higher
technologies. This is because most teachers were beginners at integrating the PBL using ICT in their
classrooms. So, they designed their lessons by letting the students create small tasks in each lesson. The
suggestion drawn from this result is that teachers should plan a big picture of the whole project by
designing their lessons for students to create small tasks in each week and then combine them as a big
project.
2) The teachers and students faced many problems about using computers to complete their tasks. The
examples of these problems were: 1) the bandwidth of the connected internet was too low; 2) there
were not enough computers and 3) the time to do the tasks were not sufficient. Many teachers solved
the problems by asking students to complete their task on paper instead of using computer programs.
However, this is not much developing the students on students’ capacity for technological application.
Therefore, the integration of subjects would be one of the ways out to solve these problems.
3) From the results, it was reflected that learners who have developed on collaborative working and
capacity for applying life skills were the low ability learners, while the high ability learners reflected
most of their development on problem-solving capacity. These reflections could be implied that the
learners who worked collaboratively in groups with mixed abilities shared their experiences. The high
ability learners were often the leaders to solve problems when the group faced problems. The low
ability learners had to learn to help the members to overcome the troubles. Therefore, the project-based
learning using ICT should proceed under the social context which could support all students to develop
themselves and others.
Suggestions for Further Study
In order to support the PBL using ICT in the classroom, it is important to develop teachers to be more
knowledgeable on the paradigm of learning based on both constructivism and constructionism. At the same
International Journal of Research in Education and Science (IJRES)
21
time, the administrators should provide the facilities and solve problems related to computers and internet
network connections for all students to create artifacts by using ICT.
The further studies on the issues of PBL using ICT might be; the learning paradigm shift , the use of computer
and network for project-based learning using ICT, the classroom setting for project-based learning using ICT,
and the process to support and monitor for project-based learning using ICT.
Acknowledgements
Research fund supported by Office of the Education Council of Thailand in 2010.
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International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
An Investigation of Eighth Grade Students’ Problem Posing Skills
(Turkey Sample)
Elif Esra ARIKAN1*, Hasan UNAL1
1
Yildiz Technical University
Abstract
To pose a problem refers to the creative activity for mathematics education. The purpose of the study was to
explore the eighth grade students’ problem posing ability. Three learning domains such as requiring four
operations, fractions and geometry were chosen for this reason. There were two classes which were coded as
class A and class B. Class A was consisted of successful students in comparison to class B in terms of
mathematical acquisition. The study has been carried out by means of qualitative research. On the other hand,
independent samples T test was used for obtaining statically inference. Moreover, chi-square test was used
whether this students’ problem posing ability is independent of mathematics topics.
Key words: Eighth grade students; Requiring four operations; Fractions; Geometry; Problem posing
Introduction
Problem posing and problem solving are accepted in the center of mathematical thinking. It can be possible to
approach a standard subject from a different standpoint by means of problem posing activity (Turhan, 2011).
Teachers usually teach mathematics as they learned in past. But it is necessary to break the chain in terms of
understanding of education for training students who are self-realized individuals (Ersoy, 2002). By reason of,
students must be had an opportunity to pose their problems and investigate deeply a relevant topic in
mathematics (Korkmaz, Gür & Ersoy, 2004). Some researchers have defended in line with their studies that
problem posing activity must be situated in mathematics curriculum (Brown & Walter, 1993; Silver & Cai,
2005). Problem posing is inventive activity to be able to use in mathematics course. Problem posing refers to a
mirror that reflects the nature and aspect of students’ mathematical experiences (VanDenBrink, 1987). A similar
result was reached by studies which were performed by primary teachers (Leung & Silver, 1997). Semantic
structure in word problems is enriched by force of problem posing for primary students (Lee, 2002). Problem
posing helps creative thinking to improve (Kilpatrick, 1987; Silver, 1997; Yuan & Sriraman, 2010). Also,
students’ problem solving abilities are improved when they examine solution of posed problem (Cai, 1998;
English, 1997; Grundmeier, 2003). On the other hand, one of the most debated topics is whether a clear link
between problem posing and problem solving. While a close relationship between problem posing and problem
solving has been found by Arıkan and Unal, Cai and Hwang based on their studies, Silver et al. and Crespo have
not found a strong correlation in their study (Arıkan & Unal, 2014; Cai & Hwang, 2002; Crespo, 2003; Silver,
Mamona-Downs & Leung, 1996).
Problem posing activity makes a sensation; enables autonomous learning; diverse and flexible thinking;
prevents misunderstanding and preconceptions; helps to deplete anxiety about mathematics learning by means
of interactive learning environment (English, 1998). In literature, one of common students’ difficulty is
organization of given situations during problem solving, they try to solve problems by coordinating initial data
excursively (Işık & Kar, 2011; Jitendra et al., 2007; Xin, 2007). Whether students have a complete
comprehending of a concept can be determined by using problem posing activity.
Recently, many mathematics teachers realized the importance of generating or reformulating a problem as well
as solving a problem in U.S. and Australia (NCTM, 2000; Skinner, 1991). Moreover, Australia Education
Association emphasized that encouraging students to pose their problems is a vital factor in education (HSU,
2006). In Japan, open-ended problems were improved for upgrading mathematics education by using problem
posing at all levels to university (Hashimoto, 1997).
*
Corresponding Author: Elif Esra ARIKAN, arikanee@gmail.com
24
Arikan & Unal
Turkey like other countries made an education reform because of globalization. Therefore, new mathematics
curriculum have been used since 2006 that it was reorganized in accordance with student centered education.
Hereunder, students have found an opportunity to pose their problems (Kılıç, 2011). Mathematical educational
programs in the sixth and eighth classes of the elementary schools; the students carry out finding a solution for
the problems requiring fractional calculations as well as composing the problems themselves, they can pose the
problems requiring a drawing of a figure during the operation of the solution.
They can solve the problems which are related with the around and the area of the planar regions as well as
composing the problems themselves (MEB, 2009). In the curriculum which was conducted by The Ministry of
National Education in Turkey, the importance of problem posing skills have been emphasized. But some
teachers think that students are not interested in problem posing because of the fact that they are getting used to
solve stereotype question as a test. As a result of this situation, teachers do not prefer to deal with problem
posing activity (Dede & Yaman, 2005).
Problem posing situations were envisaged such as free (pose a problem which is difficult), semi-structured (pose
a problem which is given by equation, photograph or figure) and structured (pose a problem which is
reconstruction from initial problem or solution of problem) (Stoyanova & Ellerton, 1996).
Ellerton’s the study was to compare eight high ability children and eight low ability children for problem
posing. As a result, more talented students posed problems were more complex in comparison with less talented
students. Therefore, Ellerton reported that there is strong correlation between problem posing and problem
solving (Ellerton, 1986).
Abu-Elwan’s the study was to develop of mathematical problem posing skills for prospective middle school
teachers. Two groups were experiment; one group was control group. While one of experiment groups’ students
posed problems by “examining textbook problems”, another experiment group’ students posed problems by
“semi-structured situation strategy”. As conclusion, semi-structured situation was found more effective strategy
to develop problem posing ability (Abu-Elwan, 1999).
With regard to math anxiety, posed a problem has been seen as motivated activity to students (Buerk, 1982;
Baxter, 2005). Brown and Walter emphasized that problem posing is to overcome math phobia in their book
“There is good reason to believe that problem generation might be a critical ingredient in confronting math
anxiety because the posing of problems or asking a questions is potentially less threatening than answering
them. The reason is in part a logical one. That is, when you ask a question, the responses “right” or “wrong” are
inappropriate, although that category is paramount for answer to questions” (Brown & Walter, 2005).
Moses, Bjork and Goldenberg (1990) highlighted that classroom climate should be prepared for problem posing
activity and mathematics teacher should encourage students to share their ideas about problem mutually. In
accordance with this purpose, the researchers suggested 4 rules to teacher as follows:




Ask students known, unknown and conditions of problem
Help students to identify features of problem
Foster students to not fear using uncertain situations and pose an easy problem
Create an environment for students to play a mathematical game which can be changed in any form.
In Dickerson’s study, five different instructional approaches were implemented for improving students’ problem
solving ability in line with purpose of doctorate thesis. Problem posing for three groups and problem solving for
two groups were executed. First group: combination of problem posing interventions which were structured,
acting-out, what-if-not and open-ended strategies, Second group: problem solving intervention by teacher, third
group: problem solving intervention by researcher, fourth group: structured problem posing implementation,
fifth group: what-if-not problem posing implementation. It was emphasized that problem posing approaches
were an effective way to raise the successful of problem solving of students. In terms of gender differences, the
results showed that while females were more successful than males in Group 2 and 3, the exact opposite was the
case for Group 1, 4 and 5 (Dickerson, 1999).
Stickles (2006) purposed to identify kind of posed problems by pre-service and in service teachers. The
participants posed problems according to statement (generation) and a given problem (re-formulation) for this
reason. Generated problems were classified as specific goal, specific goal problem-added information, specific
goal problem-initial condition manipulation, specific goal problem-initial condition manipulation, added
information, general goal problem, added information.
International Journal of Research in Education and Science (IJRES)
25
Reformulated problems were categorized as add information, change the context, combination, equivalent
wording, change the given, change the wanted, extension, simplified problem, switch given and wanted. Also,
Stickles found a strong relationship between experience and problem posing skill (Stickles, 2006).
Yuan (2009) examined a relationship between creativity and problem posing ability in doctorate thesis. In
accordance with the purpose of study, two groups of high school students from Shangai and Jiaozhou in China
and one group of high school students from U.S. participated in the research. According to result of this
research, expected relationship between creativity and problem posing ability was only found in Jiaozhou group
(Yuan, 2009).
In our study, we tried to monitor eighth grade students’ problem posing ability according to operations, fractions
and geometrical measures.
Method
Participants
Participants of the present study were 46 eighth grade students. There were two groups which were class A and
class B. Students of Class A had been more successful in mathematical problem solving compare to students of
Class B
Research Questions
This study has been shaped around two questions that were


Which subject force to students during the problem posing activity?
Is there any significant difference between class A and class B in terms of problem posing ability?
Data Collection
Three mathematics teachers and three associates in education faculty opined to en-sure the study’s reliability
and validity. The participants were presented worksheet which was three semi-structured problem situations and
then they were asked to pose three for each such as four operations, fractions and geometry problems. Also,
students were asked whether they received support apart from school. During the implementation, both of
researcher and mathematics teacher made an observation and took notes separately. Namely, observation and
written material was used for the study. Therefore, this study has been carried out as qualitative research
(Yıldırım & Şimşek, 2008). Survey method and direct observation were used for data collection tool.
Data Analysis
Descriptive analysis was used since the study was tackled in all its parts and in its natural environment.
Students’ responses were analyzed as correct or incorrect. In the final, statistical analysis (independent T test)
was applied for identifying difference between class A and class B. Students’ responses were assessed as correct
or incorrect. Moreover, chi-square test for 3x3 table was used whether problem situations and mathematics
topics are independent.
Results and Discussion
After the study was executed as paper-pencil-test, responses which belongs to class A and class B were
presented respectively in Table 1 and Table 2.
26
Arikan & Unal
Table 1. Number of posed problems by class A
Learning subdomains/semistructured problem
situations
Requiring four operations
Fractions
Geometrical measures
9x4=36
36 3=12
Pose a problem
such that its
solution is above
mentioned
9
6
10
Pose a problem
such that its
responsive is
above mentioned
9
6
2
Pose a problem according to
pattern models above
mentioned
14
9
5
When examining the number of posed problems related with according to learning subdomains, while twenty
one students’ worksheets were identified properly, seven students’ papers were not incorporated in the study
because of empty.
Four students for first situation, one student for second situation and three students for third situation answered
correctly. On the other side, five students for requiring four operations, two students for fractions and two
students for geometrical measures replied accurately. One student follows all problem situations correctly.
Table 2. Number of posed problems by class B
Learning subdomains/semistructured problem
situations
Requiring four operations
Fractions
Geometrical measures
9x4=36
36 3=12
Pose a problem
such that its
solution is above
mentioned
13
4
7
Pose a problem
such that its
responsive is
above mentioned
13
11
10
Pose a problem according to
pattern models above
mentioned
20
9
7
Twenty five of thirty one students’ papers were analyzed and the rest of them were removed from the study
because of empty. In this class, two students for first problem situation, seven students for second problem
situation and seven students for third problem situation responded rightly. On the other hand, seven students for
requiring four operations, three students for fractions and three students for geometrical measures answered
correctly.
With respect to observations of mathematics teacher and researcher, common opinions were manifested. First of
all, the most forced subject to students was fractions. Many students generated problem like that” I have 4 m
coating. If 5/3 of it is flawed, then how many meters coating do I have?”, “I have six slices pizza. If I eat 4/1,
then how many slices do I have?” and “4/3 of 150 km way I run. How many km do I run to finish?”. For this
reason, mathematics teacher expressed that students had not comprehended fractions completely. An
intervention was not executed for class A but motivation was used for class B students. For instance, class B
students exchanged ideas each other, discussed real life problems, used probability to pose problems and were
active during the implementation. Mathematics teacher emphasized “I would like to say in accordance with
tables and our observations, class A students assumed that they can generate a problem if and only if their
problem solving they had to remember problems which problem solving experiences are evoked. However,
class B students were so relax in execution. It was surprised to me that they tried to construct a problem of
probability which had been treated freshly. They were not obsessed with their problem solving experiences.
Moreover, I recognized they had not comprehended to have a grasp of fractions. Therefore I have decided to
lecture of fractions to fifth grade students by using materials”. When researcher asked to mathematics teacher
how often they implement problem posing activity during the mathematics lesson, she answered that “if we
have enough time at the end of the course, we do problem posing activity. In other words, we solely execute due
to complete course subjects on time”.
International Journal of Research in Education and Science (IJRES)
27
Indeed, the teacher’s observation about students’ comprehending of fraction is line with the result found by
Silver. Silver mentioned that problem posing enables insight into students’ understanding (Silver, 1993).
According to statistical independent samples T test, significant 2-tailed score was 0, 618 with 95% confidence
interval. Hence, there is no significant difference between class A and class B although class A students more
successful then class B students in mathematics exams. Problem posing facilitates students to study as a team
and to discuss mathematically (Schiefele, 1991). Class B studied as a team and they found themselves in
pleasure during problem posing for this reason.
It can be examined posed problems by students in terms of mathematical and linguistically and complexity.
Table 3. An example of first question according to geometry problem
A woodsman has 4 woods which each of them are 9
cm. He creates a square by means of banding
together and he put a candle in per every 3 cm.
Then, how many candles he put?
Table 4. An example of second question according to fraction problem
A dice is thrown into the air. What is the probability
that an odd number of the top surface?
Table 5. An example of third question according to fraction problem
There are ten blocks in forth step of a pattern. 2/10 of
these blocks are painted by Bordeaux. If the rest of
blocks are painted by dark blue, then what is the ratio
of dark blue blocks to full blocks?
Table 6. An example of third question according to geometry problem
Circumference of the square is 4 cm in first step of a
pattern and circumference of the squares is 12 cm in
second step of the pattern. What is the difference
between first and second steps of the pattern?
When examining problems in Table 3, 4, 5 and 6, students posed problems a lower level than expected. It may
be caused that students have not been familiar to problem posing activity. Giving chance to students for writing
their own problems, many of linguistic difficulty may be cut down (Bums & Richards, 1981; Resnick &
Resnick, 1996; Wright & Stevens, 1980). The writing aspect of problem posing assists students’ communication
skills (Burton, 1992; Matz & Lerer, 1992).
Students should have an opportunity to create their problems. Because problem posing provides autonomous
learning, a sense of ownership of mathematics, to foster creative thinking and to improve fluency in
mathematical language (Nohda, 1995).
While H0= problem posing situations and mathematics topics which are executed in this study was independent,
H1= this two qualitative variables are not independent. According to chi-square test, because of χ2=5.08, H0 has
been accepted. Therefore, it was not determined that problem posing situations and mathematics topics are two
dependent qualitative variables.
Conclusion and Implications
Eighth grade students posed word problems according to geometrical measures and fractions. Actually, in
Turkey, problem posing is a new topic. Therefore, these students generated with their sentences problems at first
time. Even if a student is not able to solve a problem, she/he can pose her/his own problem. While some
researchers found a correlation between problem solving and problem posing, the others did not find it.
Therefore, problem posing ability may be affected by other factors.
28
Arikan & Unal
One of these factors might be motivation. For example, in our study, although Class B students were not good at
problem solving, they were accomplished with the Class A. Teachers can find their students incapable to solve
a problem. For this reason, they do not prefer their students to pose a problem because of waste of time. But if
we want to our students to solve their problems, problem posing is very important activity to think creatively
and critically. Therefore, teacher should foster students to pose a problem consistently. Namely, motivation is
used for posing a problem.
Transferring of calculation based knowledge by means of teaching rules sequences creates a perception that
mathematical subjects are independent from each other. There have been two different solution finder who
solves the mathematical problems; master and apprentice. While the master reaches to a solution by means of
making an activity based on his or her conceptional knowledge for the problems he or she is working on, the
apprentice tries to ask him or herself whether he or she has been faced with the similar situation or problem
before. That's why the creation of the conception learning is to be provided and as a result of this action, the
student has to find out a required meaning for the calculation that he or she has directly been involved in.
Concerning this issue, two concepts which are fractions and length-area measures can be given as example.
Rational (fractional) numbers are abstract concept for secondary school students. When students learn new
knowledge, they build it on old knowledge (Yağbasan & Gülçiçek, 2003). In other words, the learning gap
effects mathematics instruction. Students should mingle with concrete materials related mathematics subject
instead of solving complex operations. For this reason, students need many concrete experiences by the means
of using materials such as numerical axis (fraction bars), area models (pizzas) and solid models (rainbow cubes,
orange and bread etc.) (Baki, 2008). This case is the same for geometrical measures. Geometry lessons can be
supported by virtue of computer programs such a sketchpad.
The teacher emphasized motivation factor during conversation. Motivation has been accepted as energy to
achieve the objective. Since learning requires effort, motivation plays an important role. One of components of
the motivation to learn is external motivators. Students must feel themselves to have opportunities for sharing
their responses and thoughts freely (Frith, 1997). Learning environment must be designed for students to
exchange of views and students feel relaxed during the process. In the study, motivating students affected their
problem posing performance (for class B).
Although Class B was better than Class A in terms of problem posing according to fractions and geometrical
measures, chi-square test shows that problem posing ability is independent of areas of mathematics. As
mathematics teacher mentioned that problem posing requires thinking rather than memorization because of
open-ended process. But class A students’ making sense of problem posing was related to their past experiences
with problem solving. Also, problem posing was depicted as an assessment tool by the teacher to obtain
students’ acquisitions about mathematics concepts. Similar result was specified by Lin and Leng (2008). Based
on their study, a teacher can gain pattern about his/her students’ mathematical thinking and learning by using
problem posing activity.
Mathematical experiences are necessary but not sufficient condition for problem posing. Although class A
students more capable than class B students for mathematical content knowledge, the result of the
implementations had no significant difference. Because class A students neither eager nor curious during the
problem posing activity. They only wanted to solve problems. Also they perceived problem posing as exercise
questions. There-fore they forced to themselves to remember test questions. It may be defined as a vicious cycle.
Mathematics teacher passed her opinion for problem posing that it is useful activity to identify students’
mathematical accumulations and to be used as an assessment tool. Namely, problem posing let teachers to
obtain information about students’ mathematical learning and thinking.
It must be mentioned that students’ problem posing skills are concerned with their teachers’ approaches to
posing a problem. It was explained that teachers have difficulty because of inexperience or limited experience in
problem posing activity (Rizvi, 2004). For this reason, teachers should mingle with problem posing as far as
possible. On the line, problem posing activities should be executed in education faculty. Posed problems can be
examined in terms of creativity, originality and complexity. In the present study, students were limited on
fractions. Actually, many applications are used by developing technology in schools. Using technology
simplifies teachers’ work for students’ intuitive understanding and making sense of fractions. Another notable
point is time. Problem posing activity should be allocated time in mathematics curriculum. Cross-national
studies may be carried out later on sub domain subjects like fractions and geometrical measures.
International Journal of Research in Education and Science (IJRES)
29
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International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
A Preservice Mathematics Teacher’s Beliefs about Teaching Mathematics
with Technology
Shashidhar Belbase*
Zayed University, Dubai
Abstract
This paper analyzed a preservice mathematics teacher’s beliefs about teaching mathematics with technology.
The researcher used five semi-structured task-based interviews in the problematic contexts of teaching fraction
multiplications with JavaBars, functions and limits, and geometric transformations with Geometer’s Sketchpad,
and statistical data with Excel Spreadsheet to generate data about the participant’s pedagogical-technological
beliefs. An additional unstructured interview was conducted with the same participant after his practice
teaching. The interview data were analyzed and interpreted in two layers- the first layer portrays the
participant’s voice in his narratives and the second layer portrays researcher’s voice in terms of his
interpretation of data and interconnection to literature. The analysis and interpretation generated seven major
categories of beliefs- beliefs about teaching materials, teaching strategy, bridging activities, technological tools,
mathematical concepts, and meanings, activities and transformation, and issues and challenges. Finally, the
author discusses implications of the study.
Key words: Teacher beliefs; Technology integration; Radical constructivist grounded theory
Introduction
At first, the author briefly introduces the literature on teacher beliefs about teaching mathematics with
technology and the radical constructivist grounded theory (RCGT) as a theoretical frame for the study. He
discusses methods of data generation through task-based interviews. He further discusses analysis and
interpretation of interview data by applying the principles of RCGT. He highlights some key findings in the
results and discussion followed by implications.
Many researchers (e.g., Chai, Wong & Teo 2011; Ertmer 2006; Leatham 2002 & 2007; Quinn 1998; Teo, Chai,
Hung & Lee, 2008) highlighted challenges of technology integration in mathematics education. Quinn (1998)
reported teachers’ initial beliefs toward technology use in mathematics teaching as a tool to help students to
discover concepts on their own with fun and greater motivation to learn. The next theme was changes in beliefs
in which he reported changes in the majority of the preservice teachers’ beliefs in relation to use of technology
(i.e. calculators) as tools to learn concepts and basic skills, not just to calculate. He reported their educational
experiences in schools being lack of technology in learning mathematics. Quinn (1998) clearly outlined how
mathematics teacher education courses could enhance preservice teachers’ beliefs toward the creative use of
technological tools in teaching and learning mathematics. This study showed how preservice mathematics
teachers’ embodied technological-pedagogical beliefs could be changed with well-planned and well-focused
mathematics methods course.
Leatham (2002, 2007) provided plenty of ideas about technology integration in preservice mathematics teacher
education in general and mathematics methods course in particular. His analysis seemed very deep and very
comprehensive in the sense that he captured essential themes and pertinent issues of technology integration in
mathematics education. His analysis of preservice mathematics teachers’ beliefs about teaching mathematics
with technology concluded with a nonmathematical role, real-world application, visualization of mathematics,
and exploration of mathematics. These roles of technology seemed to be powerful in teaching and learning of
mathematics. He further discussed preservice teacher beliefs in terms of formal and informal, internal and
external, subjective and inter-subjective, isolated and connected, utilitarian to developmental, temporal to
spatial, to name a few.
*
Corresponding Author: Shashidhar Belbase, belbaseshashi@gmail.com
32
Belbase
Next, Ertmer (2006) examined the relationship between teachers’ pedagogical beliefs and their technology
practices in classroom teaching and learning. She reviewed various works related to teacher beliefs in the
various subject areas and examined the pedagogical beliefs in relation to technology use. She argued that the
relationship between beliefs and teacher practices, teacher knowledge, and student outcomes are important
aspects to be considered in educational research related to teacher beliefs. She claimed that beliefs a teacher
holds may influence his or her perceptions and judgments, and finally it affects the classroom practices using
technology. She reported some researches having inconsistencies between teachers’ beliefs and their classroom
practices. Ertmer (2006) further explained that beliefs can be quite idiosyncratic, and that is why two teachers
having the similar knowledge about technology may have different beliefs about the use of technology in
teaching and learning. Researchers (e.g., Peterson, Fennema, Carpenter & Loef, 1989) supposed that teachers’
beliefs play a significant role in how it is adopted and implemented in classroom teaching and learning.
NCTM (2008) stated, “in a well-articulated mathematics program, students can use these (technological) tools
for computation, construction, and representation as they explore problems; and the use of technology also
contributes to mathematical reflection, problem identification, and decision making” (n. p.). NCTM clearly
stated its position in relation to the role of technology in mathematics education that emphasized the creative use
of technology in teaching and learning mathematics. Preservice or inservice teachers’ beliefs toward the use of
technological tools for construction can be related to efficiency of construction, learning through construction,
and process of construction. One’s beliefs about teaching mathematics with technology might be rooted in his or
her image of mathematics, past and current anxieties associated with teaching and learning of mathematics, and
his or her attitudes toward use of different tools (both manipulatives and technology) for teaching mathematics
(Belbase, 2013a).
Most of the prior studies on preservice teacher beliefs are associated with either their experiences in classroom
practices (as students and teachers) or surveys on what beliefs they hold about teaching and learning
mathematics with technology. There is a lack of study on preservice mathematics teachers’ expressed beliefs
during task-situations and idiosyncratic experiences. The studies on teacher beliefs based on recalled and
recurring memories of experiences as students and prospective teachers may not reflect their idiosyncratic
beliefs. This study aimed to explore a preservice secondary mathematics teacher’s beliefs about teaching
mathematics with technology that are formed and reformed based on the problems he encountered and
experienced he went through when he was asked to reflect on problems and his anticipation of teaching in
classroom. In this context, the research question for this study was- What beliefs does a preservice secondary
mathematics teacher hold about teaching mathematics with technology and how do those beliefs unfold with
experiences within task situations and teaching experiences?
The researcher used the radical constructivist grounded theory (RCGT) as theoretical perspective to construct
data, and analyze and interpret the participant’s beliefs about teaching mathematics with technology.
Radical Constructivist Grounded Theory: A Theoretical Frame
This research study assumed integrated epistemology of radical constructivism with grounded theory to
synthesize a radical constructivist grounded theory. There may be a variety of constructivist grounded theory
based on which constructivism integrated with grounded theory- for example, trivial constructivist grounded
theory, social constructivist grounded theory, and RCGT. In this study, the researcher adopted RCGT. Three
principles of radical constructivism assume that knowing is an active process in construction of knowledge by
the cognizing subject; his or her knowing is autopoietic (Maturana & Varela, 1980), and his or her knowing is
evaluated in relation to viability (von Glasersfeld, 1978). These principles when integrated with the principles of
grounded theory- coding, theoretical sampling, constant comparison, theoretical sensitivity, and memoing may
result into a new hybridized tool as RCGT for the study of preservice mathematics teacher’s beliefs about
teaching mathematics with technology.
Glaser and Strauss (1967) founded grounded theory method as a new paradigm in qualitative research focusing
on the discovery of a theory grounded in data. Many grounded theorists (e.g., Strauss, 1987; Strauss & Corbin,
1990, 1998; Corbin & Strauss, 2008; Glaser, 1978 & 1992; Charmaz, 2006; Clarke, 2005) used grounded theory
from different epistemological perspectives. The marriage of RCGT and grounded theory for this study made
this researcher aware of his role as a researcher, the role of the participant as a knowledge generator, and process
of grounded theory as a tool for analysis and interpretation. The RCGT provided the researcher a basis to
reconstruct the voice with “multiple registers and meanings” (Mazzei & Jackson, 2009, p. 5). The five principles
of RCGT – researcher and participant voice, symbiosis between researcher and participant, research as cognitive
function, research as an adaptive function, and praxis criteria served the guideline for the study (Belbase, 2014).
International Journal of Research in Education and Science (IJRES)
33
The first principle focuses on researcher and participant voice in the research process and product. The second
principle focuses on the mutualism of researcher and participants through task-based interactions having
cognitive gain for both. The third principle is about process of the study as a cognitive function. The research
process was a journey of interaction between the researcher and the participant through which they constructed
meaning of the participant’s beliefs based on his experiences through the task situations. The fourth principle is
about adaptation of the researcher and participants to the new situations during the interactions on task
situations. The emergence or construction of new codes and categories continued until more stable categories
emerge through the process. The process of equilibration or saturation of categories was achieved through
adaptation to additional data, new codes, and grounding categories in the subsequent interview episodes or
interactions. The fifth principle is associated with dimensions of fit and viability of the categories to the context
that portray the beliefs of the participant (Belbase, 2014). The theoretical underpinning of RCGT served as a
conceptual guide throughout the research process by bringing this researcher and the participant together as colearners, teachers and research subjects.
Methodology
Recruitment of Participant
Initially, three out of a pool of twenty preservice secondary mathematics teachers enrolled in a College of
Education at a Midwestern University in the U.S. in the fall of 2011 were recruited as the potential participants
for the study. The selection of the three participants in the study was based on their interest to volunteer in five
task-based interviews and their time available for the interviews. The research tool included five task-based
interviews when they were in mathematics methods class and one post-teaching experience interview in the
spring of 2012 with each of the three participants. The researcher analyzed and reported only one participant’s
beliefs about teaching mathematics with technology in this paper as a single-subject case analysis. The single
case was selected out of the three participants based on explicit expressions of beliefs and active participation in
the problem solving situations compared to other participants.
Task-based Interviews
The source of data for the study were a series of five 60-90 minute task-based interviews (Goldin, 2000)
conducted with a participant during the time when he was taking mathematics methods course in the fall of 2011
and one interview after his field experience in the spring of 2012. The first five interviews were based on
mathematical tasks within technological environments. The researcher designed an interview guideline with
potential interview questions from the five different content domains. The first interview was on teaching
fractions by using JavaBars, the second interview was on teaching functions and relations using Geometer’s
Sketchpad (GSP), the third interview was on teaching concepts of limit in the dynamic environment of GSP, the
fourth interview was on teaching geometric transformation using GSP, and the fifth interview was on teaching
of data using Microsoft Excel Spreadsheet. Each interview episode was video recorded. The interview after the
post field experience was an unstructured interview focusing on his experiences of teaching mathematics using
technology during his practicum.
Analysis and Interpretation
The application of RCGT as “a hybrid of radical constructivism as an epistemology and grounded theory as a
methodology” (Belbase, 2013b, n. p.) helped this researcher in conceptualizing the processes of data
construction, analysis, and interpretation. The principles of radical constructivism helped the researcher to form
an epistemological basis to understand the nature of the data and major categories or themes. The grounded
theory helped him to perform the actual analysis and interpretation through coding, categorizing, thematizing,
theoretical sampling, and theoretical memoing as cognitive and adaptive function.
There are different ways to derive major categories and formulate a substantive theory from grounded theory
based on different epistemological lenses. For example, the classical grounded theory of Glaser and Strauss
(1967) emphasizes discovery of a theory grounded in data. From the grounded theory perspective of Strauss and
Corbin (1990 & 1998) and Corbin and Strauss (2008), a theory is developed out of empirical data. According to
the constructivist grounded theory of Charmaz (2006), a theory is constructed out of the empirical data. By
applying RCGT method, key themes and a related theory are invented by the researcher out of the empirical data
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(Belbase, 2013b). The notion of discovery, development, construction, and invention relate to how a local
theory is grounded on data within different epistemological paradigms.
During the analysis and interpretation the researcher employed open, axial, and selective coding (Corbin &
Strauss, 1990 & 1998) of the interview data. At first he identified some basic concepts in the data and labeled
them as open codes. These open codes were then compared and contrasted among each other (i.e. constant
comparison) to identify major concepts from the data. The contrasting and comparing of the open codes were
helpful to identify some pivotal concepts and they were named as axial codes. He further regrouped and
condensed the axial codes into final categories as selective codes. While doing this, he used the multiple
interview episodes as means of theoretical sampling and theoretical memoing. The earlier categories were
analyzed and interpreted across the episodes. The interview notes and coding notes were used as memos to
conceptualize and reconceptualize the key categories. The process of open coding, axial coding, selective coding
and memoing with theoretical sampling formed a complex of an analytic tree. Major categories emerged as the
key constructs from the analytic tree. The interview data were then interpreted at two layers of interpretive
accounts. The first layer was participant narrative of his beliefs about teaching mathematics with technology.
The second layer was the researcher’s interpretation of participant’s beliefs about teaching mathematics with
technology. The researcher decentered the voice of the participant and his own voice through the layered
interpretive accounts in relation to beliefs on the subject of study (Pierre, 2009).
The researcher formulated the results in terms of domain specific narratives of the participant and the
categorical narratives of his interpretive accounts. The first kinds of narratives were constructed from the
participant’s expressions of beliefs during the task-based interview episodes. These narratives portray the voice
of the participant in his the first person perspective. The purpose of these narratives is to portray the participant
beliefs in a direct sense without making any interpretations. The readers may have opportunity to make their
own interpretations of his beliefs despite the fact that the researcher presented his categorical interpretations.
The categorical narratives portray what this researcher observed in the participant’s belief narratives. This
process helped this researcher to portray the participant’s beliefs about teaching mathematics with technology in
two layers – both the first person and the second person accounts.
Results and Discussion
This section presents the outcome of this study in two layers - the first layer as the first person perspective of
participant’s beliefs in his voice, and the second layer as researcher’s interpretation of participant’s beliefs as the
second person perspective. Hence, this section includes two different genres – the first-degree narrative (of
participant) and the second-degree narrative (of the researcher).
First Order Description and Analysis
The first order description and analysis consisted of integrating major belief statements in the form of
participant narrative. Following belief narratives in the participant’s voice represented his domain specific
beliefs about teaching mathematics with technology.
Beliefs about teaching fraction multiplication with JavaBars: I like fraction multiplication of non-complex
fractions that do not have mixed fraction. Cuisenaire rods, I think, would be the easiest (thing) to begin with.
From what we have said fraction as a part of a whole, we can select any one (any size of a rod) as a whole unit.
Then it might be easy to begin. I think the most difficult thing that students might have in multiplying a half time
two thirds and get a one third. Where does that one third come from? What does it mean? I think with
Cuisenaire rod we can select a larger piece that can have three different pieces of thirds (He puts three onethirds along the large piece as one unit) and we can take two of them as two thirds (He separates two of them
away.).
Use of manipulatives in the teaching of fraction multiplication may not be necessarily helpful all the time. It
depends on what the students are used to in it. If they are used to in just using Cuisenaire rods and fraction bars
or whatever, I think that at that point if you are able to design may be an applet or some kind of technology and
establish your own fraction bars of any size. Sometimes a meter-stick would be helpful to show fraction
multiplication. It can be helpful to see smaller units like hundredths. Also, picture models can accommodate
small fractions. When drawing small fraction is difficult, maybe we can use Excel Spreadsheet.
I anticipate using JavaBars for teaching fractions and fraction multiplication together with other operations.
Probably I will use those before I even use fraction bars. Because it’s free, and you don’t have to store it. It is
International Journal of Research in Education and Science (IJRES)
35
always there in the computer or Internet. I think that it is something I would use depending on if students just
needed a quick brush up. I may have to draw just pictures rather than doing it on the computer. I like this
program (JavaBars).
I agree that technological tools like JavaBars have a positive impact on conceptual understanding of fractions
compared to just use of manipulatives. I think jumping from the procedure and trying to explain the concept I
would show it to them. Even better, allowing them to use this (technology) and show it themselves. Um, that’s a
large conceptual gap. I think moving to the computer from fraction bars is just that much, especially with
JavaBars. It’s got the power to present so much more than the conceptual understanding. It has the ability to
increase the conceptual understanding.
Beliefs about teaching functions and limits with Geometer’s Sketchpad: When I think of a teaching activity
to introduce the concept of function in a class for the first time, I consider the grade level to introduce the
function concept at first. It can be probably seventh grade. I guess I would start out with some story problems.
Probably the first thing the students would attempt to make a table of values that would come from the problem.
Then I would begin with an equation whose values would come from the table. Probably a good place to start a
function is one that just uses patterns. You may be looking at arithmetically growing patterns. The students try
to find the patterns in steps one, two and three, what’s the fourth step and what’s the fifth step? As an example, I
would start with an arithmetic pattern where each term starts with one, three, five… And, I would ask them to
find next two numbers. I probably would draw the pattern in terms of blocks and ask them to draw next block.
And, the next step from there would be okay to make a table and ask how many boxes are there in each step.
Also, I would build the structure like in the figure with manipulatives, base ten blocks I suppose.
So far technology in teaching function and limit is concerned; I would use technology for bridging the concept
of the use of concrete objects and manipulatives to technology. I would just recommend any types of technology
that help in the construction of such patterns. I really like Excel Spreadsheet because you can ask students
essentially working from a written table and all they got to do is to put the table in it and graph it straight away.
They can use the Excel Spreadsheet to figure out the function from the graph.
I think GSP is definitely a good for teaching function and limit compared to other tools. At the very beginning
probably I would start with a paper and graph and then move toward technology. Especially, what we do with
drawing a line and what the slope represents. Um, for looking at linear functions at that moment I would
encourage students prepare far better for the next two steps in mathematics is do it in the paper and after
understanding what the slope of the function is they can move to the technology. Probably, I would introduce
function as a distance, function as a velocity, etc. and go to graphing a quadratic function introducing
acceleration and just preparing students with and talk about the rate of change, and talk about slope. We can
show them pretty easily here (in GSP). The great thing about it is when students ask questions or we can ask
them the questions: What happens when we increase the slope and rather than asking them to draw lines of
equations as functions and see the nature of slope in a paper, they can jump in GSP. At the start of our
discussion, I had no question in using Excel Spreadsheet. Just my personal comfort level in using Excel
Spreadsheet was very high. However, from what we have seen today I think GSP has more benefit in teaching
function and limit that really we cannot show in an Excel Spreadsheet.
I imagine that for functions specifically in algebra class, that the students may not get to full understanding of
concepts of functions, especially in seventh or eighth grades when they are in algebra class. I may have to go
through this entire process with the students one-on-one possibly in a class project if it needs. And, I am hoping
that students have this behind them the concepts. But, I imagine that these are the some of the stuff that I need to
teach. I think GSP is gonna be a great way. Maybe even if we are looking at polynomials then we can show the
functions of polynomials in GSP and show how different things change. Looking at how slope changes we can
start talking about the rate of change preparing students to move from algebra to calculus, and maybe doing a
little bit in trigonometry.
When some students are not yet ready to learn function concepts because of their weak mathematics
background, for me, the easiest way to do it is a class project with some sort of constructions that the class is
working on. And, during that time I would ask students to trade off who is doing what and help each other. Or,
I can speak to the class- okay you guys can work on this project. And, I can go to the students who need help
and work with them run through the basic processes of functions and get to the point where they need to be in
terms of conceptual understanding.
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To me, this (i.e. GSP) is probably the best tool I have seen for teaching the concept of (function &) limit. Again
the use of the software makes it far more intuitive and easy to understand situations than drawing on the board
and looking at one tangent line. This makes more sense. For next semester, I think I will not get into limits and
derivatives. What we did today would definitely benefit in my future teaching and probably I will try using these
activities very similar way what we went through. I will help students make that jump from limit to derivative
not just teaching limit.
Beliefs about teaching geometric transformations: At the beginning of teaching geometric transformations, I
think, I would begin with any of the transformations- reflection, rotation, translation, glide, and resize. Maybe, I
would begin with similar figures that are either just at different spots or they are of different size at the same
place. Probably, I would ask the students what is the difference between these two bottles. Maybe, I can place
one of the bottles in front of a mirror and ask the students- What’s the difference between these two (object and
image)? I would start from there (this) before I even talk about how to construct them and discuss what a
transformation means to them. Even before the beginning of geometric transformations, I would encourage
(students) just looking different objects and you can pull out many things in the class and how those are related.
You can even take a paper and ask them how the two pictures are related. The students may answer whether
these pictures are related with a glide or a slide or any other. What else did you see today? These might be
helpful to begin with.
I agree that I can begin from transformations we have observed from daily life to transformations in
mathematics. I would bridge the informal meaning of transformation to formal one beginning from what we did
today starting from transformation in everyday life and then talking about geometric transformations as one of
those applying with geometric properties of lines and planes and even three dimensions. With the three
dimensional example, we can go back to the water bottles (He points to the nearby water bottles on the table.).
When I anticipate my future teaching of geometric transformations probably I will use GSP. If I use any
manipulatives, that could be pattern blocks. Probably I can use a geo-board too. I agree that a geo-board might
be easy hands-on material for students to play with for a while to learn geometric transformations. You can
probably move further to use of a graph-board. Then, maybe I can move toward using GSP. I have to deal with
different comfort levels of students while using GSP to learn geometric transformations. When I think of
advantage of using GSP over other materials, I think it adds in conceptual understanding of geometric
transformations.
I think talking through and for this turning to GSP; I learned that it is a very powerful geometry-teaching tool.
We also talked last time it is a powerful tool for the algebra classroom too. I can use it in trigonometry class. So
I think that I would definitely use the GSP for class projects. It was helpful talking about how we go out
teaching with these tools more to students who are quite yet not conceptual. That means I will continue using
these things in my future teaching, not just in the teaching practicum.
Beliefs about teaching mathematics during practicum: In terms of basic teaching, the first couple of weeks of
the past semester, spring semester of 2012, were a real shock in terms of student preparedness, student
motivation, and students turning in of homework. It amazed me how the students were doing and what they were
getting by with. And, throughout the semester that challenged me. I don’t know if there was necessarily silence
out there at a bigger city than I was used to, it was a broader problem out in the city. I hope it doesn’t exist in
smaller towns yet. Throughout the semester I worked and I thought the students would turn in home works. I
spent 2-3 days in a week throughout the semester when I sat down and talked to the students how the study goes,
taking notes, and using the book as a tool. And, I asked them, “Why don’t you do homework on time?” And, we
talked about getting prepared for tough learners. I think that maybe the first time a kind of push all sides, and
second time talking and listening to them a little bit. The third time when I talked to them was after the spring
break. The last three weeks after the spring break were what I had expected in the second semester. I had
students coming to the class prepared and students turning in home works. I started with the two weeks of
teaching a little slow. Now it was back to what I expected. I think having a discussion with students and from the
standpoint of myself being successful, self-taught, learned, and tried to equip students with what helps them
getting on, what they need to see when they come to classes, not just math. I started feeling pretty good by the
end of the semester.
I didn’t know how to use GSP a lot in terms of teaching and I ended up one of the biggest usage in the past
month when I was teaching quadratics, quadratic equations, and solving functions. I just built the basic
quadratic formula with a, b, and c as parameters (i.e., ax 2+bx+c=0). We went through ten minutes of each
class. We began with- okay a is one and what kind of change do you see in b and c? What happens? Do it and
show what happens. Maybe I could take a and b animated and see how c changes. It was nice to show things
International Journal of Research in Education and Science (IJRES)
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going up and down. I think that was one of the biggest things, probably the biggest depth of learning for my
algebra students for the entire year being able to see that every day. We started factoring a quadratic equation
and graphed them. We were able to see two different lines in the graphs as the factors.
Second Order Analysis and Interpretation
The researcher analyzed and interpreted the participant’s beliefs about teaching mathematics with technology
from his perspective as the second-degree narrative. The following categories emerged with analytical and
interpretive function of RCGT process in this study.
Manipulatives and technological tools: The participant expressed his positive beliefs toward the use of
manipulatives in teaching fractions, functions and limits, and geometric transformations. Particularly, he
revealed his beliefs that fraction such as 2/3 could be represented by Cuisenaire rods, fraction strips, and even a
meter scale. For him, manipulation of fraction operations (e.g., multiplications) with Cuisenaire rods seemed a
plausible way to clarify the meaning of such operations in a visual way. He also added that the nature of fraction
multiplication determines which manipulative might be more useful in developing an understanding of the
operation. Sometimes manipulatives might not be the easiest way to develop an abstract sense of complex
fraction multiplication. Together with other manipulatives such as pattern blocks, base ten blocks and dices, he
claimed that pictures could be helpful to use as models. These different kinds of materials are suitable in making
new fraction patterns and comparing different values. He seemed to believe that moving from manipulatives to
JavaBars to teach fraction operations could help students better understand fraction multiplications. After the
interaction on fraction multiplication using JavaBars, he expressed that he anticipated using the fraction
operations with JavaBars before even using any manipulatives.
In relation to teaching of function, he seemed to believe the story problem (with two variables) as a good start.
A function could be discussed in terms of the nature of the slope when the function is plotted on a graph. He
expressed his positive belief that Excel Spreadsheet as a best tool to teach a function. He preferred the use of
tables showing relationship of two variables and lines in Excel Spreadsheet to teach students the nature of
functions. However, after the interaction with the use of GSP he appeared to find it more useful to teach
function and limit than by using an Excel Spreadsheet. He mentioned that moving back and forth between
technology and manipulatives might be more helpful than just using one of them. He claimed that both
manipulatives and technological tools were necessary for teaching geometric transformations. However, he was
not much sure about using the geo-board in teaching geometric transformations. Only one material or
technology might not be helpful in teaching geometric transformations. Sometimes worksheet, for him, seemed
easier than GSP and other times GSP seemed far better.
The effectiveness of materials, tools, and technology in teaching fractions, functions, limits, geometric
transformations, and statistical data may depend on how a teacher engages students in learning through the use
of different teaching resources. Manipulatives provide a context for teaching mathematics for conceptual
understanding. Teaching mathematics with an appropriate use of manipulatives may help in modeling situations,
making the connection between abstraction of mathematics and the real or the physical world (Heddens, 1986 as
cited in Dahl 2011). Dahl (2011) claims that the use of manipulatives may make children’s understanding of
mathematics long lasting through understanding of concepts together with procedures. However, use of
manipulatives may increase children’s reliance on such material to make sense of mathematical concepts. They
have to think at an abstract level and solve problems using algorithms and models. Manipulatives may not help
in abstract mathematical reasoning. Then, technology can be of great support in modeling abstract mathematical
problems. Many scholars (e.g., Leathami 2002; Rubin, 1999) believe that the technology plays a very influential
role in teaching and learning mathematics in a meaningful way. The effectiveness of technology in learning
mathematics and understanding concepts and solve abstract problems depends on the pedagogical philosophy
(Rubin, 1999). While using GSP in exploring the features of a quadratic function, it is imperative to engage
students in dynamic interaction among the three coefficients a, b, and c (Rubin, 1999). Some teachers seem to
be reluctant in using technology in the classroom teaching and learning of mathematics because they think that
excessive use of technological tools may diminish the conceptual understanding and procedural fluency
(Schmidt & Challahan, 1992). The participant emphasized the interface between formal and informal
mathematics through the use of manipulatives and technology.
Balancing of teaching strategies: The participant expressed his beliefs about teaching fraction multiplication
beginning with simple fractions and whole numbers. Then, he pointed to a move from multiplication of two
simple fractions, one simple fraction and a compound fraction (mixed fraction), and two mixed fractions. He
preferred using Cuisenaire rods as a starting activity while teaching fraction multiplications. He shared with the
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researcher that JavaBars could be used either at the beginning or after use of other manipulatives. Since it is a
free downloadable application, he expressed his thought of using it even before the use of any manipulatives. It
seems that what to use in teaching of mathematics as a resource may depend on how it is used, for what purpose
it is used, and for how long. For him, it was always a balancing of teaching strategy.
While teaching functions and limits, he pointed to step size for different slope as an important thing to consider,
as an important variable. Different step size meant, to him, different values of change in x for certain values of
y, different values of y for certain x, and different values of x for different values of y. That means he positioned
his strategy that a starting point in using GSP could be from a rate of change. He preferred an initial activity
with paper and pencil and manipulatives while working with most of the mathematical problems including
fraction multiplications and functions. He then suggested the use of technology after students have some basic
skills. His beliefs on ‘asking questions’ while teaching about functions and slopes using technology showed that
he preferred working with students and interacting with them. At this point, his beliefs seemed to be positive
toward integrated teaching by bringing students of different ability groups together and using different
manipulatives and technological tools. His anticipation of balancing of teaching with different approaches
focused on class projects with construction activities and problem solving in groups with trading off among
students and helping each other. He claimed that he preferred engaging the class in tasks and situations and
helping a group of students that is struggling with the problem. While working with such students, he anticipates
beginning from basics and grouping them in different ability groups with different tasks. He focused on
collaborative learning from each other in the class by grouping of students who need additional help through
expert groups within the same class. At this point, his beliefs seem to be aligned to differentiated instruction.
Concerning the order of teaching geometric transformations, he did not have any specific choice in such order to
teach reflection, rotation, or translation. However, he clarified that he would pull out ideas from a comparison of
two similar objects and their position to begin instruction of geometric transformation. He seemed to prefer
differentiated instruction through discussion sessions within different ability groups. He claimed that he spent
some time to talk to the students about their problems at the beginning of his teaching. This talk helped him
address their problems in mathematics. After addressing students’ problems, he felt comfortable to engage all
students in group activities. He also seemed able to engage all the students in learning functions with the use of
GSP in the school computer laboratory (during his practice teaching). For him, working with students on any
problem was a balancing act between moving to and from what they already know and what they do not know.
The participant articulated a need for bridging students’ prior knowledge to new areas of teaching and learning
in mathematics. He emphasized different ways of bridging procedures and concepts through the use of
manipulatives, diagrams, and technological tools. For example, a data-table could be useful to introduce a
function. He claimed that the table could be linked with an equation and a pattern as a way to start teaching a
function. For him, the activity of finding unknowns from a pattern could help one to generalize a particular
concept of a function. Also, he related different examples of functions from everyday life to standard definition.
While doing this, he focused on the relation of a concept of fraction, function, limit, geometric transformation
and data with understanding procedures and the concepts. He expressed his strong beliefs in moving from
‘algebra to pre-calculus’ by the use of GSP while discussing functions and limits. He purported to begin the
teaching of geometric transformation from an informal pair of objects, as one is a replica of other as object and
image and differentiate them with mathematical relations and reasoning. He emphasized an order of teaching
function and transformation by building foundational knowledge at first. He claimed that the use of informal
transformation might help in understanding some basic features even before teaching of formal geometric
transformations. He suggested a transition from manipulative, graphs, and pictures to technology (e.g., GSP);
however, he seemed flexible in this order. His notion of bridging through everyday examples of various
mathematical phenomena seemed extended further to an integrated and contextual teaching of mathematics.
Integrated approach has the potential for enhancing the scope and power of mathematics teaching and learning
by bridging unknown to known and vice-versa. The debate still is ongoing about how to integrate. One idea of
integration through unifying concepts (a concept that cuts across most branches of mathematics) and integration
by merging areas of mathematics might be a good idea, but we have to pay attention toward optimal integration.
The optimal integration of curriculum and pedagogy incorporates themes and builds an integrated curriculum
around those themes. There are five essential roots for integration- quantitative literacy, mathematics
preparation, mental development, technology, and culture (Kennedy, 2003). Other roots for integration: core
curriculum root, competition root, social and political root, and cultural root are also important players in
shaping school mathematics curriculum and its integration across other disciplines. Another viable approach for
integration comes from Burns and Sattes’s (1995) evolutionary stages of curriculum integration in terms of
content, instruction, assessment, and classroom culture. Berlin (2003) emphasizes integration that should feature
real-world or mathematical problem situations that meant to be engaging, interesting, appealing, and relevant to
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students in their targeted grade levels. However, integration should not be just putting things from different
fields together, but the process of seeking connections should be a generative one – greater the number of
opportunities that students have to make connections, the more likely they will be able to search for future
connections themselves (Clement & Sowder, 2003).
Balancing of teaching strategy by providing all students equal learning opportunities is a great challenge to a
mathematics teacher. A teacher should have knowledge of integrating technology, manipulatives, and other
resources in the classroom teaching and learning environment. Integration of technology in mathematics class is
a complicated one due to cost, time, and an appropriate use. “Teaching with technology is complicated further
considering the challenges newer technologies present to teachers” (Koehler & Mishra, 2008). A teacher has to
balance his or her teaching with technology considering various factors – nature, applicability, stability, visual
and perceptual effects, both cognitive and social affordability, and flexibility. Mishra and Koehler (2006) and
Koehler and Mishra (2008) introduced the framework of technological pedagogical content knowledge
(TPACK) that integrates technological pedagogical knowledge, technological content knowledge, and
pedagogical content knowledge. This framework could be utilized in the balancing of teaching strategies.
However, having the knowledge of pedagogical knowledge, technological content knowledge, and pedagogical
content knowledge does not guarantee that balance. It is more of an art of a teacher to bring the knowledge into
action and maintain a balance between integration and differentiation of his or her instruction.
Working with technology: The participant’s view in relation to use of technological tools in teaching
mathematics seemed developmental from less use to more use and procedural to conceptual. In the beginning,
he seemed not aware of any specific technological tools for teaching fraction multiplication. When he was
introduced with the use of JavaBars, he mentioned that he would use this tool for teaching different fraction
operations even before the use of other manipulatives. He claimed that JavaBars is a free applet and also it is
easily available for download from the Internet. For him, it was an easy-to-operate tool while teaching fraction
operations in general and fraction multiplication in particular. His first preference of Excel Spreadsheet for
teaching function showed a naïve understanding of the broader aspect of the application of technological tools in
such an abstract mathematical concept. He seemed to be aware of using Excel Spreadsheet in teaching function
through tables and graphs. He revealed his beliefs about technological tools saying that any kind of technology
that could help in the construction of a pattern would be good to use in teaching mathematics. His preference to
Excel Spreadsheet for graph and table relation and the nature of a function from the graph explicated his prior
beliefs about use of technological tools in teaching a function. When the researcher introduced the use of GSP
for the concept of slope and reasoning about function with a slope in GSP, this discussion had a positive
influence in the participant’s idiosyncratic beliefs about the use of the tool for teaching a function. His initial
belief toward Excel Spreadsheet as a most viable technological tool for teaching and learning functions changed
with his experiences with GSP. Although he seemed confident in using Excel Spreadsheet for teaching a
function, after the interaction between the researcher and the participant about using GSP for teaching function
and some activities of plotting function graphs and observing the subtle changes of ∆x and ∆y in linear and nonlinear functions, the participant claimed that the GSP was more useful to teach functions than the Excel
Spreadsheet. He had a prior background of working with GSP; nonetheless, he seemed not having an
understanding of using GSP for teaching function and limit.
The participant claimed that GSP is a powerful teaching tool that can be used not only in teaching geometry, but
also in pre-calculus and trigonometry. For him, the greatest benefit to students by using GSP was conceptual
learning. He also anticipated using the tool in his future teaching beyond practicum. To him, the use of GSP in
teaching limit and derivative was more intuitive than without using it. He shared with the researcher that GSP
helped him in making sense of limit and meaning of a derivative. His anticipation of using GSP and other tools
in his future teaching related to his sense of ownership to the technology. His beliefs seemed to be consistent
with earlier research findings.
Leatham (2002, 2007) highlighted the degree of ownership of technological tools in teaching mathematics by
the four preservice mathematics teachers. He discussed the phases – entry, adoption, adaptation, appropriation,
and invention – that his research participants moved through while they were participants to his study. Not all of
the participants had these ownership levels. He found only one out of the four participants reached the level of
invention with the use of computers. Entry phase was the time when they were first introduced to the
technology in the context of teaching mathematics. Adoption phase is the one in which the preservice teachers
begin to use technology themselves and see how it functions in the context of mathematics. Adaptation is the
phase in which the preservice teachers make connections between technological tools and different
mathematical contents and teaching. Next phase, appropriation is the phase in which the preservice teachers
begin to make an appropriate choice of technological tools that fits with the teaching of content. In this phase,
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they gain more independence in the selection and use of technological tools. In the last phase, invention, they
begin to use technological tools to learn mathematics. They try to explore new technologies that are suitable for
different content teaching. They develop their own model to use the technological tools in teaching mathematics
(Leatham, 2002). In this study, the participant seemed to be confident in the use of JavaBars, GSP, and Excel
Spreadsheet in teaching fractions, functions and limits, geometric transformations, and statistical data. He
appeared to be in the appropriation phase at the end of five task-based interviews and later when he used GSP in
his classroom teaching of a quadratic function, he seemed to be in the invention phase. Hence, the task-based
interviews provided the participant an opportunity to observe his idiosyncratic beliefs about teaching
mathematics with technology and change those beliefs toward positive direction favoring toward constructivist
approach of technology integration in mathematics classes.
Technology for mathematical concepts and meanings: The participant preferred some hands-on activities due
to easiness to begin fraction multiplication or function or geometric transformation. For him, multiplication of a
fraction by a whole number was an easy step. He appeared to think that many students perform fraction
multiplication without knowing the meaning of the operation. There might be different meanings of
multiplication. Multiplication of two whole numbers can produce a result that is greater than or equal to the
larger of them. Multiplication of a fraction with a whole number or fraction by another fraction might be
different depending on the nature of the factions involved in the process. The meaning of fraction multiplication
depends on context or nature of numbers involved in the process. To clarify the meaning and operation of
fraction multiplication, he preferred using Cuisenaire rods and fraction strips at the beginning. However, he
seemed to consider that fraction multiplication is easy to understand the process and concept with JavaBars. A
major problem in understanding a fraction operation is related to understanding of what is constructed and why
it is constructed the way it is constructed either using manipulative or JavaBars. Here, construction is associated
with the use of fraction multiplication as an operator to get a result.
In relation to teaching a function, he seemed to believe that it was easy to see the slope of a line in the XYplane. He preferred introducing slopes of simple linear functions for conceptual understanding. He then claimed
that GSP is helpful in making sense of a, b and c in a linear function ax+by=c in the graph and a quadratic
function ax2+bx+c = 0. He appeared to think that students could have a hard time in understanding the concept
of function in the beginning, but it might be easy with the use of GSP. One could begin with the definition of
function and examples of different functions of everyday life. Then visualization of a function with the use of
GSP may make a better sense than just working with algebraic functions by making implicit operations explicit
with visualization. The participant also claimed that a geometric transformation could be considered as a part of
the function. The reflection, rotation, translation, and dilation could be considered as different kinds of operators
as functions. These operators are visible with technology such as GSP that helps in making sense of those
function operations. The participant claimed that students with visual learning might have a deeper
understanding of a function or limit with the use of GSP. He also accepted that procedures and concepts could
be well bridged through the use of technology. Sometimes there can be a large conceptual gap between what
they already learned and what they are going to learn. In such a case, for the participant, a teacher could
introduce simple examples of slope and the concept of slope in a line. Then he or she could use GSP for the
exploration of slope.
Leatham (2002, 2007) discussed the different roles of technological tools – motivational, procedural, and
conceptual roles. Technological tools can motivate both the teacher and students to orient into the exploration
and discussion. The tools such as GSP, JavaBars, and Excel Spreadsheet may enhance procedural computations,
constructions, and problem solving. The greatest thing that technological tools can serve to the teaching and
learning is conceptual understanding. The conceptual learning of mathematics may be achieved with
visualization, demonstration, illustration, exploration, connection, and extension (Leatham, 2002 & 2007).
While using GSP in teaching and learning of functions and limits, the idea of instantaneous change can be
extended to the idea of derivative. The participant’s beliefs seemed to be influenced by the multiple roles of
GSP in relation to conceptual understanding of functions and limits and geometric transformations by making
implicit mathematical operations explicit.
Transformative teaching activities: The participant seemed to believe that he could establish his own fraction
bar for teaching fraction multiplication. This way he wouldn’t have to rely on supplies from outside, and he
might get rid of dependence on school administration to acquire teaching materials. He claimed that he could
make fraction strips by paper cutting or drawing. Also, he appeared to think of constructing different kinds of
charts and diagrams to model different fraction multiplications. These materials seemed to supplement the
technological tools such as JavaBars. However, he anticipated using JavaBars in teaching fraction operations
International Journal of Research in Education and Science (IJRES)
41
even before the use of manipulatives. He anticipated teaching a function at different levels of abstraction
depending on the grade level.
He considered that the best place to start teaching function is to discuss it with a pattern and do some
construction. The pattern of numbers and construction of different charts was valuable lessons during his
teaching. He seemed to consider that the use of box and whisker plot was helpful to see many things at a time
for interpretation of data. This kind of chart was helpful in conceptual learning of central tendency and
dispersion at a time.
He appeared to believe that his teaching at the beginning and by the end of practicum was different because of
an increased level of confidence, group work with students, and overall progress made by the students by the
end of his teaching practice. He anticipated focusing on conceptual learning with a vision toward improved
learning through students’ active participation. He seemed very positive about the use of technological tools
(e.g., GSP, JavaBars, Excel Spreadsheets and others) in future teaching of fraction, function and limit, geometric
transformation, and data. His transition from limit to derivative through the use of the GSP was quite
encouraging and one of the most remarkable changes in his idiosyncratic beliefs about use of technology for
teaching mathematics. He anticipated creating a better teaching and learning environment. For this process, he
wished to talk to the students and try to understand where they got problems. He interacted with the students in
relation to their role in classroom and assignments. He mentioned that the first talk was not much successful, but
the second talk was a little more successful than the first one. The third talk turned into a very successful in the
sense that he was able to see the students’ following his advice. He felt very contented with the transformation
of students’ attitudes in the class within the short period during his practice teaching in a school. The
transformation of students gave him a sense of accomplishment. This change portrays that teaching as a nonlinear process of dealing with content and context bringing the students into pedagogy of caring (Hackenberg,
2010).
The transformative teaching and learning of mathematics with technology can be achieved through
understanding of teacher responsibility, student responsibility, and the goal of teaching and learning of
mathematics with technology (Leatham, 2002). The shift in teacher’s roles and responsibilities together with
students’ roles and responsibilities leads to a transformation. This shift should be a positive one, and it should be
linked to a greater motivation, enthusiasm, and change of beliefs and actions. The preservice teachers’ adoption
of the technological tools as early, middle, late, interested adopters and non-adopters may influence the degree
of transformation in their beliefs and practices of using the tools in their future teaching (Keren-Kolb, 2010).
The participant’s anticipated adoption of the technological tools (i.e. JavaBars, GSP and Excel Spreadsheet) in
teaching certain contents seems to be at the level of interested adopter. He thinks that he will use technological
tools in his future teaching of mathematics to transform his teaching and students’ learning focusing both
procedures and concepts.
Some issues and challenges: The participant considered that there were misconceptions about multiplication of
fractions. General misconception may arise from students’ beliefs that multiplication makes the product bigger
than the multiplicands. In the case of fraction multiplication, the case might not be the same. When a whole
number is multiplied by a fraction less than one, then the product is less than the number. Likewise, he seemed
to believe that there might be a misconception about the use of technology. This misconception might arise from
the use of technology as an alternative to manipulatives. However, he considered that a technology should not
replace any manipulatives and other hands-on activities. They should complement each other. Diagnosing a
misconception and correcting such misconception about fraction operations, functions and limits,
transformations, and data were important considerations for teaching mathematics. He also seemed to believe
that there might be some challenges associated with the use of GSP at the beginning because students might not
be familiar with computer or functions of the GSP tools. Such challenges depended on contexts in the class and
students’ prior experiences with technology. Sometimes students might engage in off-task activities in computer
while solving a problem using the GSP. Students might not pay attention to the constructive use of GSP, but
they might spend time on other stuffs beyond what they were expected to do. This issue might be a problem
according to the participant.
Some students even might struggle with the meaning of the function and limit. This kind of problem might arise
due to the past experiences with procedural learning. The participant claimed that a teacher should be able to
identify these problems in the class. These problems might be related to lack of student preparedness, lack of
motivation, and lack of homework turning in on time. He faced these problems at the beginning of his teaching
during practicum, which greatly surprised him when students did not work on assignments. For him, these were
some of the great challenges for the preservice teachers when they go for practice teaching. He thought that
42
Belbase
there were more such problems in bigger towns than in smaller cities. More problems in the bigger cities might
be due to wider cultural variations and family background of students. At the beginning of his teaching, this
kind experience was a shock for him. This kind of issue was not just specific to his class. Those problems might
be related to many factors including access to technology in schools and cost associated with it in terms of time
and resources and student motivation in education as a whole, not just mathematics.
One of the greatest challenges to present day teaching and learning of mathematics with technology is
underutilization of available technology in schools (Cuban, 2002). The participant’s frustration at the beginning
of teaching grows further with the problem of underutilization of the computer lab in the school. He requested
the school administration for installing GSP on the computers, but it was not materialized until he bought GSP
package himself and installed it in some of the computers in the lab and his classroom. Another challenge is that
many preservice and inservice mathematics teachers use everyday technology for the personal use, but they do
not use any of them for classroom teaching (Keren-Kolb, 2010). This problem also indicates toward the
challenge of teacher resistance toward the use of technological tools in teaching mathematics. “The one
challenge that is not easily addressed by the introduction of everyday and other technologies, however, is the
most difficult: educators’ reluctance to meaningfully integrate new technology tools into school learning”
(Keren-Kolb, 2010, p. 12, this author added the italicized part). Although the participant did not explicitly
indicate toward these challenges, his implicit beliefs about issues of technological tools and challenges of using
those tools in the teaching of mathematics revealed some of these elements. More of his beliefs were related to
challenges of enhancing conceptual learning, proper use of technology, and students’ attitudinal concerns.
Implications of the study
Some implications of this study were synthesized from the categorical findings and discussions - interface
between formal and informal mathematics, trade-off between differentiated and integrated teaching, forming
beliefs with learning to use tools, making implicit operations explicit, and a nonlinear process of understanding
pedagogy through practice.
Interface between formal and informal mathematics: The first category on ‘manipulatives and technological
tools’ show an interface between formal and informal mathematics. Technological tools (i.e. digital tools such
as JavaBars, GSP and Excel Spreadsheet) could provide a context to bridge formal and informal nature of
mathematics teaching and learning. Technology integration in mathematics education may develop positive
beliefs toward the use of different digital tools together with manipulatives in teaching fractions, functions and
limits, and geometric transformations. Some mathematical concepts could be taught with Cuisenaire rods,
fraction strips, a meter scale, pattern blocks, base ten blocks and dices and other concepts could be instructed
with JavaBars, Excel Spreadsheet, and GSP. In fact, any mathematics concept could be taught using both kinds
of materials (manipulatives and digital tools) because these tools function as an interface between formal and
informal mathematics.
Interface between differentiated and integrated teaching: The second categorical finding on ‘balancing of
teaching strategies’ points to possible trade-off between differentiated and integrated teaching of mathematics.
Differentiated teaching is related to treating different students differently based on their ability to learn and
make progress on their own pace. The teacher may form a group of students based on their learning ability in the
class. Then he or she may provide them different tasks or problems to solve. Some may use manipulatives, and
others may use technological tools in the process of learning any mathematical concept. The teacher can apply
different approaches focusing on class projects with construction activities and problem solving in groups with
trading off among students and helping each other. He or she can engage the class in tasks and situations and
help a group of students that is struggling with the problem. Integrated teaching can connect students’ prior
knowledge to new areas of teaching and learning in mathematics. He or she may emphasize different ways of
bridging procedures and concepts through the use of manipulatives, diagrams, and technological tools. The
process of integration may go through unifying concepts by merging areas of mathematics with integration of
curriculum and pedagogy. Hence, use of technological tools (e.g., JavaBars, Excel Spreadsheet and GSP) may
provide an interface between differentiated and integrated teaching of mathematics.
Forming beliefs with learning to use tools: The third categorical finding ‘working with technology’ relates to
forming beliefs with learning to use tools. This study showed that when the participant learned to use the tools
(e.g., JavaBars and GSP) for the first time, the experience formed idiosyncratic beliefs about the integration of
tools in teaching mathematics. The preservice teachers may have limited experience of using technological tools
in their prior mathematics classes. When they come to experience new tools and techniques integrated with
International Journal of Research in Education and Science (IJRES)
43
mathematics concepts that themselves struggled through before the use of technological tools, a new experience
challenges their earlier conceptual understanding and subsequently may challenge their existing beliefs. Hence,
positive experience of learning to use new technological tools integrated with mathematics may help the
preservice or inservice teachers to form new beliefs or modify their existing beliefs about teaching mathematics.
Power of technological tools to make implicit meanings explicit may empower teachers and help them in
forming new beliefs.
Making implicit operations explicit: The fifth categorical finding ‘technology for mathematical concepts and
meanings’ is associated with making implicit mathematical operations explicit. The use of JavaBars for fraction
operations, GSP for operations of geometric transformations and functions, and Spreadsheet in the operation of
central tendencies and dispersions provided the participant and the researcher to make the inherent mathematical
operations explicit through constructions of models. Mathematical processes related to fraction multiplications,
limits and functions, and geometric transformations had operator roles that were invisible (implicit), but the use
of the digital tools (e.g. JavaBars and GSP) revealed what was happening in those operations by making them
explicit. These were the ‘aha’ moments for both the researcher and the participant when they were able to
experience the essences of those operations within the dynamic environment of the technological tools. Hence,
making implicit operations explicit could have a great significance of technology integration in mathematics
teaching.
A nonlinear process of understanding pedagogy through practice: The sixth and the seventh categorical
findings ‘transformative teaching activities and some issues and challenges’ are associated with a nonlinear
process of understanding pedagogy through practice. Preservice mathematics teachers have a limited
opportunity for practical pedagogy of mathematics when they are in content and method courses. The content
courses in mathematics provide them knowledge of mathematical contents and the courses of teaching
mathematics provide them knowledge of mathematical pedagogy, technology, and theory. Their knowledge of
content and pedagogy through these courses seem to be ineffective in forming and changing their beliefs.
However, when they go for practice teaching, they have an opportunity to implement their knowledge of
content, pedagogy, and technology. The field experience may provide them real time experience of classroom
complexities dealing with students of different interest, nature, and motivation. Sometimes, their expectations
do not meet in the classes, and they may go through situations of frustrations and dissatisfaction from their
teaching. Their self-reflection on teaching learning process in the complexities may help them understand the
non-linearity of theory and practice of using technological tools. They may understand that building a positive
and a caring relationship is the first thing before even use of any sophisticated tools for teaching mathematics.
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International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
Gender Differences in Mathematics Achievement and Retention Scores: A
Case of Problem-Based Learning Method
1
John T. Ajai1*, Benjamin I. Imoko2
Taraba State University, Nigeria, 2 Benue State University, Nigeria
Abstract
This study was undertaken to assess gender differences in mathematics achievement and retention by using
Problem-Based Learning (PBL). The design of the study was pre–posttest quasi-experimental. Four hundred and
twenty eight senior secondary one (SS I) students using multistage sampling from ten grant-aided and
government schools were involved in the study. Two hundred and sixty one male students and one hundred and
sixty seven female students were taught algebra using PBL method of instruction. Algebra Achievement Test
(AAT) constructed by the researchers was the main instrument used for data collection. Two hypotheses were
raised for the study and tested using t-test at .05 level of significance. The study revealed that male and female
students taught algebra using PBL did not significantly differ in achievement and retention scores, thereby
revealing that male and female students are capable of competing and collaborating in mathematics. In addition,
this finding showed that performance is a function of orientation, not gender. The studies recommend the use of
PBL by mathematics teachers to overcome the male image of mathematics and enhance students’ (male and
female) achievement and retention.
Key words: Problem-based learning; Gender; Achievement; Retention; Mathematics; Algebra
Introduction
Many people do not know that mathematics is more than what is taught at school, and different from what most
people think it is. The students have a wrong image of mathematics- that mathematics is many formulae to
learn, without knowing why; mathematics is a never changing, not lively subject; something for nerds and
loners, and thus, maybe, also something for boys and men and not for girls and women. Gender is a set of
characteristics distinguishing between male and female, particularly in the cases of men and women. Depending
on the context, the discriminating characteristics vary from sex to social role to gender identity. Gender
differences in mathematics achievement and ability has remained a source of concern as scientists seek to
address the under-representation of women at the highest levels of mathematics, physical sciences and
engineering (Asante, 2010).
Literature Review
Literature about gender and academic performance in mathematics exist with different views and findings.
Studies conducted in countries of the North have shown that boys performed better than girls in mathematics
(Fennema, 2000; Kaiser-Messmer, 1994; Muthukrishna, 2010). Asante (2010) cited studies (Fox, Brody &
Tobin, 1980; Hedges & Nowell, 1995; Peterson & Fennema, 1985; Randhawa, 1994) showing that boys
generally achieved higher than girls on standardized math tests. However, an interesting body of international
literature suggests that female students perform better than male students (Arnot, David & Weiner 1999; Hydea
& Mertzb, 2009). A large scale study in the U.S.A. by Hydea & Mertzb (2009) revealed that girls have reached
parity with boys in mathematics performance, including at high school where a gap existed in earlier decades.
They affirmed that girls are doing better than boys even for tasks that require complex problem solving.
The Second Southern and Eastern Africa Consortium for Monitoring Education Quality (SACMEQ) Survey
(2000-2002) by International Institute for Educational Planning (HEP)-UNESCO (2004) shows no significant
gender differences among students in South Africa. The same study shows that girls scored significantly higher
than boys only in Seychelles. On the other hand, in Tanzania, Kenya, Mozambique, Zanzibar and Malawi, boys
scored significantly higher than girls did. In the other school systems, including the ones in South Africa, the
differences were not significant.
*
Corresponding Author: John T. Ajai, jtajai@gmail.com
46
Ajai & Imoko
An alternate body of research has shown that the gender differences in mathematical performance are
diminishing (Frost, Hyde, & Fennema, 1994; Hyde, Fennema & Lamon, 1990). Perie, Moran, and Lutkus
(2005) found that the gap has been narrowing in the United States of America. Research in Australia indicates
that gender differences in mathematics achievement are reducing and shifting (Forgasz, Leder, & Vale, 2000).
Vale (2009) found that many studies conducted between 2000 and 2004 in Australasia showed no significant
differences in achievement in mathematics between male and female students, though males were more likely to
obtain higher mean scores.
Internationally, researchers have undertaken studies in various contexts to examine factors that influence
gendered achievement in mathematics. Many of such studies have focused on factors related to differences in
the performance of boys and girls in mathematics (Abiam & Odok, 2006; Mahlomaholo & Sematle, 2005;
Opolot-Okurut, 2005; Zhu, 2007). Feminist researchers have tried to make meaning of the experiences of girls
and boys in the mathematics classrooms, and to interpret male-female power relations (Jungwirth, 1991; Waiden
& Walkerdine, 1985). Their findings revealed that girls are often marginalized and given subordinate status in
the mathematics class. The findings suggest that perceptions of teachers are that girls’ performances in
mathematics are dependent on rote learning, hard work and perseverance rather than natural talent, flexibility
and risk taking which are the learning styles of boys.
Gender differences in mathematics teaching, learning and achievement have also been explained on the basis of
gender differences in cognition and brain lateralization (Fennema & Leder, 1990). In a similar argument,
Paechter (1998) argues that male and female students do experience the world in different ways. Firstly, they are
differently positioned in society. The second is their different learning styles and how they perceive and process
reality. These researchers emphasize that most mathematics classroom discourse is organized to accommodate
male learning patterns, hence their high achievement in mathematics. Mutemeri and Mygweni (2005) argue that
the idea that mathematics is for boys may result in low motivation in girls and could widen the gender gap in
mathematics achievement in favor of boys.
Boaler (1997) is of the view that the different learning goals of girls and boys leave girls at a disadvantage in
competitive environments. Boys and girls preferred a mathematics curriculum that enabled them to work at their
own pace as their reasoning was different. Girls value experiences that allow them to think and develop their
own ideas, as their aim is to gain understanding. Boys, on the other hand, emphasize speed and accuracy and see
these as indicators of success. Boys are able to function well in a competitive environment of textbook based
mathematics learning.
Other important factors that emerge in research on gender and mathematics are cultural, family influences,
socio-economic status of parents, as well as cultural and traditional influences (Kaino & Salani, 2004). Asante
(2010) citing Collins, Kenway and McLeod (2000) argued that schools establish symbolic oppositions between
male and female students through gendering of knowledge and defining of certain subjects as masculine. In
contrast, female students are conditioned in the society to believe that mathematics is a male subject, and it is
acceptable for them to drop it. Studies done in Botswana by Finn (1980), Duncan (1989), and Marope (1992)
cited in Kaino (2001) indicated that cultural expectations of society could result in differences in performance
between girls and boys in mathematics. In Nigeria, it has been argued that nurture entrenches male dominance
over the female gender (Bassey, Joshua & Asim, 2007).
The above review suggests that many factors may be associated with the gender gap, including issues such as
classroom interactions, students’ attitudes, students’ interest and self-esteem, teachers’ gendered attitudes,
curricular materials, beliefs, social and cultural norms. These differences put together have implications for the
kind of instructional procedures that are to be adopted for setting up an appropriate teaching and learning
environment for mathematics instruction that is suitable for both genders. The choice of gender as variable for
this study is predicated on the current world trend and research emphasis on gender issues. The millennium
declaration of September 2000 (United Nations, 2000) has as its goals the promotion of gender equity, the
empowerment of women and the elimination of gender inequality in basic and secondary education and at all
levels by 2015. Mathematics is a science subject and some gender-based science researchers (Howes, 2002;
Sinnes, 2006) have reported that females in principle will produce exactly the same scientific knowledge as
males, if sufficient rigor is undertaken in scientific inquiry. Though the issue of gender inequality in science,
technology, and mathematics education (STME) is global, it is believed that bridging gender gap is one major
way of achieving egalitarianism and enhancing human development. There is need therefore to give boys and
girls exactly the same opportunities and challenges. It is in the light of the above that this study used problembased learning (PBL) to explore the issue of a gender difference in academic achievement and retention at
secondary school level in Benue State of Nigeria.
International Journal of Research in Education and Science (IJRES)
47
Problem-Based Learning (PBL)
PBL is a term used for a range of pedagogical approaches that encourage students to learn through the structured
exploration of a research problem. It is an active learning strategy which enables the student to become aware
of and determine his/her problem solving ability and learning needs, to be able to make knowledge operative
and to perform group works “in the face of real life problems” (Akınoğlu & Tandoğan 2007). The PBL method
requires students to become responsible for their own learning. The PBL teacher is a facilitator of student
learning, and his/her interventions diminish as students progressively take on responsibility for their own
learning processes. This method is characteristically carried out in small, facilitated groups and takes advantage
of the social aspect of learning through discussion, problem solving, and study with peers (Hmelo-Silver, 2004).
The facilitator guides students in the learning process, pushing them to think deeply, and models the kinds of
questions that students need to be asking themselves, thus forming a cognitive apprenticeship.
In PBL, class activities are constructed around a problem or problems. The instructor no long lectures. Instead,
when the instructor integrates PBL into the course, students are empowered to take a responsible role in their
learning. The instructor is not the authoritative source of information and knowledge. Students have to take the
initiatives to inquire and learn; and the instructor only guide, probe and support students’ initiatives. What
students learn during their self-directed learning is applied back to the problem with re-analysis and resolution.
Problems are used as a stimulus for students to start the learning process. The students reason through the
problem and find out what they already knew and what they should know in order to solve the problem. It is
through this active and reflective thinking process that students become responsible for their own learning.
Through the application of their knowledge to the problem, the students test and integrate what they are
learning. In general, PBL aims to motivate students to participate in the learning process and to help foster
problem solving skills.
The Key Research Question
Is there a gender gap in mathematics achievement and retention? The study tested the following hypotheses at
.05 level of significance.
1.
2.
Male and female students taught algebra using problem-based learning (PBL) approach do not differ
significantly in their achievement mean scores.
Male and female students taught algebra using problem-based learning (PBL) approach do not differ
significantly in their retention mean scores.
Method
The pre-test and post-test quasi-experimental design was adopted for the study. The sample consists of 261 male
and 167 female senior secondary school one (SS1) students in 10 secondary schools across education zone B of
Benue State of Nigeria. The students and schools were selected through multistage sampling. The students were
taught algebra by using problem-based learning method. Algebra Achievement Test (SAAT), constructed by the
researcher was used to collect data. AAT which was validated by experts in mathematics and science education
has 25 multiple choices (each with four options) and 7 essay items constructed from SS1 mathematics
curriculum and was scored out of 100 marks. Using Kuder–Richardson (KR – 21) formula, the internal
consistency of the multiple-choice items was found to be 0.80. Similarly, an inter-rater index of 0.863 was
obtained to test the internal consistency of the essay items using Kendall’s coefficient of concordance. The inter
rater scores were further tested for significance of the mean scores difference using paired-samples t-test
analysis. This indicated no significant difference between rater 1 (M=48.87, SD=13.09) and rater 2 (M=48.66,
SD=12.957), t (60)=1.82, p=.074.
Problem-based learning lessons based on the SS2 mathematics curriculum were with topics in change of subject
formulae, substitution in formulae, direct, inverse, joint and partial variations, factorization, formation and word
problems in quadratic equations were used. Mathematics teachers of three years’ experience and above were
trained by the researcher to facilitate the problem-based learning. The training exercise was based on the
purpose of the study, the topics to be taught, the use of the lesson plans, the use of the AAT as well as general
conduct of the study. It was ensured that all the teachers used equal length of time (four weeks) to facilitate
learning of the topics. Throughout the exercise, the researcher went round to supervise and ensure smooth
learning in all classes. Independent t-test was used to test the hypotheses at 0.05 level of significance.
48
Ajai & Imoko
Results
As seen from Table 1, the post-test achievement mean score of the female students is 47.03 with standard
deviation of 8.91, while that of the male is 45.62 with standard deviation of 9.68. The difference between the
pre-test and post-test mean scores of the male students is 28.30 and that of the female students is 28.78. The
difference between post-test mean scores of male and female students is 1.41. Although this difference is in
favor of female students, the difference between the pre-test and post-test of those is not quite much. The
implication is that there is not much difference between the achievement scores of male and female students.
Table 1. Achievement mean scores, standard deviations and t-test value of subjects for gender in PBL
Gender
N
Pre-Test
Post-test
Mean
t
df
p
̅
̅
𝛿
𝛿
𝑋
𝑋
Female
167 18.25 3.46 47.03 8.91
28.78
1.553
426
0.121
Male
261 17.32 4.34 45.62 9.68
28.30
Mean Diff.
0.93
1.41
Gender factor is not significant. This is evident from Table 1 with t(426)=1.553, p=.121. The implication is that
there is no significant difference between the achievement mean scores of male and female students taught
algebra using PBL strategy. Thus, the hypothesis of no significant difference is not rejected.
As shown in Table 2, the female students had a retention mean score of 50.72 with standard deviation of 8.25,
while the male students had 50.29 with standard deviation scores of 9.68. The difference between the retention
mean scores and post-test of the female students is 3.69, while the difference for male students is 4.67. The
difference between the retention mean scores of the male and female students is 0.43 and it is in favor of the
female students. This shows that the female students retained knowledge just a little more than male students.
Table 2. Retention mean scores, standard deviations and t-test value of subjects for gender in PBL
Gender
N
Retention-test
Post-test
Mean
t
df
p
̅
̅
𝛿
𝛿
𝑋
𝑋
Female
167
47.03
8.91 50.72 8.25
3.69
1.553
426
0.121
Male
261
45.62
9.68 50.29 9.68
4.67
Mean Diff.
1.41
0.43
The t-ratio for gender factor as shown in Table 2 is 0.508 which is not significant (p= .611). This indicates that
there is no significant difference between the retention mean scores of male and female students taught algebra
using PBL. Thus, the hypothesis of no significant difference between the retention mean scores of male and
female students is not rejected.
Discussion and Implications
The study found that female students outperformed their male counterparts (both post-test and retention), though
the difference is not statistically significant. The findings of this study is consistent with the one in the United
States of America by Hydea and Mertzb (2009) which says girls have reached parity with boys in mathematics.
This is an indication that girls can do better than boys in tasks that require complex problems, such as PBL,
which the present study used. This finding is also in agreement with the narrow gender gap in achievement in
U.S.A (Perie, Moran & Luktus, 2005) and in Australia (Forgasz, Leder & Vale, 2000). This is however at
variance with Ogunkunle (2007), in Nigeria, where part of the findings established significant difference in
favor of males and another part in favor of the females. It also disagrees with earlier studies of Fennema (2000)
and Asante (2010) which showed significant gender differences.
The reason for the equal performance of male and female students may not be unconnected with the fact that
both see themselves as equals and capable of competing and collaborating in classroom activities. This
affirmation is opposed to the views of Fennema and Leder (1990) that gender difference in mathematics is based
International Journal of Research in Education and Science (IJRES)
49
on gender differences in cognition and brain lateralization. The approach of PBL to learning and teaching
showed that performance in mathematics is a function of orientation rather than gender.
Conclusion and Recommendations
It is evident from this study that students’ achievement and retention in algebra are not dependent on gender, but
function of method. Both sexes are capable of competing and collaborating in classroom activities. It must be
stressed that this was a case study. Further research would need to be undertaken to examine the trends that
emerged in this study in greater depth. A sounder approach would be to examine situational factors that may be
influencing gender differences, for example, classroom cultures, teacher attitudes, parental and teacher attitudes
and others.
The study suggests that there is a need to give boys and girls exactly the same opportunities and challenges in
the mathematics class. PBL should therefore be used as an additional teaching strategy to other traditional
methods of teaching mathematics. Male and female students need to compete, collaborate and gain from one
another in mathematics teaching and learning. Thus, mathematics teachers are enjoined to use PBL as an
instructional approach that will foster greater healthy rivalry in mathematics instruction.
Teacher professional development programs should make more concerted efforts to advise teachers about the
ways in which to approach the teaching of mathematics to avoid disadvantaging particular groups of girls or
boys. Mathematics teaching and evaluation strategies should be gender bias-free. This way, males and females
will tend to see themselves as equals capable of competing and collaborating in classroom activities. Male and
female teachers should work jointly with boys and girls and adopt a more socially just and inclusive approach to
creating equal opportunities for all students.
If this method, proposed by this study, is adopted in mathematics teaching and learning, it will boost the
performance of students in skills acquisition, problem solving ability and development of the right type of
attitude toward mathematics as a subject. Guidance and counselling machineries in the school should be
energized to encourage more female students’ active participation in effective mathematics learning. Female
students should be informed that mathematics could be studied and passed just like any other subjects and that
the subject is an essential tool, a prerequisite for further education in a host of vocations.
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International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
Non-mathematics Students’ Reasoning in Calculus Tasks
Ljerka Jukić Matić*
University of Osijek, Croatia
Abstract
This paper investigates the reasoning of first year non-mathematics students in non-routine calculus tasks. The
students in this study were accustomed to imitative reasoning from their primary and secondary education. In
order to move from imitative reasoning toward more creative reasoning, non-routine tasks were implemented as
an explicit part of the students’ calculus course. We examined the reasoning of six students in the middle of the
calculus course and at the end of the course. The analyzed data showed that the students’ reasoning differed in
the middle of the course and after having passed the course, in terms of having more characteristics of creative
reasoning. In addition, we found several negative met-befores and met-afters affecting the students’ knowledge
and interfering with their reasoning.
Key words: Reasoning; Non-mathematics students; Calculus; Non-routine tasks
Introduction
The tasks that are given to students in a mathematics course can be categorized as routine and non-routine tasks.
The main difference between a non-routine and a routine task is that in a non-routine task the solver has to, at
least partially, construct his/her solution method, while in a routine task, the method is already known by the
solver or provided by an external source such as the book or the teacher (Lithner, 2012). We can ask ourselves
what competencies students in tertiary education are developing when they are given the same types of tasks in
exams as they have met on their mathematics courses. When it comes to exam requirements, students do expect
this kind of situation to occur. Their expectations come from the previous exam papers accessible to them, and it
is usually part of a didactical contract between students and lecturers (ibid.). The reasoning employed in such
situations Lithner (2003) calls imitative reasoning. This kind of reasoning is founded on copying task solutions,
for example by looking at a textbook example or remembering an algorithm. Examining final exams from the
introductory calculus courses at Swedish universities, Bergqvist (2007) found that most of the tasks can be
solved using imitative reasoning. Teachers were concerned with the exam pass rate; therefore, the majority
demanded imitative reasoning (Bergqvist, 2012). The situation is no different in Croatian universities. There are
no published studies, but browsing through exams and course material accessible on the web pages of various
Croatian universities, one can reach a similar conclusion, especially when it comes to calculus courses for nonmathematics students.
Lithner (2008) points out that even after many years of research, students still perform inefficient rote thinking
and rely on imitative reasoning. The main problem with such reasoning is that students do not develop the
conceptual knowledge necessary for learning different aspects of mathematics (Lithner, 2004). Cox (1994)
argued that many first-year university students obtain good grades by concentrating on routine topics, instead of
aiming at a deep understanding of fundamental facts. While many teachers are trying to reduce the complexity
of mathematical concepts and processes, students are trying to cope with curriculum goals so they often use
quicker short-cut strategies to learning and passing exams (Schoenfeld, 1991).
The research conducted by Glasnović Gracin (2011) showed that teachers at primary and secondary educational
levels in Croatia rely heavily on textbooks especially for practicing taught subject matter. In an analysis of
textbook exercises, Glasnović Gracin (2011) showed a predominance of operation activities on the reproduction
or simple-connections level i.e. dominance of routine tasks. These findings showed that only imitative reasoning
is necessary for solving mathematics tasks at lower educational levels. In this study, we examined the reasoning
of several non-mathematics students who have been exposed to non-routine calculus tasks on a traditional
calculus course. A detailed description of the course can be found in the methodology section. The purpose of
the study was to examine what characterizes students’ reasoning when faced with a problematic situation.
*
Corresponding Author: Ljerka Jukić Matić, ljukic@mathos.hr
52
Matić
Theoretical Framework
In this section we will describe and discuss the theoretical frameworks for our study and analysis, and present
our research questions.
The Reasoning Framework
In mathematics education literature, many definitions of the term reasoning can be found. Lithner (2008)
defined reasoning as the line of thought that is adopted to produce assertions and to reach conclusions when
solving tasks. It is not necessarily based on formal logic, nor restricted to proof. It may even be incorrect as long
as there are some sensible reasons (to the reasoner) supporting it. Lithner (ibid.) differentiates the thinking
process from the product of that process. The product of the thinking processes, the way of understanding, can
be observed as behavior, but conclusions made about the cognitive processes involved in that behavior, would
still be to some degree, speculative. In this study, mathematical reasoning is a mental act and the purpose is to
investigate the students’ way of thinking.
There are two types of mathematical reasoning: imitative and creative reasoning (Lithner, ibid.). Everything that
includes rote- learning reasoning is, in fact, imitative reasoning, and the opposite reasoning is creative
reasoning. The basic idea of creative reasoning or creative mathematically founded reasoning is the creation of
new and well-founded task solutions. Creative mathematically founded reasoning fulfills the following criteria:
novelty, plausibility, flexibility and mathematical foundation. Novelty includes new reasoning sequence that is
created or recreated if forgotten. Plausibility can be described as using arguments to support the strategy choice
and/or strategy implementation. The choice of strategy can be supported by predictive arguments i.e. will the
strategy solve the difficulty, and strategy implementation can be supported by verificative argumentation i.e. did
the strategy solve the difficulty. Flexibility admits different approaches and adaptations to the situation and it
does not suffer from fixation that hinders the progress. Mathematical foundation means that the argumentation is
founded on the intrinsic mathematical properties of the component involved in the reasoning. Creative reasoning
does not have to be a challenge in terms of problem-solving, but conceptual understanding is deeply anchored in
it, unlike in imitative reasoning. This means that the construction and anchored argumentation in creative
reasoning is impossible to do without considering the meaning of the components involved in the reasoning
(Lithner, ibid).
Imitative reasoning is a term that describes several different types of superficial reasoning. In Memorized
Reasoning (MR), the strategy choice is founded on recalling an answer and the strategy implementation consists
of writing this answer down with no further consideration. Algorithmic Reasoning (AR) is implemented when
the strategy choice involves recalling a certain algorithm (set of rules) for solving a given problem. The strategy
implementation is trivial, straightforward once the rules are recalled. AR has several variants: familiar
algorithmic reasoning, delimiting algorithmic reasoning and guided algorithmic reasoning. The strategy choice
in Familiar AR is founded on recognizing the task as being familiar, which can be solved by a corresponding
known algorithm. In Delimiting AR, an algorithm is chosen from a set that is delimited through surface relation
with the task. Following the algorithm carries out the implementation of the strategy. If this implementation
does not produce the desired outcome, the algorithm is abandoned, and a new one is chosen. In Guided AR, the
reasoning is mainly guided by two types of sources that are external to the task. In person-guided AR, a teacher
pilots the student’s solution. Within text-guided AR, in the task to be solved, the strategy choice is founded on
identifying similar surface properties to those in the text source (e.g. a textbook).
Met-befores and Met-afters
A met-before is a mental construct that a person uses at a given time based on prior experiences (Tall, 2006).
Using met-befores can sometimes be an advantage when the person is learning a new mathematical concept, and
sometimes it can be an obstacle that causes severe difficulties. Hence, met-befores affect the learning of new
concepts, but new mathematical concepts may also affect older knowledge. Such mental constructs are called
met-afters (Lima & Tall, 2008). Met-afters are those experiences met at a later time that affect the retention of
old knowledge. Met-afters can also be both positive and negative; the negative effect of a met-after can be an
indication of the fragility or inconsistency of previously learned knowledge.
New knowledge that builds on previous knowledge is much better remembered, but concepts that do not fit into
earlier experience are learned temporarily and easily forgotten or not learned at all. According to McGowen and
Tall (2010), this can be observed when a student, for instance, is interested only in algorithmic reasoning,
relying on well-established procedures or algorithms. If there is no conceptual meaning, this kind of knowledge
International Journal of Research in Education and Science (IJRES)
53
is stored improperly and is very fragile when the person tries to adapt it to a new situation. This previous
knowledge makes it difficult to understand new subject matter. The student is trying to distinguish among
accessible rules and is trying to imbed new knowledge into his fragmented knowledge structure.
Research Question
Hiebert (2003) argues that students learn when they are given an opportunity to learn, thus we expected that, in
the long run, the non-routine tasks incorporated in the traditional calculus course would have a positive
influence on students’ reasoning. Therefore, we formed the following research questions: How do average nonmathematics students reason when given non-routine calculus tasks? What characterizes non-mathematics
students’ reasoning in the middle of the calculus course and after passing the calculus course?
The research reported here should be seen as exploratory, and the conclusions are of relevance to the Croatian
context and to other similar university contexts.
Methodology
Participants and Context
This study was conducted at one university in Croatia, and the participants were first-year civil engineering
students. In order to move from the “vicious circle”, where university calculus courses promote procedural
knowledge and imitative reasoning (Tall, 1997), non-routine tasks were implemented in the calculus course for
civil engineering students. At this university, calculus courses are designed according to the needs of the
specific study program. This means that calculus courses are not shared but each study program has its own
calculus course. Selden, Mason & Selden (1998) suggested that the non-routine tasks should be implemented as
an explicit part of the curriculum in traditional calculus courses, not at the end, but throughout the course in the
exercises sessions or in the homework. This group of civil engineering students was given several non-routine
tasks for homework in each exercise session. The students had to hand in the homework to the teaching assistant
in the following exercise session as evidence that they actually were solving it, but the solution did not have to
be correct. Sometimes the homework tasks were solved in the next exercise session, and sometimes the solution
was only commented on. The routine calculus tasks, which were given to the students in the course, required the
application of some procedure that was shown to them either by the lecturer or teaching assistant, while the nonroutine tasks were more oriented toward conceptual understanding. For instance, the routine tasks that the
students solved in exercise sessions required the evaluation of function limit at some point e.g. “Find
lim𝑥→1
1−𝑥 3
1−𝑥
”, while the corresponding non-routine task given for homework had the following form “Is there
an a such that lim𝑥→3
𝑥 2 +𝑥−𝑎𝑥−𝑎+4
𝑥 2 −2𝑥−3
exists? Explain your answer.”
In order to pass the calculus course, the students in this study program have to pass both a written exam and an
oral exam. Moreover, the written exam consists of a mid-term exam and a final exam. In the written exam
students have to solve various tasks, the oral exam has an emphasis on mathematical theory. The participants for
this study were chosen according to their scores obtained in the calculus mid-term exam. They scored around
70% in the mid-term exam, and we would classify them as average students, i.e. students who possessed some
knowledge, were far from failing the exam, but also were not close to excellent scores. Their scores represent
the most common results in the calculus exam among civil engineering students.
Method
The empirical data was collected from six task-based interviews. The students were interviewed in pairs. Arksey
& Knight (1999) argue that this method is better for establishing an atmosphere of confidence with two students
being interviewed at the same time and because the interviewees may ‘fill in gaps’ for each other. Also, the
students’ interaction may be of interest. Schoenfeld (1985) gives support for this kind of interview, stating that
two-person protocols often provide better insight into and information about students’ reasoning and knowledge.
The students were interviewed on two occasions: in the middle of the course, just after the mid-term exam, and
after passing the course. In the interview, the participants were given specific tasks designed in collaboration
with the course’s lecturer and teaching assistant. At the start of the interview, the participants were instructed to
talk to each other when solving the task, to say out loud what they are thinking at that moment, not to plan what
to say, and to behaves though they are alone in the room working together on their homework or an assignment.
Similar directions were recommended by Ericsson & Simon (1993) in order to encourage the students to think
54
Matić
out loud. During the first minutes of the interview, the interviewer just reminded the participants to keep talking
if they were silent for a while. If the students struggled with the given tasks for more than several minutes, the
interviewer asked direct questions to try to get the students to explain what they were doing and why they were
doing it.
Participation in the study was voluntary and the students had the right to withdraw from the study at any time,
we believe therefore that the students invested a significant amount of effort into solving the tasks given. The
students in this study will be referred to by their initials in order to assure their privacy. The interviews were
video-taped, transcribed and analyzed together with the students’ written work.
Tasks
Before the interviews took place, we examined the participants’ results and solutions from the mid-term exam
and from the final exam in which they had to solve several routine tasks from differential calculus. The tasks in
the mid-term contained the function given with a concrete algebraic expression, and the tasks required students
to:
- calculate the limit of the function at the point,
- examine whether or not the function is continuous at the point
- determine extreme values for the given function.
The task in the final exam asked students to investigate all properties (domain, zero points, continuity,
asymptotes, extrema, intervals of decrease and increase, the point of inflection, intervals of concavity) of a
𝑥2
certain function given with algebraic expression (e.g. 𝑓(𝑥) = 2 ) and to draw the graph of the function
𝑥 −1
accordingly.The tasks in these two exams can be characterized as routine tasks since students had met and
practiced similar tasks during the calculus course, and they should have been able to employ a familiar
algorithm in order to solve them.
In the interviews, the students were given tasks that were designed in conjunction with the course’s lecturer and
teaching assistant. The guidelines in designing the tasks were that they should differ slightly from the textbook
exercises. According to Selden et al. (1998), such tasks can become significant problems, and thus become nonroutine tasks. Therefore, the tasks contained concepts that students had met, used and learnt on the calculus
course, they were not a challenge in terms of problem-solving, but they did not have a concrete algebraic
expression for the given function.
The following tasks were given to the students:
1. It is given function 𝑓: 𝑅 →R. Let 𝑓′(𝑥0 ) = 0for only one point𝑥0 . Also let𝑥0 > 0and𝑓(𝑥0 )<0.
Iflim𝑥→+∞ 𝑓(𝑥) = lim𝑥→−∞ 𝑓(𝑥) = +∞, how many zero points does function fhave?
2. Sketch the graph of a function f which satisfies the following conditions:
a. 𝑓 is discontinuous at 𝑥 = 0 and 𝑓 (0) = 1;
b. 𝑓’’(𝑥) < 0 for all 𝑥 < 0 and 𝑓’’ (𝑥) > 0 for all𝑥 > 0;
c.
𝑓’(−1) = 0 and𝑓’(𝑥)  0 for 𝑥

− 1.
The first task was given to students after the mid-term exam, and the second task was given to them after they
had passed the course. The students were given sheets of papers for use in solving the tasks. They were able to
use their notebooks if they felt they needed to look something up. According to the lecturer and teaching
assistant, it was unlikely that the students had encountered such tasks in the course, however in order to be able
to determine whether a task is routine or not, it is not sufficient to examine the properties of the task alone, but
the relation between the task and the solver has to be also considered (Schoenfeld, 1985).
Data Analysis
The students’ work was videotaped because their gestures, tone of voice and the interaction between the
students play an important part in their behavior and can tell us something about their mathematical reasoning.
However, in this paper we will focus mainly on the verbal expressions, thus the main data for analysis consisted
of the transcribed interviews, supplemented with the written work that the students produced during the
interview. The interviews were transcribed by the author. As the research questions in this study deal with
characterizing students’ mathematical reasoning, the transcriptions were primarily focused on verbal and written
mathematical communication. Therefore, we examined the students' arguments, guesses, and conjectures related
to mathematics, produced in written or oral form during the interview.
International Journal of Research in Education and Science (IJRES)
55
The analysis for each interview was conducted in the following steps. First we provide a description of the data
from the interviews. Then follows an interpretation of the data with the aim of understanding the central parts of
the reasoning. The reasoning was characterized with the help of the reasoning framework. First, we classified
the reasoning sequence as either creative reasoning or imitative reasoning. If the reasoning sequence was
classified as imitative reasoning, we tried to determine whether the reasoning sequence was memorized or
algorithmic. In addition, we examined the students’ erroneous hypotheses and conclusions to detect mental
structures that affected their knowledge base.
Lithner (2008) proposes that a reasoning structure is carried out in four parts: a task is met, a strategy choice is
made, the strategy is implemented and a conclusion is obtained. The strategy choice and conclusion of the
reasoning sequence were identified in the transcripts, which enabled us to identify separate reasoning sequences.
Strategy choice includes recalling a procedure, selecting from accessible procedures, constructing a new
procedure or simply guessing. After the strategy has been implemented, a conclusion is reached as the product
of strategy implementation. It can be both incorrect and incomplete.
Validity and Reliability
The study in this paper is a qualitative one, so it is impossible to generalize findings. Here the generalizability
can be replaced by the notion of ‘fittingness’ i.e. “the degree to which the situation matches other situations in
which we are interested” (Schofield 1990, p. 207). On the other hand, Goetz and Le Compte (1984) use the
notion ‘translatability’ and the notion of ‘comparability’. Translatability refers to a clear description of one's
theoretical stance and research techniques, i.e. whether the theoretical frames and research techniques are
understood by other researchers in the same field. Comparability is the degree to which the parts of a study are
sufficiently well described and defined so that other researchers can use the results of the study as a basis for
comparison (Goetz & Le Compte 1984, p. 228). ‘Thick descriptions’ are, therefore, vital for others to be able to
determine if the attributes compared are relevant (Kvale 1996). Therefore, we have aimed at making the process
transparent by providing a good amount of detail about our study.
Results
Reasoning in Task 1
The work of the students will be presented, each followed by a summary of the reasoning characteristics. The
interview was translated from Croatian, ‘…’ denotes a pause, and [] some action, implicitly or explicitly shown.
Students D and P
Description:
Task 1 was given to students D and P when they were in the middle of the calculus course. When presented with
the task, the students sat in silence for a while and then student P said:
“It is equal with zero so… When we search for zero point of the function, it is in fact the first
derivative. Here it says 𝑓 ′ (𝑥0 ) = 0, therefore it means that we have only one zero point. And it’s 0.
This point is in the origin of coordinate system.”
Student D remained silent while student P was explaining his solution; however, his facial expression implied
that he was not sure that this solution was right. The students then were silent for some time, so the interviewer
began asking some questions. The interviewer asked the students if they had ever met derivatives and where
derivatives were used in the calculus course. Student P identified𝑓’(𝑥0 ) = 0 again as the method for
calculating a zero point of the function, and when the interviewer explained that in fact 𝑥0 represents critical
point of the function, student P stuck with his interpretation. He remembered that they used second derivative
“for calculating minimum and maximum”. However, student D then intervened:
D: I think 𝑓 ′ (𝑥0 ) = 0 says𝑥0 is the point of inflection.
[silence for several minutes]
I: Hm…Let’s try this way...Where is this point? What else is given?
D: It’s in the origin.
I: Do you have any other information?
P: I don’t understand. I’ve never met a task like this before.
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D: It’s too difficult. I know how to calculate zero point for the function but this way no… We are not
mathematics students.
I: But you’ve met tasks without calculation in the course, haven’t you?
P: Yes.
[silence]
The students gave up solving the tasks, with no further attempt to examine other given conditions.
Interpretation:
The students’ reasoning in Task 1 had an erroneous base from the start which was an obstacle preventing them
from moving beyond the first condition. At the beginning, student P used familiar AR when he concluded that
the function in fact has one zero point. Here, he relied on superficial property, namely the expression where the
function 𝑓 was equal with the zero and concluded that there is only one zero point (𝑥0 ) without investigating
other information given in the task. When the interviewer intervened, it made no difference. It seemed that
student P had a somewhat vague understanding of the derivatives and their properties. He recalled that the
second derivative was used when they calculated if the critical point was minimum or maximum, but since this
was not a task with calculation he was unable to use his knowledge flexibly. Student D used imitative reasoning
when he concluded that 𝑥0 is the point of inflection. He relied on recalling something he had seen before, not
giving any arguments as to why he reached this conclusion. The fact that the students experienced significant
frustration because they could not interpret the given condition in the right way was reflected in their reluctance
to examine other conditions. Students D and P claimed that the task was too difficult pointing out the fact they
had never met a task phrased this way before, and that the task was created for mathematics students since there
were no concrete numbers to work with. They pointed out that they knew how to calculate derivatives and zero
points. The students’ emphasis on “calculation” and their comment that they “have never met tasks like this
before” indicate dependence on algorithmic reasoning i.e. mimicking the same procedure.
Students N and M
Description:
Students N and M read the task silently and afterwards they commented on the amount of written text in the
task. Their whole reasoning process was very short, but they also experienced some difficulties in the process:
N: Too much text [laughs].
M: Hm, well…
N: How many zero points does function 𝑓 have?
M: So for only one x we have 𝑓’(𝑥0 ) = 0… Aha, it means that function has only one extreme.
N: Zero point.
M: Zero point and extreme are not the same.
N: Oh, yes, yes… It’s about extreme values.
M: This extreme value is here [shows first quadrant] …
N: No... 𝑥0 is here [point at positive part of the x axis]… and 𝑓(𝑥0 ) is negative…. so the point is in
fourth quadrant.
The students had the idea to separate function limit in two parts and to see what each part represents. Student M
said that “function values go to the positive infinity for all positive 𝑥-values”, and student N concluded the same
for negative 𝑥-values. Both students gave an answer to the question. The interviewer asked them how they came
up with the solution.
N: I made a figure of the graph in my head connecting conditions.
M: Yes, yes, I imagined that. It all fits.
Interpretation:
Students M and N solved the given task without any assistance or prompting from the interviewer. The students
experienced some problematic situations in the beginning, regarding the position and type of the given point:
student N identified 𝑥0 as the zero point, and student M situated the point in the wrong part of the coordinate
system, namely the first quadrant. However, solving the task together was valuable for the students’ reasoning
sequence: they helped each other; one member of the pair steered the reasoning sequence on the right track. The
students together reached a meaningful conclusion after separating the function limit in two parts and examined
functions values of the given function. At the end, the students verified their solution. Their reasoning had the
characteristic of being creative, regardless of initial problems. They were able to adapt to the problematic
situations. Apparently the reasoning sequence was new to them; their strategy choice and implementation were
supported by predictive and verificative argumentation. The arguments that were used to reach the conclusion
were based on sound mathematical properties. After the interview the students said that the task looked difficult
at first, but when they look at it now it seems easy.
International Journal of Research in Education and Science (IJRES)
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Students B and J
Description:
Student B read Task 1 out loud, and both students had the same reaction to it, that they were intimidated by the
amount of data in the task. When they began to interpret the given data, student B concluded that 𝑓’(𝑥0 ) = 0 is
a derivative of the constant, but student J corrected him:
J: No, that's critical point.
B: Hm…
[silence]
The interviewer asked the students to express out loud what they were thinking at that moment, but the students
stayed silent for several minutes. In order to stimulate the reasoning process, the interviewer prompted the
students:
I: Go on... What seems to be the problem?
B: Don’t know what to do with the data.
I: What about drawing what is given in the task?
Even with this prompt, the students were not able to move further in the reasoning sequence. The interviewer
started to ask leading questions:
I: Where does the point (𝑥0 , 𝑓(𝑥0 )) lie?
B: In the second quadrant [pointing at the fourth quadrant] because 𝑥0 is positive and that is on the right
from the origin on 𝑥- axis, and 𝑓(𝑥0 ) is negative and that is down on 𝑦-axis.
I: What about next piece of data? What does it say?
B: That everything goes to infinity.
At the end of the task, student B concluded that he can sketch parabola from the given data but he was uncertain
how to draw it because there was only one point given in the task, and he did not have the formula of the
function. However, student J took the initiative:
J: [takes pencil and draws] You do not need another point. See? If you connect all conditions, you get a
graph.
B: Oh… and there are two zero points
Interpretation:
Here in Task 1, the students met several problematic situations, such as the wrong interpretation of symbols
presented in the task or the inability to move beyond their inference about the amount of text and data in the
task. The students’ strategy choices are made or at least influenced by the interviewer, and the reasoning
sequence was guided by the interviewer. Besides giving a prompt for drawing data in the coordinate system, the
interviewer asked what a certain piece of data represents. Students B and J alternated in data interpretation, but
neither of them was independent in the reasoning process. The students drew the point (𝑥0 , 𝑓(𝑥0 )) in the correct
quadrant, and then concluded that the function values go to infinity. Student B could not draw a graph of the
function because he did not connect the given conditions. Here he was fixed on one specific strategy choice,
namely, drawing a graph of a function with only one point given. Consequently, that hindered his reasoning
sequence. On the other hand, student J made an adequate graph of the function according to given data and
verified her solution. If we examine parts of the students’ reasoning sequence, we can detect components of
(local) creative reasoning, but more so in student J’s reasoning than in student B’s. Using their conceptual
understanding, examining intrinsic mathematical properties of the concepts in the tasks, the students obtained
the solution. But the final conclusion could not have been reached without the interviewer’s initial help and
supportive questioning; therefore, it is not certain that the students would have been able to solve the task on
their own. After the interview was concluded the students said that they were used to calculation in the tasks,
and that they would avoid this type of task if it came up in the exam.
Reasoning in Task 2
The work of students will be presented, each followed by summary of the reasoning characteristics. The
interview was translated from Croatian, ‘…’ denotes a pause, and [] some action, implicitly or explicitly shown.
Students D and P
Description:
The initial behavior of students D and P was similar as in Task 1, but this time student D was involved in
solving the task from the start. Both students decided to read the task, and to interpret each condition:
P: [draws coordinate system] This means that function in 0 would cross coordinate axes. [draws small
circle in the origin of coordinate system]
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D: Second condition....We used this when we calculated minimum and maximum.
[silence]
The students stopped and it seemed as though they did not know what to do next, so after several minutes of
silence, the interviewer asked them to explain the role of the first and second derivative when examining
properties of the function. The students remained silent, so the interviewer tried another approach asking:
I (interviewer): How did you calculate intervals of concavity for a concrete function?
P: Hm …It’s concave upward when the second derivative is less than zero.
D: Hm...no... I think is concave downward in that case.
P: [draws graph of function where certain parts are concave upward and concave downward] So let’s
look at parabola… second derivative… greater than zero here where [it] is concave upward, and
concave downward here [pointing to the figure]
[silence]
I: What about the next condition? What does −1 represent?
P: It's zero point.
D: Hmm… no it's maximum. See here at the figure [pointing to upper figure]… the derivative is zero in
the critical point and then it follows that it’s maximum from other given conditions
[silence]
I: And now, can you incorporate all condition in your sketch?
P: No, I can’t make a figure in my head.
D: [silence]
Interpretation:
As in Task 1, students D and P experienced significant difficulties in Task 2. Their initial reasoning can be
characterized as familiar AR – based on recalling something they had calculated many times. The students’
uncertainty prevented them from proceeding further. After the interviewer’s intervention, the students were able
to move on, but they showed that they were quite dependent on external guidance in their reasoning. In some
situations, the students were able to develop their ideas. And in some situations, the interviewer took the
initiative, for instance, when the students had to connect second derivative and intervals of concavity for some
concrete function. It was only after this prompt that the students’ reasoning sequence continued. From the
students’ behavior, it seems that this reasoning is new to them, but their reasoning sequence is not entirely
flexible in a new and problematic situation, even though in many parts, it is mathematically founded, i.e.
arguments given are based on intrinsic mathematical properties of concepts involved in the tasks. Local creative
reasoning can be found in part of the reasoning sequence, for instance, when student P uses a concrete function,
namely a quadratic function to reconstruct his knowledge on the second derivative and its relation to concavity.
Or when student D reached a conclusion about a given point, and verified it using sound mathematical
arguments. But in the end, the students were not able to reach a final conclusion. Their strategy choice was
dependent on the interviewer’s hints and lead, and it is unlikely that the students would solve the task on their
own. However, after the interview the students said that all the concepts in the task were indeed familiar to
them, but they preferred tasks with calculation.
Students N and M
Description:
After several months, when faced with Task 2, students N and M employed the same strategy as they had in
Task 1. They read the task silently and then they started talking to each other and interpreting conditions. First
of all, student N decided to draw the coordinate system. Then student M said that the function 𝑓had a hole in 0
and student N sketched that hole in the coordinate system as an empty circle at the point (0,0) and drew a full
black circle at the point (0,1). Then they interpreted condition b. :
M: Second condition... It could be … [silence]
I (interviewer): Where have you used the second derivative?
M: When something is convergent and divergent….but I don’t remember when you use the first thing
and when the second.
N: For intervals of decrease or increase.
M: [draws a figure of curves being concave upward and concave downward] That is the first derivative,
the second is used when something converges or diverges.
N: Ok, you mean concave upward and downward…Discontinuity is here where the function changes its
shape.
International Journal of Research in Education and Science (IJRES)
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M: Well, no… that would mean… when 𝑓’’(𝑥) > 0 [writes on the paper], it is concave upward, when
𝑓’’(𝑥) < 0[writes on the paper], it is concave downward.
N: See, here where 𝑦 = 1, it fits… it goes like this [corrects figure, draws curve looking like parabola,
having minimum]
M and N switched to condition c. identifying −1as the only extreme value of this function. They changed their
figure according to a new condition, and again student M was a bit puzzled as to whether the new figure was the
correct solution of the task. Student N explained that the new figure fulfills all the conditions:
M: So it’s ok that we have two parts?
N: The function has discontinuity in 0.
M: Hm…I tried to imagine something I have seen before, and now I see it doesn’t make sense.
Interpretation:
In Task 2, students M and N chose the following strategy: examine each condition separately and draw them in
the figure. The chosen strategy was implemented, and the students adjusted the figure in each step to correspond
to the given conditions. From the interview, it seems that the reasoning sequence was new to the students. Also
they showed a certain degree of flexibility and relied on the intrinsic properties of concepts involved in the task.
Student M could not remember the proper terminology, but he sketched his “divergent and convergent
functions” which were in fact figures representing curves being concave upward and concave downward. This
enabled student N to give the “right name” for function property. When student N pointed out that discontinuity
should be incorporated in the drawing, and it should be where the function was changing its shape from concave
downward to concave upward, it had the effect of disturbing student M’s reasoning sequence. He became
puzzled by the outcome. Similarly, he expressed his doubts at the end of the solution process, because the graph
did not look like any other graph he had ever seen.
When a conclusion was reached, student N verified their solution. His reasoning sequence had many
characteristics of being creative, especially in terms of flexibility in the problematic situation of adjusting the
figure according to the given conditions. From the interview it can be seen that the reasoning sequence was
entirely new to the reasoners. Some parts of student M’s reasoning sequence also have characteristics of being
creative, for instance, when he reconstructed his knowledge for property of concavity. But his uncertainty with
the final product of solving indicates a switch to familiar AR; seeking security in something familiar.
After the interview, the students said they like challenges and they would probably attempt to solve this kind of
task in the exam. They just needed to “connect the dots” which indicates the use of more creative reasoning in
the solution process.
Students B and J
Description:
In Task 2, students B and J first decided to draw the coordinate system, and then they read the rest of the task.
As he was reading condition a. out loud student B noticed there were some points in the text and drew them in
the coordinate system. The points he marked had the following coordinates (1,0) and (0,0). Student B said that
condition b. was about intervals of increase and decrease of that function, and then he stopped, looking puzzled.
During that time, student J was silent, examining the task. Since both students did not say a word for some time,
the interviewer intervened asking questions about the given condition. Student B corrected his own drawing by
marking point (0,1) instead of point (1,0), but he interpreted condition b. again in the same manner. This time
student J got involved and corrected him:
I (interviewer): Are you sure you drew it correctly?
B: Well...no [corrects his drawing]
I: What about condition b.?
B: We had this for maximum and minimum. When second derivative is less than zero, than we have
maximum. When second derivative is greater than zero, we have minimum.
J: No. This [condition] says when the function is concave upward and when the function is concave
downward…
[silence]
I: Where the function is concave upward and where the function is concave downward?
J: On the left side of 𝑥-axis is concave upward and on the right side is concave downward.
[silence for a while]
Students B and J discussed what shapes of concave upward and concave downward looked like, trying to decide
what shape parabola 𝑦 = 𝑥 2 has. After this discussion, student J drew the graph of function that satisfied
condition b. The students moved to condition c. However, this new information caused confusion when the
students tried to incorporate it into other data they had.
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[silence]
I: What does the third condition say?
J: That’s critical point.
I: Where?
J: On the negative part of the 𝑥-axis.
I: What property does function have there?
J: It’s concave upward
I: So what kind of point do we have there?
J: Hm … It’s maximum.[puts the pencil down]
I: Is this the solution?
B: Yes.
Interpretation:
At the beginning, the students were reluctant to express their thinking out loud and to continue their reasoning,
which can be inferred from the silent gaps in the process. However, we would not say that the students lacked
the resources or knowledge needed for the tasks. The students used arguments that are mathematically founded
to provide validity of their conclusions. Their reasoning was anchored in intrinsic properties of the components
in the reasoning: the relation between the property of function being concave upward or downward and the
shape of parabola to determine what shape a given function has, or the relation between the critical point and the
shape of the function to conclude what extreme value is given. Here in Task 2, the students’ reasoning had many
characteristics of being creative mathematically founded reasoning, but the gentle guidance of the interviewer
was needed to stimulate the reasoning sequence. At the end of the task, the students did not adjust their figure so
that all conditions were met i.e. they did not verify their conclusion. The drawing represented the graph of the
function with two extrema; having maximum on the left side of x-axis, and having minimum on the right side of
x-axis.
After the interview, the students concluded that the task was not hard at all. Student J claimed that in the
beginning the unfamiliar situation looked daunting, and student B agreed with student J. This indicates a change
in their reasoning compared to Task 1.
Met- befores and Met-afters
Several negative met-befores and met-afters were detected in the knowledge of students D and P, which
hindered their reasoning sequence. For example, student P interpreted 𝑥0 in 𝑓’(𝑥0 ) = 0 and−1 in𝑓’(−1) = 0
as a zero point of the function. The calculation of zero points of a function is frequently performed in high
school mathematics, wherefore student P disregarded a sign for the derivative and identified this expression with
commonly seen expression 𝑓(𝑥0 ) = 0 and this triggered familiar AR. On the other hand, in their calculus
course, the students learned about the concepts of critical points and extrema before the concept of point of
inflection. Therefore identifying the expression 𝑓’(𝑥0 ) = 0 in Task 1 as a property for the point of inflection
can be considered as a met-after which, we believe, influenced student D’s reasoning.
We identified certain negative met-befores in the knowledge structure of students M and N. In Task 1, those
were the interpretation of 𝑥0 as zero point in 𝑓 ′ (𝑥0 ) = 0 (student N), and placing the point (𝑥0 , 𝑓(𝑥0 )) in the
first quadrant (student M). The first met-before is similar to the case of students D and P, and the latter can be
connected with the presentation of many function graphs in textbooks and lectures. Usually, when the lecturer
(either at university or secondary school) draws a graph of arbitrary function as an example to show a property,
it is mainly placed in the first quadrant, or the major part of the graph is drawn there. In Task 2, student N
identified the second derivative with properties of increase or decrease of the function. In the students’ calculus
course, usage of the first derivative for determining intervals of increase and decrease is taught before the
second derivative and its connection with concavity. This met-before indicates that concepts related to the first
and second derivatives were not properly understood. But, it seems that this mental construct did not have a
negative effect on solving the tasks in calculus exams, because the students did pass the course.
Some negative met-befores were found in the knowledge structure of student B. At the end of Task 1, student B
identified the graph of the function as the graph of well-known quadratic function. This interpretation of the
obtained graph is not necessarily a problematic met-before, but it can certainly have a negative impact in other
situations. Here we see another met-before as much more problematic and that is the need for a “formula” i.e. a
concrete expression to help the student to draw “parabola”. According to Tall (2006), seeking a function that a
person has already met and the need for formula hinders the development of advanced mathematical thinking.
Another negative met-before is the interpretation of condition b., namely connecting 𝑓’’(𝑥) < 0 with the
property of increase or decrease of some function, which together with the interpretation of 𝑓’(𝑥0 ) = 0 as the
International Journal of Research in Education and Science (IJRES)
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derivative of the constant, indicates that the student did not quite understand the topic of the first derivative and
related concepts of extrema and intervals of increase and decrease. However, the student did pass the calculus
course with an average grade, indicating that these problematic met-befores were not evident in the reasoning
that was required in the calculus exams.
Discussion and Conclusion
In this study, we wanted to investigate how average non-mathematics students reason when faced with nonroutine calculus tasks and what characterizes their reasoning. Throughout their primary and secondary
mathematics education, Croatian students frequently use imitative reasoning (Glasnović Gracin, 2011), and this
is also the case in many university calculus courses. However, the students in this study had non-routine tasks
implemented in their university calculus course. According to Hiebert (2003), students learn when they are
given an opportunity to learn, so we expected that the incorporation of non-routine tasks into the course
curriculum would have a positive effect and influence on the students’ reasoning in terms of making it more
creative. Even though we did not design the study as an experiment nor did we have a control group to which
we could compare the effect of the implementation of non-routine tasks into the course, the results of this study
can still provide valuable insights into students’ reasoning.
First, we identified whether the tasks in the study were routine or not. The reasoning of students in this study
showed that the tasks given to them in the interview do not belong in the category of routine tasks. On the
contrary, the students had many difficulties, met many problematic situations when solving the tasks and did not
identify them as the type of tasks they had previously been exposed to. These tasks did not represent problems
in terms of Schoenfeld (1985), but can be characterized as moderately non-routine tasks. The students did not
already know a method to solve the tasks, and the tasks were dependent on their background. Although students
had learnt and used concepts that appeared in the tasks, not providing a concrete algebraic expression for the
functions made the tasks problematic.
The results showed that the students’ reasoning differed in the middle of the course and after passing the course.
The students’ reasoning had become more creative by the end of the course, although they still showed a
tendency to be guided in the reasoning sequence. The absences of computation in tasks prevented students D
and P from relying on procedures and using algorithmic reasoning to which they were accustomed. However, if
we compare their reasoning as examined on two separate occasions, in the middle of the course (Task 1), and
after the course (Task 2), we can see some positive shifts in the reasoning sequence. Reasoning in the second
task is in some part local creative reasoning since the students did use argumentation and considered intrinsic
properties of problematic components together. On both occasions, students M and N had some difficulties
when solving the tasks, but their reasoning had many characteristics of being creative. They corrected each
other, and consequently, produced a valid solution for the tasks. But the question that remained unanswered is
whether students would be able to solve the tasks alone. When it comes to students B and J, student J showed
many characteristics of creative reasoning even in Task 1. Student B’s reasoning had improved by Task 2, but
on both occasions the students needed supportive guidance.
On the other hand, what the students had met before and after learning a mathematical topic had a wide impact
on their reasoning. Negative met-befores and met-afters prevented students from proceeding in solving the task,
or led them to imitative reasoning. However, our intention is not to classify all met-befores and met-afters that
might appear in the students’ knowledge and that might inhibit creative reasoning, but to caution to their
existence. The students in our study passed the calculus course, which indicates that those met-befores and metafters did not prevent them from successfully solving tasks that required imitative reasoning. Here we based our
conclusion also on the students’ mid-term and final exams which we examined prior to each interview session.
When each interview was finished, the students commented on the tasks that they had been given. Besides
students D and P in Task 1, other students were surprised that the tasks were not very difficult in the end, that
they indeed contained concepts that they had learnt and used in the calculus course. In the interview after the
mid-term exam, the students claimed that they probably would not even try to solve such a task in the exam. The
main reason was fear of an unfamiliar situation and being accustomed to calculation, i.e. using familiar
algorithmic reasoning. However, after passing the course, the students changed their attitude a bit. They said
they might try to solve this kind of “task without calculation” if they encountered it in the exam. Even students
D and P softened their attitude. We believe that the non-routine tasks implemented throughout the traditional
calculus course had a positive effect, making a small shift in the student’s reasoning. We believe that our
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findings are valuable since the study investigated the reasoning of average non-mathematics students in a
traditional calculus course.
When faced with new situations, students tend to look for something familiar, and usually seek a remedy in the
form of imitative reasoning, like searching the textbook for similar solutions or recollection of similar tasks (e.g.
Boesen, Lithner & Palm, 2010; Haavold, 2011). Selden et al. (1998) pointed out that students lack tentative
solution starts, i.e. general ideas for beginning the process of finding a solution, and that, together with mental
constructs of met-befores and met-afters, provides a significant obstacle for creative reasoning. We argue that
mathematics educators and lecturers should take this into consideration when teaching students. On the other
hand, creative reasoning is beneficial in checking the quality of students’ long-term knowledge. In imitative
reasoning, students do not consider the intrinsic properties of the objects they are reasoning about, and
frequently they rely on well-established procedure, mimicking, almost unconsciously, its every step (Lithner,
2012). Even though imitative reasoning provides a reduction of complexity in the course requirements, students
do not construct appropriate meaning in such a process. The remedy is not avoidance of non-routine tasks, but
quite the opposite, facing students with new situations. The non-routine tasks and creative reasoning can
uncover negative met-befores and met-afters which students are oblivious to when they perform imitative
reasoning. This uncovering is important for the sequencing courses that build upon previous courses, i.e. where
new knowledge is building up previous acquired and mastered concepts.
But is it possible that non-routine tasks become more visible in a calculus course for non-mathematics students,
so that they are not only part of the homework, but part of the exercise sessions? And to whom does this matter?
There are no simple answers to these questions. In the calculus course that our participants took, the non-routine
tasks were implemented mainly in the homework. The course syllabus is overloaded, and it is difficult to
explicitly deal with non-routine task on regular basis. During the course, the students showed resistance toward
non-routine tasks that required a greater investment of their time than the usual routine tasks. But we as
educators argue that it does matter, because we want to build up a work force that is able to adapt to the
demands of today’s business and economy. We believe that flexible thinking and creative reasoning are part of
this ability. There is also no simple answer to this question from the point of view of the students. Students, not
only in this study program but in many other science and technical study programs, have many requirements in
the courses more closely related to their profession. They usually lack the time for deeper engagement in
mathematics, they want to pass the mathematics course and at the same time they would like to know how to
apply the gained mathematical knowledge (Jukić Matić, 2014). But not engaging students in creative reasoning
gives them the illusion of understanding and leaves them with the idea that mathematics is about
“prescriptions”, i.e. algorithms.
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Didactique et de Sciences Cognitives, 11, 195-215.
International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
Relieving of Misconceptions of Derivative Concept with Derive
Abdullah Kaplan1*, Mesut Ozturk2, Mehmet Fatih Ocal3
Ataturk University, Turkey, 2Bayburt University, Turkey, 3Agri İbrahim Cecen University, Turkey
1
Abstract
The purpose of this study is to determine students' learning levels in derivative subjects and their
misconceptions. In addition, this study aims to compared to the effects of the computer based instruction and
traditional instruction in resolving these misconceptions. For this purpose, 12th grade 70 students were chosen
from high schools in Ağrı city with simple random sampling method. With the pre-test results, the
misconceptions were determined and these misconceptions were tried to be relieving in two groups of students
with computer based instruction and traditional instruction, separately. The result of the study showed that both
the computer based instruction by using Derive software and traditional instruction methods were effective in
resolving misconceptions that students constructed. However, it was found that the computer based instruction
was more effective than traditional in relieving them.
Key words: Derivative; Derivative misconceptions; Computer Algebra System; Computer based instruction
Introduction
In today’s world, computer technology developed sharply and it provided the development of instructional
technologies, too. Therefore, there is an increasing change and development process in different areas (Yildiz,
2012). This technology influenced individual’s thinking power and powered the systematic knowledge.
Therefore, it brought a period of mental production [MEB, 2005]. As it influenced many disciplines,
instructional technologies also influenced the mathematics. It provides students with better understanding of
mathematical concepts by means of computers’ dynamic representations in innovative mathematics education
(Bottino & Kynigos, 2009). It helps students to learn the interaction concepts between students and computer
(Balacheff, 1993).
Computers in education environments are classified into three groups as computer controlled instruction,
computer based instruction and computer assisted instruction (Köse, 2009; Uşun, 2004; Yıldırım Kayabaş,
2007). In this study, computer assisted instruction will be taken into account and the data will be evaluated
accordingly. In general, computer assisted instruction can be defined as using computer in learning environment
(Baki, 2001; Tatlı, 2009).
Some of the productions in instructional technologies are CAS (Computer Algebra System) and DGS (Dynamic
Geometry Software). These technologies are used in mathematics instructions frequently (Ersoy & Baki, 2004).
High level computer software which performs numeric and symbolic operations, and draws graphics is called as
computer algebra system. Some of the computer assisted software such as Derive, Sac, Theorist, Converge,
Macsyma, Reduce, Magma, Maple, Axiom, Mathematica and similar ones are examples of computer algebra
systems ([MEB,2005]; Ersoy, 2003; Ersoy & Baki, 2004; Zotos, 2008).
According to Ersoy and Baki (2004), Derive, one of the CAS software, is easy to use and it has a feature to use
advanced calculator. The studies about CAS begins in 1970s and although they developed separately, studies
about CAS is combined with the same research area of artificial intelligence (Zotos, 2008).In this study, derive,
one of the instructional technology software: CAS, is used appropriate to programming approach.
Misconception
Misconception can be defined as the perception (conception) that is far from the consensus of the experts’
perceptions for a specific subject (Zembat, 2010). İşleyen, Tatar, Akgün, Soylu, and Işık (2010) asserted that
misconceptions are not simple mistakes and students have a tendency to insist on repeating the same mistakes
*
Corresponding Author: Abdullah Kaplan, kaplan5866@hotmail.com
International Journal of Research in Education and Science (IJRES)
65
(Zembat, 2010). Mistake can be considered as errors in learning while misconceptions can be considered as
factors that blocks the learning (Keçeli, 2007; Ubuz, 1999). Misconceptions appears while students do not learn
the concepts comprehensively or while they make incorrect reasoning if they learn the concept incorrectly
(Umay & Kaf, 2005).
According to İsmail (1993), incorrect construction of the knowledge results in problem which is called
misconception in the further times. In addition, the hardness of some subjects may also causes misconceptions
for students. Moreover, Baki and Güveli (2008) states that there are few number of subjects that do not cause
misconception in our mathematics curriculum. Students need to gather old one to new knowledge for
constructing their intellect. That’s why it is important that investigate old knowledge which existing and
relieving of misconceptions (Güneş, Dilek, Demir, Hoplan & Çelikoğlu, 2010).
Having recommend these three sequence for relieving of misconception: first of three investigate the gap of
knowledge and have got misconceptions of students. Secondly enhanced materials and learning method
according to attainment for relieving the gap and misconceptions. Lastly try to relieving misconceptions and gap
knowledge by means of developed materials and learning method (Büyükkasap, Düzgün, Ertuğrul & Samancı,
1998; Kaplan, Altaylı & Öztürk, 2014).
Derivative
There are many different definitions for derivative. Based on the limit, derivative is limit of the ratio to increase
in the independent variable of the increase in the function while change of independent variable approaches to
zero (Balcı, 2012). From the geometric meaning of the derivative, the derivative of the function is equal to the
tangent of the angle between the indicated point of the function and x-axes that is the slope of the tangent
(Karadeniz, 2003). Hugges-Hallet et al. (1996) defined the derivative using relationship between average speed
and average change rate. They used the first movement and average change ratio concepts. The derivative is
defined as the slope of tangent line at a point on the curve (as cited in Berry & Nyman, 2003).
The derivative subject is a basic subject for many areas. Some of the areas are in solving various numeric
questions in numeric analysis in mathematics (İbrahimoğlu & Bayram, 2008), velocity, acceleration and
experimental time series in physics, population increase in biology ([MEB, 2005]; Yılmaz & Güler, 2006),
marginal concepts in economy (Balcı, 2012), and many other science branches (Gür & Barak, 2007).
When examine of literature show that there are several studies investigate of misconceptions toward derivative
subject (Aksoy, 2007; Amit & Vinner, 1990; Bezuidenhout, 1998; Bingölbali, 2010; Gür & Barak, 2007;
Ferrini-Mundy & Graham, 1991; Hӓhkiöniemi, 2005; Orton, 1980; 1983; Özkan & Ünal, 2009; Ubuz, 2001;
2007). Seen these studies that students have got misconceptions such as given derivative function of derivative
in a point, considered as if derivative function of tangent equations, given tangent equations of derivative in a
point. Some of cause of this misconception is incomprehension geometric interpretations of derivative, not
understanding relation between limit and derivative etc. based on derivative definition (Ubuz, 2001; 2007).
According to Zembat (2010), for a specific subject, trying to relieving misconceptions is as important as
determining misconceptions for teachers. Therefore, teachers should have high level of pedagogical content
knowledge and they should relieving the obstacles that may blocks students’ conceptions (Özmantar &
Bingölbali, 2009). Concepts and knowledge sometimes become easy and concrete. However, they sometimes
become hard to understand and abstract. There is a need for some models for students to learn abstract
mathematical concepts to concretize them. For these purpose, the learning environment should be enriched with
concrete materials ([MEB,2005]; Bottino & Kynigos, 2009; Ersoy & Baki, 2004; Kaplan, 2005). Different
methods and materials can be used to relieving misconceptions that students constructed. One of these materials
is computers and computer software which are appropriate to today’s technology.
The Significance of This Study
The Project of Fatih maybe the largest of altered for have carrying more forward Turkish Educational System in
recent years. Succeed of this project will be possible encourage of teachers via made study in this field. In this
connection made each study regard with utilize of computers in educational system is important, because they
will provide contribute to development of education. This study is seen one of the studies supported The Project
of Fatih.
66
Kaplan, Ozturk & Ocal
Reason of selected of software of Derive is attracted interested of students that have got specifications easy
accessible, drawing graphs, solving equivalent, taking derivative and the integral. Moreover use of this software
is not difficult. This study in conducted with the expectations of being a guide for secondary school mathematics
teachers in derivative subject, of encouraging to increase the number of software in related areas in education,
and of introducing different software in universities.
In this study, the effectiveness of computer assisted instruction method is investigated. The study will determine
the effectiveness of computer assisted instruction in resolving misconceptions. This study investigated what
kind of misconceptions students have in derivative subject and whether “Derive”, one of the CAS software, is
effective in resolving misconceptions in derivative concept and if it is effective, whether there is a significant
difference between it and traditional instruction method.
Method
This study was conducted to nonequivalent groups posttest-only design by using quasi-experiment design. The
groups were subjected to pre-test and post-test. In this design, random assignment cannot be applied. The groups
were tried to be matched according to some criteria prepared before and based on specific variables. The
matched groups are randomly assigned as experimental and control group. Definitely, it is impossible to
consider the groups as equal. The quasi-experiment method can be designed as applying or not applying pre-test
(Büyüköztürk, Kılıç - Çakmak, Akgün, Karadeniz, & Demirel, 2010; Çepni, 2010; Fraenkel, Wallen & Hyun,
2012; McMillan & Schumacher, 2014). In this study, two mathematics classrooms were chosen for applications.
Both classes’ teachers were the same. First of all, pre-test was applied to both groups. Then, one of them was
randomly selected as experiment group while the other was as control group. The subject was presented to
experiment group by using Derive, one of the CAS software. In the control group, the subject was presented
with traditional direct lecturing method.
Sampling
For choosing the sample of this study, simple random sampling method, one of the random sampling methods,
is used. With this method, high schools are listed and two schools were chosen randomly. In addition, one of
these schools was chosen randomly for pilot study and the other was chosen for administering the final form of
the research tool. For the last application, 12th grade two classes of the school were assigned as experiment and
control groups. The sample of this study was 12th grade 70 students in two different high schools in Ağrı city.
Table 1. The Distribution of the Students in the Study
Test Administration
Number of the Student
Pilot Study
32
Final Administration
38
Total
70
Data Collection Tool
To collect data, a question pool is prepared. It includes questions used in the previous literature (Çelik, 2000;
Gür & Barak, 2007), some questions asked in the university entrance exam, the activities in the secondary
school mathematics curriculum prepared by National Ministry of Education and some questions prepared by
different teachers. Among those questions, 12 open ended questions were selected based on four attainments
which are appropriate to mathematics course for 12th grade secondary school mathematics curriculum. These
questions were selected by two teachers. According to the results of the pilot study, some questions were revised
by choosing different questions in the question pool. In this process, the attainments were not changed. Then,
the final version of data collection tool was prepared for pre-test and post-test. For validity concerns, the tool
was subjected to four expert in this area and five teachers. Lawshe content validity coefficient with view of
expert is measurement 0.77. This level is sufficient for content validity (Lawshe, 1975). The pilot study was
conducted to satisfy the reliability of the study and the reliability coefficient (Cronbach Alpha) of the test was
found to be 0.85. Then, there was no need to change any questions to increase the reliability of the test. This
reliability level is considered to be highly reliable according to (Kayış, 2009; Field, 2009). This data collection
tool was administered in the application school as pre-test (to determine the misconceptions) and as post-test.
In this study, attainments related to derivative subject and numbers of questions for each attainment were
presented in the table below for satisfying the validity of the test.
International Journal of Research in Education and Science (IJRES)
67
Table 2. Table of specifications for attainment
Attainment
No
Attainments
Number
of
Questions
2
1.
Finds the derivative of a function at a specific point by using the definition of the
derivative.
2.
Determine the domain of the derivative of a function.
1
3.
Explains the derivative concept with the help of geometric applications.
Explains the relation between distance of an object which moves through a line in ttime and its velocity and acceleration at t with examples.
2
5.
Explains the derivative concept with the help of physical applications.
2
6.
Writes the tangent and normal equations of a graph of a function at a point
2
4.
3
Experiment Process and Data Collection
At the beginning of the study, the students’ misconceptions were tried to be determined. Then, the pre-test was
administered to both experiment and control group. After determining students’ misconceptions in derivative
subject, misconceptions were tried to be relieved by using traditional method (by means of paper-pencil) in
control group. For the experiment group, on the other hand, computer assisted instruction method is
administered by means of Derive, one of the CAS software, as an instruction tool. In this study is used to learn
through program (Baki, 2002). In teaching process, the students had opportunity to work on the same program
as an individual. First of all, the subject was presented without using computer and examples were given. Then,
each of students the same examples were solved with the computer software again and controlled their correct in
own computer. Program called monitor and students see reel of program, and then they uncover logic, algorithm
and formula there. At the end of the research were applied as post-tests measure.
The Analysis of the Data
Descriptive statistics is used in analyses of collected data in this study. Descriptive statistics utilize summarized
of results. If study that univariate use statistical technics such as frequency, percentile and mean, analysis of data
is more clear and understandable (McMillan & Schumacher, 2014).
Students’ answer to questions for each attainment in pre-test and post-test were evaluated and four categories
were constructed accordingly. These are “understanding”, “incorrect understanding”, “not understanding” and
“non-response” (Çepni, Bayraktar, Yeşilyurt, & Coştu, 2001). In the results of this study, types of
misconceptions that students have were tried to be determined with this categorization. Then, whether they
coincide with the misconceptions mentioned in the literate was investigated. The results of the pre-test and posttest were compared with MS Excel and whether the instruction method used to resolve the determined
misconceptions was effective was evaluated. With two groups’ post-test results, superiorities of “Derive”, one
of the CAS software, and traditional instruction methods were compared. In the data analysis, the findings are
given in the table according to the attainment numbers. There are six attainment for this study.
The Limitations of This Study
The purpose of this study is relieving of misconceptions which derivative subject of junior high school students.
Some students that achievement highly level is not participant this study, because students prepare the License
Placement Exam while carry out the study. This state is limitations for this study. In addition to sampling
numbers is quite a few and some schools unable to fit subject of derivate on plan, thus they couldn’t participant
in this study. This is secondly limitations. Data that derived in this study is applied only descriptive statistic and
it is block generally of the study. This related with statistical result validity.
Results
The table below presents the pre-test results of the experiment and control group according to the attainments
and related questions.
68
Kaplan, Ozturk & Ocal
f
E*
%
C*
E*
1
4
1
21
2
13
11
68
2
1
4
1
21
1
3
4
16
3
2
2
4
11
1
11
9
58
4
2
8
9
42
3
7
9
37
1
16
8
84
5
2
1
1
5
1
6
7
32
6
2
1
7
5
E*: Experiment Group (n=19)
C*: Control Group (n=19)
1
Incorrect Understanding
(Misconceptions)
Understanding
Questions
Attainment No
Table 3. Pre-test results of the experiment and control group
f
Not Understanding
%
f
Non- Response
%
f
%
C*
E*
C*
E*
C*
E*
C*
E*
C*
E*
C*
E*
C*
5
58
5
21
21
47
47
47
42
5
37
37
14
2
11
5
2
4
2
1
0
2
3
6
15
3
10
5
7
4
5
3
5
2
5
3
74
11
58
26
11
21
11
5
0
11
16
32
79
16
53
26
37
21
26
16
26
11
26
16
0
1
3
5
4
1
4
6
2
2
8
6
0
3
2
5
6
2
1
2
4
5
3
3
0
5
16
26
21
5
21
32
11
11
42
32
0
16
11
26
32
11
5
11
21
26
16
16
1
3
1
6
11
3
5
5
1
14
2
6
3
2
6
5
2
4
4
5
2
11
4
6
5
16
5
32
58
16
26
26
5
74
11
32
16
11
32
26
11
21
21
26
11
58
21
32
For the first attainment related to definition of the derivative, there were two questions. Students in both
experiment and control group were expected to find the derivative of the function by using definition of the
derivative. Students solved the questions by using practical derivative rule; however, they experienced difficulty
when they try to solve it by using the definition of the derivative. In the first question, students were expected to
write the definition of the derivative by themselves. Majority of the students solved this question by using the
practical derivative rule but not the definition of it. In the second question, on the other hand, the definition of
the derivative was presented and they were expected to explore that the definition of the derivative is actually
the derivative of the function. Majority of the students correctly solved this question. However, 11 % of
experiment group students and 16 % of control group students fell into misconceptions. Students’
misconception was rooted from doing operations by memorization. In addition, they did not know the definition
of the derivative. These findings were also determined in the Gür and Barak’s (2007) study.
In the second attainment, there is a question which seems like an easy one. The derivative of the function in the
question can easily be taken if the domain of the function is not considered. However, if the domain is taken into
account, the derivative of the function does not exist. Since majority of the students does not take the domain of
the function into account, they made misconceptions. 58 % of experiment group students and 53 % of control
group students made misconceptions in this question.
Third attainment is related to geometric meaning of the derivative. First question about this concept requires to
solve it by equalizing the derivative of the function with the slope of the tangent at that point. Students who
have misconception tried to solve this question by equalizing the derivative of the function with the equation of
line that the function is tangent. In the second question, especially experiment group students could not solve the
question correctly. Students who have misconceptions tried to use minimum point in order to find the distance
of the parabola to line. The fourth attainment is composed of three questions that necessitates from students to
find slope by using the first derivative of the function. In these questions, students experienced misconceptions
because they thought the derivative of the function as the slope of the tangent.
There were two questions for fifth attainment that are related to physical interpretation of the derivative. First
question is related to finding the instant velocity of a moving object. 84 % of students in experiment group
correctly solved this question. In addition, there was no student who has misconception in this question.
Students who have misconception in control group generally did not know general meaning instant velocity as
derivative and they tried to find the instant velocity by substituting the value of the given second in the function.
74 % of experiment group students and 58 % of control group students could not answer the second question.
International Journal of Research in Education and Science (IJRES)
69
The source of the misconception in this question is that students tried to use Pythagoras theorem and special
triangle equations in order to find the shortest period of time instead of simply finding the derivative.
Attainment No
The sixth attainment is related to the slope of the tangent and normal. The first question is asked for the slope of
the tangent, while the second question is related to the slope of the normal. In the first question, some of the
students made misconception because they incorrectly know the practical derivative rules while some others
thought the first derivative as the slope of the tangent. In the second question, on the other hand, students could
not differentiate the difference between the slopes of tangent and the normal. Some other students also had
misconceptions because they learnt the practical derivative rules incorrectly (Gür & Barak, 2007).
1
2
3
4
5
6
Table 4. Pre-test and post- test results of control group students
Incorrect
Not Understanding
Understanding
Non - Response
Understanding
(Misconception)
%
%
%
%
Pre-test
Post-test
Pre-test
Post-Test
Pre-Test
Post-Test
Pre-Test
Post-Test
32
5
21
47
24
37
71
63
55
74
50
61
47
53
32
21
18
21
18
26
16
11
13
16
8
11
29
9
24
16
0
0
8
2
3
3
13
32
18
23
34
26
11
11
21
14
34
21
According to the pre-test and post-test results, there was a decrease among the students who received traditional
instruction method for all attainments in the number of students who had misconceptions. In addition, there was
an increase in the “understanding” category for all attainments.
Pre-test and post-test results for the first attainment revealed that there is 39% increase in “understanding” level
and there is 29 % decrease for students’ misconceptions. Moreover, decrease in “not understanding” and “nonresponse” categories was observed for the first attainment. Decrease for all categories except “understanding”
category was observed in the second attainment’s questions. 58 % increase occurred in “understanding”
category. For the questions of third attainment, there was a 3 % increase in “non-response” category. The levels
of “misconception” and “not understanding” decreased, but there is 24 % increase in “understanding” category.
In the fourth attainment, students’ misconceptions decreased with 10 % and their “understanding” level
increased with 27 %. Moreover, there is a decrease observed in the “not understanding” and “non-response”
categories. The increase level in “understanding” category of the fifth attainment was 26 %. The percentage of
“non-response” did not change, but there were decreases in “misconception” and “not understanding”
categories. The decrease was observed in “misconception”, “not understanding” and “non-response” categories
in the sixth attainment. On the other hand, there was a 24 % increase in “understanding” category.
Attainment No
Table 5. Pre-test and post-test results in experiment group
Incorrect Understanding
Not Understanding
Understanding
(Misconception)
%
%
%
Pre-test
Post-test
Pre-test
Post-Test
Pre-Test
Post-Test
Pre-Test
Post-Test
1
2
3
4
5
6
45
21
13
46
45
18
79
84
55
84
63
87
42
58
18
12
5
24
21
11
13
5
0
3
3
16
24
19
11
37
0
5
18
7
8
3
11
5
45
23
39
21
0
0
13
4
29
5
Non - Response
%
According to the pre-test and post-test results, there was a decrease among the students who received computer
assisted method for all attainments in the number of students who had misconceptions. In addition, there was an
increase in the “understanding” category for all attainments. In addition, it is important to state that any
70
Kaplan, Ozturk & Ocal
misconception was not determined in two questions of the post-test related to physical interpretation of
derivative for the fifth attainment.
There is a 24 % increase in “understanding” category while there is a 21 % decrease in “misconception”
category for the first attainment. For this attainment, the post-test results revealed that there is no “not
understanding” and “non-response” categories observed from students’ responses. In the second attainment,
“understanding” category increased by 63 %. In addition, the students’ misconceptions’ decreased by 47 %.
Moreover, there was no student in “non-response” category for this attainment. While 45 % of student could not
correctly solve the question for the third attainment in the pre-test, it is important to mention that students in
“incorrect understanding” decreased to 13 % and those in “understanding” category increased to 42 % in the
post-test results. There was decrease in “misconception”, “not understanding” and “not understanding”
categories for the fourth attainment, there was 38 % increase in “understanding” category. In the fifth
attainment, there was no student in “misconception” category observed. While there was decrease in “not
understanding” and “non-response” categories, “understanding” category had an increase of 18 %. There was 69
% increase in “understanding” category for the sixth attainment. In addition, there was decrease in
“misconception or incorrect understanding”, “not understanding” and “non-response” categories.
1
2
3
4
5
6
Question
Attainment No
Table 6. Post-test results for experiment and control group
1
2
1
1
2
1
2
3
1
2
1
2
Incorrect
Understanding
(Misconception)
Understanding
f
E*
12
18
16
12
9
16
16
16
18
6
17
16
%
C*
11
16
12
12
9
14
15
13
14
5
13
10
E*
63
95
84
63
47
84
84
84
95
32
89
84
f
C*
58
84
63
63
47
74
79
68
74
26
68
53
E*
7
1
2
1
4
3
0
0
0
0
1
1
Not Understanding
%
C
*7
0
5
2
4
3
1
2
1
4
2
4
E*
37
5
11
5
21
16
0
0
0
0
5
5
f
C*
37
0
26
11
21
16
5
11
5
21
11
21
E*
0
0
1
3
4
0
2
2
0
3
0
1
Non - Response
%
C*
0
0
0
2
1
0
0
1
0
1
1
0
E*
0
0
5
16
21
0
11
11
0
16
0
5
f
C*
0
0
0
11
5
0
0
5
0
5
5
0
E*
0
0
0
3
2
0
1
1
1
10
1
1
%
C
*
1
E
*
0
C
*
5
3
2
3
5
2
3
3
4
9
3
5
0
0
16
11
0
5
5
5
53
5
5
16
11
16
26
11
16
16
21
47
16
26
When comparing the post-test results, the “understanding” level of experiment group is higher than that of
control group for all attainments except the third attainment. In general, there was a decrease in percentages of
“misconception” category.
When investigating the questions in the first attainment, number of students in experiment group who correctly
solved both first and second questions is higher than number of students in control group. For the questions in
this attainment, there was no student who did not give any response in experiment group. On the other hand,
there was a few numbers of students in control group. Instead of using the definition of the derivative, students
who had misconception in the first question tried to solve it by using the practical derivative rules as observed in
the pre-test. In the second question, on the other hand, only one student in experiment group had
misconceptions. The source of this misconception is that student did not know the definition of the derivative
correctly. No student was observed in the control group who had misconception. On the other hand, there were
three students who did not give any answer. In the second attainment question, some of the students resolved the
misconception of not considering the domain of the function while taking its derivative. However, some other
resisted on doing the same misconception.
In the question for the third attainment related to geometric interpretation of the derivative, the percentages of
students in control and experiment groups were equal in “understanding” category. While 58 % of students in
experiment group could not answer the second question in the pre-test, the percentage decreased to 11 % in the
post-test. In the second question, the levels of misconception for both group were equal, while control group’s
International Journal of Research in Education and Science (IJRES)
71
percentage in “not understanding” category is lower than experiment group’s. In the first question, on the other
hand, 11 % of control group students and 5 % of experiment group had misconception. For both groups, the
“understanding” level was 63 %. In the three questions for the fourth attainment, the number of students in
experiment group was higher than that of students in control group. There was no student in experiment group
who had misconception in the second and third questions. On the other hand, the numbers of students who had
misconception in both control and experiment groups were equal.
In the two questions for the fifth attainment related to physical interpretation of derivative, number of student in
the experiment group in “understanding” level is higher than that of control group. There was no student who
had misconception in experiment group for both of these questions, while 5 % of students had misconception in
the first question and 21 % of them had misconception in the control group. However, 53 % of experiment
group students did not give any response to second question, while 47 % of control group students did not solve
the question. For two of the question asked for the sixth attainment, the percentage of experiment group students
is higher than that of control group students in “understanding” category. On the other hand, the percentage of
control group students in “misconception” category is higher than the other.
Discussion and Conclusion
In this study, the purpose was to determine the misconception in derivative concept and to resolve it with Derive
software. The results are indicated below:
1.
2.
3.
4.
5.
6.
7.
Students have misconceptions that they were unable to use the operations with the definition of the
derivative (Gür & Barak, 2007; Hӓhkiöniemi, 2005; Orton, 1980),
Students have misconceptions that they did not consider the intervals while finding the derivative of the
functions (Orton, 1980),
Students experienced difficulty in doing the geometric interpretation of derivative and they had
misconception by thinking the distance (Ferrini-Mundy & Graham, 1991; Ubuz, 2001),
The misconception rooted from thinking the derivative of the function as the derivative at a specific
point (Amit & Vinner, 1990; Orton, 1980; Özkan & Ünal, 2009; Ubuz, 2007),
They have misconception rooted from not knowing the general derivative rules (Gür & Barak, 2007),
They have misconceptions rooted from not knowing the physical interpretation of the derivative
(Bezuidenhout, 1998; Bingölbali, 2010),
They cannot construct a relation between the slopes of the tangent and the normal (Bingölbali, 2010;
Ferrini-Mundy & Graham, 1991; Gür & Barak, 2007; Orton, 1983; Ubuz, 2001).
The subjects were presented to students with the Derive software in order to resolve students’ misconceptions.
Therefore, they received computer assisted instruction. Traditional instruction method was administered to the
control group. After the comparison of both groups, both methods were found to be effective in resolving
students’ misconceptions. However, the computer assisted instruction method was more effective than the other.
Especially for the questions related to visualization, the physical and geometric interpretations of the derivative,
as it is expected, students who received instruction with Derive software were more successful to resolve the
misconceptions. This is because software increased students’ interpretation power (Bingölbali, 2010). In
addition, students who cannot give any answer to questions in the pre-test passed to the other categories and at
least they tried to solve the questions. Considering that students have misconceptions due to incorrect reasoning
about the question (Kaplan & Ozturk, 2012), students’ learning level was observed to increase.
It was observed that students experienced difficulty especially in physical and geometric interpretations of the
derivative. Some of the reasons for students’ difficulties are that 12th grade mathematics curriculum is very
dense, that instead of studying the geometric interpretations of the derivative, students generally give attention
to the derivative’s practical rules, that students experience in visualization of the derivative due to
insufficiencies in classroom environment, and that even some schools have computer facilities, the necessary
software and experienced staffs do not exist in schools.
Recommendations
Some mathematical concepts are hard to learn and they require to be concretized. To teach such hard concepts,
technology should be used. Therefore, students should construct or explore their own knowledge by means of
72
Kaplan, Ozturk & Ocal
such technology. So, the learning can be permanent. In education environment, classrooms should be satisfied
with computers and visualization tools such as CAS and DGS software, so students should benefit from them.
The researchers in this area can develop teaching environment that does not allow to the construction of
misconceptions in students’ mind especially in the hard subjects to learn such as derivative and integration. To
resolve the misconceptions in students’ mind, some instruction strategies should be used such as active learning
strategies, cooperative learning, project based instruction, analogy, concept maps, concept cartoons, conceptual
change texts which are the examples of effective instruction methods and approaches. In addition, difficult
subjects should be investigated to find students’ misconceptions with the clinical qualitative researches, and the
findings should be investigated in detail to find the reasons for students’ misconceptions observed.
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International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
Implementation of Structured Inquiry Based Model Learning Toward
Students’ Understanding of Geometry
Kalbin Salim*, Dayang Hjh Tiawa
Universiti Teknologi, Malaysia
Abstract
The purpose this study is implementation a structured inquiry learning model in instruction geometry. The
model used is a model with a quasi-experimental study amounted to two classes of samples selected from the
population of the ten classes with cluster random sampling technique. Data collection tool consists of a test item
questionnaire understanding of geometry and geometric perception of students towards learning using
interactive learning technology. Post test data analysis begins with understanding the concept of geometry such
analysis prerequisite test for normality test using Kolmogorov-Smirnov test and homogeneity test data using
Levene test. Two mean difference test used was Kruskal Wallis k-independent samples of assisted program
SPSS version 16. The results showed that students' understanding of the concept of geometry which acquire
learning with structured inquiry model is significantly higher than the understanding of the concept of students
who received conventional learning. Understanding the concept of geometry students with interactive learning
technology is better than the understanding of the concept of students with conventional learning. There is a
positive interaction between the applications of structured inquiry learning model by using interactive learning
technology for students' understanding of the concept of geometry.
Key words: Inquiry Model structure; Understanding concepts; Geometry; Interactive learning technology
Introduction
Geometry is a part of mathematics which deals with the relationship between points, lines, angles, and wake
field space (P4TK Matematika, 2008). Travers et all (1987) states that: "Geometry is the study of the
relationships Among points, lines, angles, surfaces, and solids ". There are two kinds of knowledge in geometry
is flat geometry and the geometry of space. At the high school mathematics curriculum in grade 10 will explain
about learning the geometry of space. Builds space on its basis derived from concrete objects to make the
process of abstraction and idealization (Nola. R, 2005). Abstraction is the process of attention and determine the
nature, attributes, or special characteristics that are important in a geometrical object. Idealization is a process
considers everything from concrete objects that ideal. In other words look a solid object that is equal to the
actual shape (Paivio. A, 2013).
Johnson and Rising (1972) states that: "Mathematics is a creation of the human mind, concerned primarily with
ideas, processes, and reasoning." Which means that mathematics is a creation of the human mind which is
essentially related to ideas, processes , and reasoning. As stated before, the process of idealization and
abstraction from concrete objects, humans develop knowledge that relates to the real objects are given special
names cube. In the learning process, the high school students who are still in the stage of concrete operations
(based on Piaget opinion) is very difficult to catch the nature or particular characteristics of the cube, as he has 6
pieces of square-shaped field side. Therefore, approaches and learning strategy rests on the notion that
understanding a concept or self-constructed knowledge (constructed) by the students. This means, a formula, a
concept or principle in the geometry of space, should be rediscovered by students under the guidance of teachers
(guided reinvention). Conditioned learning students to rediscover, making them accustomed to investigate and
find something, and it will also be useful in other fields as well as in everyday of life.
Based on interviews with a number of high school mathematics teachers in the Province Kepulauan Riau , that
there are several factors as causes of students' learning difficulties in understanding the concepts of geometry.
Among these are (1) the number of concepts that are abstract geometry; (2) learning model geometry applied
was based on the assumption that knowledge can be transferred intact from the mind of the teacher to the
student's mind; (3) the evaluation system, the teacher just focus on the assessment of the results of formative and
*
Corresponding Author: Kalbin Salim, kalbin_utm@yahoo.com
76
Salim & Tiawa
summative examinations; (4) quizzes with feedback from students rarely implemented in learning mathematics;
(5) students who take geometry learning situation varies greatly with different cognitive abilities. They differ in
terms of learning preferences, prior knowledge, intelligence, motivation, learning pace and in the other case.
According Santyasa (2007), "To improve the quality of the processes and outcomes of learning, the learning
experts have suggested the use structured inquiry learning paradigm for teaching and learning activities in the
classroom". With the changes in the learning paradigm change center (focus) learning from a teacher-centered
learning to student-centered learning. In other words, when teaching in the classroom, teachers should strive to
create a learning environment that can be students learning, to encourage students to learn, or provide an
opportunity for students to participate actively construct concepts learned. Conditions of learning which make
students only receive materials from teachers, notes, and memorize it should be changed to the new knowledge
search, looking through the guidance (structured inquiry), find knowledge actively so as to increase
understanding of not only the memory of it (Paivio, A, 2013). There are several models of learning which is
based on the paradigm of constructivism, in the search for new knowledge and build such a model of PBL
(Problem Based Learning), CTL (Contextual Teaching and Learning), inquiry training models, conceptual
change learning model, and the model group investigation. "But learning model in accordance with the
characteristics of the learning material and the character of the students in the class should provide a greater
contribution to the development of student learning" (Santyasa, 2007).
Therefore, the concepts in the learning material a lot of abstract geometry in this case it is realized that there
needs to be related to the daily life of students, as well as the situation of students of different cognitive abilities,
so in this study used a model of structured inquiry-based learning approach. Inquiry structured approach used in
this study is expected to help students improve their understanding of the concept of geometry. In addition,
structured inquiry approach is also expected to develop intellectual thinking skills and other skills such as asking
questions and finding answers originated skills of their curiosity. Thus they will be familiar such as rigorous
science, diligent, objective, respect the opinions of others and creative (Joyce, B.et.al, 2000).
Structured inquiry model is a model that promotes the involvement of learners actively and creatively in the
search for, examine, formulate concepts and principles of geometry and to encourage students to develop
intellectually and skill in solving the problem. In the structured inquiry model student-centered learning, so that
students can actively participating in the learning process. According to Sanjaya (2009), the main objective of
the strategy is the development of inquiry structured thinking skills-oriented learning process. Criteria for
success of the learning process by using the inquiry model structure is not determined by the understanding of
the learning material but the extent to which students are active search for and find something. Structured
inquiry model emphasizes on the development of cognitive, affective and psychomotor balanced manner so that
through this model of learning more meaningful.
In relation to learning goemetry, students are directed to formulate a hypothesis, perform experiments related to
the learning material geometry and make conclusions to answer the hypothesis. In studying and developing the
science needs to be supported by students' attitudes in students. Student attitudes related to skill groups in the
field of the requirements for the learning process. So in essence the attitude of students is a tendency or impulse
to behave and thinking in accordance with the model expected.
The attitude of students consists of curiosity (curiosity), flexibility (flexibility), critical reflection (critical
student attitudes), and honest. Students who have high student attitudes will have fluency in thinking that they
will be motivated to excel and have a strong commitment to achieve success in learning. In studying the
geometry of the learning material presented by the use of structured inquiry model, students' attitudes are very
influential in the learning process. Frequent case is the lack of a critical attitude and less eager students in
learning mathematics, students' attitudes like this are not expected to arise in the implementation of learning in
the classroom or laboratory. Thus, in the present study attitudes students need to be a review.
Based on the description set forth above, the research by applying the model structured inquiry which can
increase students' understanding of the concept becomes better and higher so that student achievement in
learning material geometry for the better. The main problem in this study is whether the use of structured
inquiry learning model can further enhance the students' understanding of the concept than the use of
conventional learning models in terms of attitudes and perceptions of students in the geometry learning material.
The purpose of research to determine the ratio of students who are given understanding of the concept of
learning by inquiry model and conventional structured, knowing students' perception of the interaction between
the application of the learning model with an attitude of students in learning geometry.
International Journal of Research in Education and Science (IJRES)
77
Methodology
This research includes quasi-experimental research with pretest and posttest quasi design. Data analysis was
performed to obtain the increased understanding of the concept in terms of attitudes and perceptions of students.
In this study, subjects were given further treatment was measured as a result of the treatment. In experimental
models, researchers are free to determine the design of experiments (Arikunto, 2006). This study used a factorial
design as shown in Table 1 below:
Table 1. Factorial Research Design
Attitudes and
perceptions of students
High (Y1)
Low (Y2)
Learning Model
Inquiry Model structure Conventional (X2)
(X1)
(X1Y1)
(X2Y1)
(X1Y2)
(X2Y2)
Table 1 is a factorial design study to see the effect and interaction learning model, students 'attitudes and
perceptions of students' understanding of concepts in geometry material. Learning model that compared the
structured inquiry model (X1) and conventional (X2). Students' attitudes and perceptions of students categorized
into high (Y1) and low (Y2). Samples were SMAN 3 Tanjungpinang who are following the lesson material
geometry. The sample was selected 2 class from a population of 10 class with cluster random sampling
technique (Suparno, 2007). The number of samples included in this study were 58 people consisting of 29
people experimental class and 29 people control class. There are two types of instruments that are used to
retrieve data that research; 1) test understanding of concepts shaped geometry multiple choice (multiple choice)
questions numbered 20 held at the end of the test, and 2) a questionnaire attitudes and perceptions of students
who are 20 questions to categorize the students who have the attitudes and perceptions of students of high and
low.
Instruments tested then further analyzed the data selected are good questions to be used in research. Analysis
instruments include test validity, level of difficulty, distinguishing, and reliability of the instrument. Test
conducted research about the validity of using the product moment correlation formula (Arifin, 2009). Level of
difficulty, distinguishing features and reliability problems were analyzed using the equation proposed by
Arikunto (2006). Questions used in the study have valid criteria with high reliability.
Prior to determine differences in learning outcomes of students who were given learning with structured inquiry
model and conventional learning model, first tested the ability of the students in the class early experimental and
control classes. If the initial ability of students in the two classes together, the study design did not use pretest
and posttest design but simply use the posttest-only design. The same initial Capability impact on treatment
outcome illustrates the differences of learning outcomes only due to the difference in treatment in applying the
learning model. Initial capability data taken from the tests earlier learning material. The results of the initial test
students' abilities in the experimental classes and control classes can be seen in Table 2 below:
Table 2. Initial student capability test results
Statistics
Value
df
51.000
T stat
0.2313
P(T<=t) two tail
0.7852
T critical two tail
2.0322
Based on Table 2, note that the p-value of 0.7852 and is greater than the 0.05 significance level research. It
shows that there are differences in average ability students between classes beginning the experimental and
control classes before being given treatment in both classes. Thus, the research design used to see the effect of
the application of learning models to the students' understanding of concepts in this study using a posttest-only
control design. Research data processing begins with the test requirements analysis in the form of normality and
homogeneity test data. Normality test aims to determine whether the data taken from the population that is
normally distributed or not normal. The test is performed using SPSS version 17 with choice test test used is the
Kolmogorov-Smirnov. While the data homogeneity test was conducted to determine whether the two sets of
data are homogeneous or inhomogeneous. Homogeneity test using the Levene test programming are also
processed using SPSS version 17.
Testing the significance of differences between the mean level of conceptual understanding between the
experimental group and the control group performed statistically. Prerequisite test results analysis show that
there is not normally distributed data or homogeneous so that the statistics used are non-parametric statistical
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Salim & Tiawa
test and the two mean difference used was Kruskal Wallis k-independent samples. Two mean difference test was
performed using SPSS version 17.
Results
Data were described on the application of inquiry learning model structure on learning materials goemetri
include; test data on the students' understanding of concepts goemetri material, and data attitudes and
perceptions of students. Then performed a comparative analysis of students' understanding of concepts taught
with structured inquiry learning model with the conventional model by considering the attitude of the students.
Description of student understanding concepts in geometry Matter Data acquisition value students'
understanding of the concept of the geometry of the experimental class and control classes are presented in
Table 3.
Table 3. Description of value understanding conceptual geometry
Experiment Class
Control Class
Number (N)
29.00
29.00
Max Value
85.50
71.70
Min Value
65.10
49.80
The mean of value
78.62
65.21
Standard Deviation
7.39
7.19
Based on Table 3, it can be seen that the average value of students 'understanding of geometric concepts in the
experimental class at 78.62 while the average value of students' understanding of geometric concepts in control
class is 65.21. This shows that the average value of students 'understanding of geometric concepts in
experimental class is higher than average value of students' understanding of the concept of geometry on the
control class. The maximum value of understanding the concept of three-dimensional geometry of students in
the experimental class (85.50) is higher than the maximum value of understanding the concept of threedimensional geometry in the control class (71.70). Judging from the standard deviation, the standard deviation
of the data on the students 'understanding of the concept of the experimental class (7.39) is not much different
than the standard deviation of the data on the students' understanding of the concept of class control (7,19). It
shows that the students 'understanding of data distribution in the experimental class with the same homogeneous
distribution of the data on the students' understanding of the concept of the control class.
Description of attitudes and perceptions of students data acquisition attitude scores of students students students
in experimental classes and control classes were taken before treatment is given. Indicators include the students'
attitudes and perceptions; rigor, discipline, creativity, expression / ideas, attitudes curious, critical and
responsible. Description student attitudes students score data are presented in Table 4.
Table 4. Data description of attitudes and perceptions of students' scores
Experiment Class
Control Class
Number (N)
29.00
29.00
Max Value
105.00
86.00
Min Value
76.00
63.00
The mean of value
91.12
74.52
Standard Deviation
8.09
8.73
In Table 4 it can be seen that the average scores of students in the classroom student attitudes experimental/
structured inquiry model (91.12) is higher than the average scores of students at grade student attitude control /
conventional model (74.52). Maximum score students' attitudes of students who were given structured inquiry
model (105.00) is higher than the maximum score attitudes students taught with the conventional model (86.00).
Likewise with the second lowest value of different classes namely 76.00 (class of structured inquiry model) and
63.00 (class experimental model). The standard deviation of the experimental class / structured inquiry model
(8.09) is lower than the standard deviation of the control class/ conventional model (8.73). These results suggest
students are more likely to have such a good student attitudes more thoroughly, more critical in the classroom
learning with structured inquiry model than in a conventional classroom.
Furthermore, the category of attitudes and perceptions of students of high and low on both classes can be seen in
Figure 1.
student persentation
International Journal of Research in Education and Science (IJRES)
80
60
40
62,07
55,17
37,93
79
44,83
high
20
low
0
experiment group
control group
group research
Figure 1. Number of students by category attitude students
In Figure 1, it can be seen that the percentage of students who have high student attitudes in the experimental
class 63.03% greater than the percentage of students who have high attitude and perception 55.17% in the
control classes. The number of students who have low student attitudes and perceptions of the experimental
class as much as 37.93% of the students and the students who have low student perceptions on attitude control
class as much as 44.83% of the students.
Description Conceptual Understanding of Geometry Based on Students' Attitudes and Perceptions
The data is reviewed student conceptual understanding than students' attitudes and perceptions of high and low
are presented in Table 5.
Table 5. The description of the student conceptual understanding by attitudes and perceptions of students of
high and low
Attitudes and perceptions low
attitudes and perceptions height
Number (N)
34
24
Max Score
88.75
75.65
Score Min
50.51
54.23
The mean score
72.41
69.78
Standard Deviation
0.10
4.20
In Table 5 it can be seen that the average value of understanding the concept of students who have the attitudes
and perceptions of students 'high (72.41) is not much different than the average of students' understanding of
concepts that have low student attitudes (71.97). However, the maximum value of understanding the concept of
students who have high student attitudes (88.75) is higher than the maximum value of understanding the concept
of students who have low student attitudes (75.65).
Based on the standard deviation, the distribution of the value of understanding the concept of students with
student attitudes and perceptions category height closer to the average value of the class compared with the
distribution of the value of understanding the concepts and perceptions of students with categories attitudes and
perceptions of students is low. It is shown from the standard deviation scores of students with students 'attitudes
and perceptions of high (0.10) is smaller than the standard deviation scores of students with students' attitudes
and perceptions of low (4.20).
Prerequisite Test Results Analysis
Test requirements analysis consists of tests of normality and homogeneity test data. Prerequisite test analysis is
performed to determine the type of comparative test statistics that will be used. Parametric statistics used in the
comparative test if the data is normally distributed and homogeneous, but if the data is not normally distributed
or not homogeneous, the statistics used in the comparative test is non-parametric statistics. The data will be
statistically tested students' understanding of the concept is the data in terms of attitudes and perceptions of
students in the control class and experimental class. Normality test is performed to determine whether the
samples come from populations with normal distribution or normal distribution. Normality test data used in this
study is Kolmogorof-Smirnov test using SPSS version 17.
Dependent variable data including data entered into the understanding of the concept of the dependent variable
list and then the data is free (learning model) and moderator variables (attitudes and perceptions of students) is
inserted into the factor list. If the probability value or significance value calculation data is greater than 0.05
(Sig.> 0.05), the data are derived from normally distributed populations, otherwise the value of the probability
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Salim & Tiawa
or significance value calculation data if it is smaller than 0.05 (Sig. <0.05), the data are derived from
populations that are not normally distributed. Summary of the results of the data normality test conceptual
understanding based model of learning and attitudes of students who count with the Kolmogorov-Smirnov test
are presented in Table 6 and Table 7.
Table 6. Results of normality test data research based on classroom research
Independent variable
Research group
Kolmogorov-Smirnov
Statistic
df
Sig.
understanding of concepts
Experiment group
0.225
28
0.002
Control group
0.310
28
0.000
[α = 5% (0.05)]
Based on the test results data normality (Kolmogorov-Smirnov test) using SPSS version 17 are shown in Table
6 and Table 7, it can be seen that the value of the significance of the data shows all the data were not normally
distributed groups. It can be seen from the value of significance (sig) the results of the calculations in the table.
Significance calculation of the entire group of data either by grouping class research (Table 6) and by the
attitude of the student's perception (Table 7) are all less than 5% significant level (sig. <0.05). It shows that the
data are not normally distributed.
Table 7. Results of normality test data research based on students' perceptions attitudes
Independent variable
Attitude and perception
Kolmogorov-Smirnov
of student
Statistic
df
Sig.
understanding of concepts
High
0.172
33
0.035
low
0.254
23
0.008
[α = 5% (0.05)]
Test of homogeneity in this study conducted to see whether the data dependent variable (understanding
concepts) based on the model of learning and attitudes of students in this study are homogeneous or
inhomogeneous. Homogeneity test data used in this study is the Levene test based on the average value (based
on the mean) data using SPSS version 16. The dependent variable data that is inserted into the understanding of
the concept of the dependent variable list and then the data is free (research grade) and moderator variables
(attitudes and perceptions of students) is inserted into the factor list. If the probability value or significance
value calculation data is greater than 0.05 (Sig.> 0.05) then the data is homogeneous,
otherwise the value of the probability or significance value calculation data if it is smaller than 0.05 (Sig.
<0.05), then the data is not homogeneous. Summary of test results with the Levene test for normality of data
presented in Table 8.
Factor List
(Independent Variabel)
Table 8. Results of homogeneity test data research
Dependen List
Levene Test
Group research
Attitude and student perseption
[α = 0:05 (5%)]
Understanding of concept
Understanding of concept
Statistic
df1
df2
Sig.
0.512
35.23
1
1
58
58
0.873
0.000
Based on the test results of data homogeneity (Levene test) using SPSS 17 are shown in Table 8, it can be seen
that the value of the significance of the data shows that there are several groups of homogeneous data, but there
are also sets of data are not homogeneous. It can be seen from the value of significance (sig) the results of the
calculations in the table. Homogeneous groups of data based on Table 7 that the data of understanding the
concept of class based research group. It can be seen from the value of the significance of these data (the data in
the last column of the third row) which has a significance value calculation (sig) larger (.873) from the research
significance level (α) used is 0.05. While based on the attitudes and perceptions of students (high and low), the
data are not homogeneous understanding of the concept. That is because the empirical significance (0,000) in
the last column of the fourth line, is smaller than the significance level of research (α) used is 0.05.
Comparative Test Results Review (Hypothesis)
Test requirements analysis showed that most of the research data berdistribus not normal and homogeneous, so
that inter-group comparative test data in This research uses non-parametric statistical Kruskal-Wallis test sample
International Journal of Research in Education and Science (IJRES)
81
t-independent. Comparative test performed with SPSS version 17. Prior to the comparative test, it will first
discuss the research hypothesis to provide answers while on the formulation of research problems.
a. First hypothesis
H0: there is no difference in the students' understanding of the concept of getting learning with structured
inquiry model with understanding the concept of learning by students who received conventional models.
Ha: There are differences in the students' understanding of the concept of getting learning with structured
inquiry model with understanding the concept of learning by students who received conventional models.
b. Second Hypothesis
H0: there is no difference in students' understanding of concepts that have high student attitudes and the
attitudes of students is low. Ha: There are differences in students' understanding of concepts that have high
student attitudes and the attitudes of students is low.
c. Third Hypothesis
H0: there is no interaction between learning model with students 'attitudes and perceptions of students
understanding concepts. Ha: There is no interaction between learning model with attitude and perception of
students understanding of concepts.
Inter-group comparative test data is performed using the Kruskal-Wallis test k-independet sample with SPSS
version 16. The zero hypothesis (H0) is accepted if the significance value calculation (sig.) Is greater than the
significance level used in this research is α = 5% (0.05). However, null hypothesis (H0) is rejected and the
alternative hypothesis (Ha) is accepted if the value of the calculation of significance (sig.) Is smaller than the
significance level used in this research is α = 5% (0.05). Summary of the results of the research hypothesis
testing using the Kruskal-Wallis test can be seen in Table 9.
Table 9. Summary of comparative test (Test of hypothesis)
Hypothesis
Kruskal Wallis
Chi-Square
df
Asym Sig.
First Hypothesis
35.20
1
0.000
Second Hypothesis
0.737
1
0.445
Third Hypothesis
38.265
3
0.000
Based on the summary of the results of hypothesis testing in Table 9, it is known that the value of the
significance of data calculation (Asym sig.) For the first and third hypothesis is smaller than the significance
level of 5% (0.05), so the null hypothesis is rejected and the alternative hypothesis is accepted. Whereas in the
second hypothesis, the significant value of data calculation (Asym sig.) Is greater than the significance level of
5% (0.05), so the null hypothesis is accepted and the alternative hypothesis is rejected. It can be concluded as
follows:
a.
b.
c.
There are significant differences between students' understanding of concepts that get structured
inquiry learning with understanding the concept of learning by students who received conventional
models.
There was no significant difference between students' understanding of the concepts that have high
student attitudes with low student attitudes.
There is interaction between students learning model with an attitude of students Based on the results
of comparative tests, the first hypothesis is known that there are differences in students' understanding
of concepts significantly the gain learning with structured inquiry model with understanding the
concept of learning by students who received conventional models.
When viewed from the mean value of understanding the concept of Table 3, it can be concluded that students
gain understanding of the concept of learning with structured inquiry model is better than the students who get
the understanding of the concept of learning with the conventional model. Understanding the concept of better
student after a given learning with structured inquiry model the impact of the advantages of the application of
structured inquiry model. Barlow in Muhibbin (2005) stated that the inquiry-based learning structured more
emphasis on the use of the intellectual process of learners in acquiring knowledge by finding and organizing
concepts and principles into an order of importance according to the student. Thus, it can be said in the inquiry
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Salim & Tiawa
process structured student trying to find a concept of knowledge in the learning material by using their
intellectual thus more memorable learning for students and last long in the memory. So that more students can
make understand the subject matter of geometry and specific learning makes understanding the concept of
students in geometry for the better material. The results of this study also found by Nurhayati (2011), which
indicates that the application of structured inquiry model can improve student learning outcomes in the material
geometry. In addition to the use of models of learning, other things that can affect student learning outcomes are
student perceptions dam attitude in the face of math. According Baharuddin in Ulum (2007), "Attitudes and
perceptions of students is basically the tendency of individuals to act or behave in solving a problem
systematically. Attitudes and perceptions of high student should have an impact on learning outcomes better.
Although in this study, the value of understanding the concept of students who have high student attitudes
slightly higher or about the same compared to the value of understanding the concept of students who have low
student attitudes (Table 5), but these results did not differ significantly. This can be seen from the comparative
test in Table 9.
The results also demonstrate the interaction between class learning (structured inquiry model and the
conventional model) with the attitudes and perceptions of students (high and low) on the understanding of the
concept of the students on the material geometry. In order to understand the interaction of the further analysis by
describing the graph in Figure 2.
80
60
40
20
0
experiment group
high
control group
low
Figure 2. Graph of interaction between students' attitudes and perceptions understanding the concept of students
on the learning material geometry
In Figure 2, it can be seen that the average value of the marginal understanding of the concept of geometry
students in the experimental class in general higher than the average marginal understanding of the concept of
geometry students in the control class. This suggests that learning by using structured inquiry model can
improve the understanding of the concept compared to conventional models (lecture and textbook). If the terms
of the attitudes and perceptions of students, students 'understanding of concepts is higher in classes taught with
structured inquiry model compared to the students' understanding of the concepts that have attitudes and
perceptions of students are low.
It shows that, with the attitudes and perceptions of high student students may be able to follow the learning
process that uses a structured inquiry model. Attitudes and perceptions of student like curiosity, creative and
proactive indispensable in the process of structured inquiry as to understand the learning material geometry.
Students who have a high attitude and perception has more high creativity in learning geometry. Just as seen in
conventional learning class students who have high student attitudes and perceptions more able to follow the
learning compared to students who have a low student attitudes.
Conclusion
Based on research that has been done, it can be concluded that there are differences in understanding of the
concept of learning by students who obtain structured inquiry model with understanding the concept of learning
by students who received conventional models. When viewed from the mean value, the students gain
understanding of the concept of learning with structured inquiry model is better than the understanding of the
concept of learning by students who received conventional models in the learning geometry (Bentley, 2012).
Understanding the concept of students who have high student attitudes better than the understanding of the
concept of students who have a low student perceptions attitude however, mastery of concepts based on high
and low students' attitudes did not differ significantly. There is interaction between structured inquiry learning
model with attitudes and perceptions of students is higher.
International Journal of Research in Education and Science (IJRES)
83
References
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Arikunto, S. (2006). Prosedur Penelitian Suatu Pendekatan Praktek. Jakarta: Rineka Cipta.
Bentley, T. (2012). Learning beyond the classroom: Education for a changing world. Routledge.
Johnson, D.A. & Rising, G.R.(1972). Guidelines for teaching mathematics. Jakarta: Departemen Pendidikan dan
Kebudayaan.
Joyce, B., Weil, M. & Calhoun, E. (2000). Models of Teaching. Boston: Allyn and Bacon.
Muhibbin, Syah. (2005). Psikologi Pendekatan dengan Pendekatan Baru. Edisi Revisi.Bandung: Remaja
Rosdakarya.
Nola, R. (2005). Pendula, models, constructivism and reality. In The Pendulum (pp. 237-265). Springer
Netherlands.
Nurhayati. (2011). Penggunaan Model Pembelajaran Berbasis Masalah Melalui Model Inkuiri dan Eksperimen
Ditinjau dari Kreativitas dan Sikap siswa Siswa Pendidikan Geometri STKIP PGRI Pontianak pada
Pembelajaran Geometri. Tesis PPs UNS Surakarta: tidak diterbitkan.
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Santyasa, I W. (2007). Landasan Konseptual Media Pembelajaran. Universitas Pendidikan Ganesha. Makalah.
Suparno, Paul. (2007). Metodologi Pembelajaran Geometri Kontruktivisme dan Menyenangkan. Yogyakarta:
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Ulum, B. (2007). Sikap siswa (http://blogbahrul.wordpress.com).
International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
Students’ Understanding of the Definite Integral Concept
Derar Serhan*
Emirates College for Advanced Education, United Arab Emirates
Abstract
This study investigated students’ procedural and conceptual knowledge of the definite integral.Twenty five
students enrolled in one section of an undergraduate Calculus II class participated in this study. Data were
collected from a test that was conducted during the fourth week of the semester. The test aimed at collecting
information about the students' procedural and conceptual knowledge of the definite integral.. The results
indicated that the students’ most dominant knowledge was the procedural one, and that they had limited
understanding of the definite integral. Their abilities to represent the concept in different ways were limited;
they could represent the concept using only two different representations.
Key words: Calculus; Definite Integral; Mathematics Education; Understanding
Introduction
University students encounter many Calculus concepts during their college education. These concepts are used
not only in Mathematics classes, but also in Chemistry, Engineering, Physics, as well as many others. The
definite integral concept is one of the main concepts introduced in Calculus and important for students to
master. Many research studies indicated that students have difficulties with the concept of the definite integral as
well as the concepts of function, limit, and derivative (Grundmeier, Hansen, & Sousa, 2006; Mahir, 2009;
Orton, 1984; Serhan, 2009; Tall & Vinner, 1981; Tall, 1987).
A major tenet of understanding is the capacity to make connections between conceptual and procedural
knowledge. Hiebert & Lefevre (1986) defined conceptual knowledge as "knowledge that is rich in relationships.
It can be thought of as a connected web of knowledge" (p. 3). Students' conceptual knowledge is developed by
the construction of relationships between pieces of information. On the other hand, procedural knowledge, is
divided into two distinct parts "One part is composed of the formal language, or symbol representation system,
of mathematics. The other part consists of the algorithms, or rules, for completing mathematical tasks"(p. 6).
According to Hiebert & Lefevre (1986) understanding takes place when new information is connected through
appropriate relationships to existing knowledge. It is very important to link conceptual and procedural
knowledge; students who are able to link the two together are able to develop a strong mathematical knowledge.
While students who are deficient in either kind of knowledge, or who developed both of them as separate
entities are not fully competent in dealing with mathematics concepts.
According to Tall & Vinner (1981) the concept image consists of all the mental pictures that are associated with
a given concept in the individual’s mind. The student’s image, that is developed based on his/her own
experiences with the concept, consists of everything a student associates with the concept; symbols, words,
pictures, etc. It takes years of all kinds of accumulated experiences to build that image which changes as the
individual matures and encounters new stimuli. Whenever the student processes new information, change may
occur on existing concept images. New difficulties, conceptions, and misconceptions may be encountered and
may cause uncertainty or conflicts with some parts of the concept image. However, the concept definition “is a
form of words used to specify the concept” (Tall & Vinner, 1981, p. 152). This may be a definition the student
has learned, or it may be a personal reconstruction of a definition by the student. The concept definition is then
the form of words the student uses for his/her own explanations of his/her (evoked) concept images (Tall &
Vinner, 1981).
One of the key concepts in Calculus is the definite integral of a function. While learning this concept, students
encounter Riemann sums, limits, derivatives, area, and many other concepts. To have a good understanding of
the definite integral, students should be able to make connections between all of these concepts as indicated by
*
Corresponding Author: Derar Serhan, dserhan@ecae.ac.ae
International Journal of Research in Education and Science (IJRES)
85
Hiebert & Lefevre (1986). Research in student understanding of these concepts informs us of common
misconceptions students hold about integration as well as common computational errors that students encounter.
Research also sheds information about the ways that students tend to view integration and points towards
suggested instructional methods. The present study adds to the research in the field of investigating students’
procedural and conceptual understanding and concept images of the definite integral. The research investigating
students’ understanding of the definite integral indicated that students associate the definite integral with finding
the area under the curve (Mahir, 2009; Rasslan & Tall, 2002; Sealey, 2006). In other studies, researchers found
that students were good at computing the integral (Abdul-Rahman, 2005; Ferrini-Mundy & Graham, 1994;
Orton, 1983; Pettersson & Scheja, 2008)
In a study that explored students' understanding of symbolic and verbal definitions of the definite integral,
Grundmeier et al. (2006) found that a large number of students were able to evaluate the integrals correctly but
had difficulty with both the symbolic and verbal definitions of a definite integral. Mahir (2009) administered a
questionnaire that aimed at investigating students’ understandings of the concepts associated with the definite
integral as well as their computation skills. Mahir found that students did not have a good conceptual
understanding of integration. Also Rasslan and Tall (2002) used a survey that included a question involving a
definite integral in which the function crossed the horizontal axis (changing the sign). They found out that many
students did not know how to calculate the area when the function changed its sign.
In summary, the research on definite integrals found that student knowledge was limited to procedural
knowledge since they were good at computing the integral but had difficulty explaining the negative area as well
as connecting the different representations of the definite integral. The present study was designed to investigate
students’ procedural and conceptual knowledge of the definite integral by examining their understanding
through the concepts that they form in association with the definite integral. Based on literature research, no
similar study was conducted in the UAE; therefore, the undertaking here is important and makes significant
contribution to the field. It is important for instructors to be aware of how and what students learn in their
introductory calculus courses so they may optimize student understanding. This study serves as a stepping stone
for further investigation of student learning in calculus.
Research Questions
The main aim of this study was to examine students’ procedural and conceptual knowledge of the definite
integral. The study aimed at addressing the following research questions:
1.
2.
3.
Which is the most dominant knowledge of the definite integral for students is it procedural
knowledge or conceptual knowledge?
Are students capable of dealing with negative areas and explaining their answers?
What concept images do Calculus II students associate with the definite integral concept?
Method
Participants
The participants in this study were 25 undergraduate students enrolled in a Calculus II course at a major
university in the United Arab Emirates. Students in this class had already been introduced to anti-derivatives,
definite integrals, Fundamental Theorem of Calculus, integration by substitution, area between curves, and
improper integrals in a Calculus I course that they finished the previous semester. The concepts in this class
were introduced to students verbally, algebraically, graphically and numerically.
Procedure
At the beginning of the semester, the researcher explained to the instructor the purpose of the study. Data were
collected from a test that was conducted during the fourth week of the semester. The test aimed at collecting as
much information as possible about the students' procedural and conceptual knowledge of the definite integral.
The test consisted of six questions that focused on students’ procedural and conceptual knowledge. The
questions were distributed as follows: the first two questions were computational in nature, asking the students
to evaluate given definite integrals; the third question focused on graphical representation especially of the
negative area; the fourth question was about connecting derivatives with definite integrals given the graph of the
derivative of a function; the fifth question was about improper integrals, and the last question was the following:
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Serhan
“What does

b
a
f ( x)dx
mean to you? Give as much details as you can.”
The first four questions were designed to examine students’ procedural knowledge and whether or not students
were capable of recognizing and connecting the different representations of integrals. The third question, in
particular, examined students’ understanding of negative area when the function changes its sign. The fifth
question was designed to examine if students would be able to recognize improper integrals when the integral is
undefined over the given interval. The aim of the last question was to evoke students’ concept images of the
definite integral concept and to check if students were able to make associations between the different
representations of the definite integral.
Analysis and Results
The study followed an in-depth analysis that focused on the students’ responses to each question. The objective
of the data analysis was to investigate students’ procedural and conceptual knowledge of the definite integral.
Students’ responses to the first five questions were given a numerical grade and were analyzed to get a good
understanding of students’ conceptual knowledge. Students’ responses to the sixth question were organized in
categories.
Most of the participants in this study were able to compute the given definite integrals in the first two questions,;
21 out of 25 participants (84%) were able to give a correct answer for both questions, while the other four
students were able to give a correct answer to only one of the two questions.
For the third question, the area above the x-axis between a and b and the area below the x-axis between b and c

c
f ( x)dx
were given. Based on these given areas students were asked to find a
and explain their answers. Out
of the 25 participants, 18 (72%) were able to come up with the correct answer. Only five of these 18 students
were able to explain their answers correctly. Following are some of the students’ explanations:
a.
b.
Because it is under the x-axis, the area is negative.
First area – second area
c.

b
b
d.

a
a
Adx
Adx
e.
Area =
f.

b
a
+

c
b
Bdx
c
 Bdx
+
b

c
a
f ( x)dx
f ( x)dx
-
c
a
f ( x)dx
Based on the graph of the derivative of a function that was given in the fourth question, students were
asked to find the critical points of the function. Only five out of the 25 students (20%) were able to come up
with the correct critical points.
The fifth question focused on one type of improper integral in which the integral was undefined over the given
interval. None of the students were able to figure out the answer to this question, most of them, 20 out of 25
stated the given answer was wrong and evaluated the integral and came up with different inaccurate values.

b
f ( x)dx
Students’ responses to the last question; “What does a
mean to you? Give as much details as you
can”, were categorized into the following four categories :’Area”, ‘Integral’, “Antiderivative” and “Unclear”.
These categories were establised based on students’ responses only. Table 1, provides a summary of students’
reponses within these four categories.
International Journal of Research in Education and Science (IJRES)
87
Table 1. Categories mentioned by students
Category
Number of Students
Area: The area between the curve and he x-axis from a to b
12
Integral: The integral of the function f(x) from a to b
7
Antiderivative
4
Unclear or no answer
7
Table 2 gives details of the number of students who mentioned multiple categories.
Table 2. Multiple categories mentioned by students
Number of Categories Mentioned by Students
Number of Students
0
7
1
11
2
6
Discussion and Conclusion
The main aim of this study was to examine students’ procedural and conceptual knowledge of the definite
integral. Based on the previous results, all students worked on the first two questions and most of them were
able to evaluate the given definite integrals. In addition to that, students were able to evaluate the integral based
on the given negative area. However, they had difficulty explaining their answers. It was difficult for them to
connect the derivative and the integral in addition to dealing with improper integrals. The results of this study
agree with with Orton’s findings that students have the ability to use procedural knowledge and solve
integration problems but have a limited understanidng of the basic concepts of integration.
Based on the categories of the last question, area was the most popular image of the definite integral that was
used by students. This agrees with the findings of previous research studies (Grundmeier, Hansen, & Sousa,
2006; Mahir, 2009; Rasslan & Tall, 2002; Sealey, 2006). The findings of the study also show that students had
difficulty explaining the negative area, which is also similar to findings of other research studies (Orton, 1983;
Rasslan & Tall, 2002). The second popular image that was given by students of 
b
a
f ( x)dx
indicated that this
was the integral of the function f(x) from a to b. This points to a mere language understanding of 
b
a
f ( x)dx
.
The results of this study indicated that students had limited understanding of the definite integral. Only six
students were able to come up with more than one representation of the concept. And none of them was able to
come up with more than two representations. The most dominant knowledge of the integral for students was
procedural knowledge. It is very important for students to develop the ability to make connections between
different representations. Every opportunity in the classroom should be used to help students accomplish that.
In addition to that, the results of this study revealed that none of the students mentioned the Riemann sum in
their image of the definite integral concept and that only a few of them mentioned the antiderivative in their
images. Therefore, it is important for instructors to review the way that this concept is presented and taught in
class. It is clear from the results of this study that there needs to be more emphasis on the multiple
representations and their connections with the concept definition. In addition to that, there should also be more
emphasis on the Riemann sum and how students may use it to enhance their structural understanding of the
definite integral.
This study was not designed to study a specific concept image of the definite integral; rather its aim was to
investigate students’ procedural and conceptual understanding of the definite integral concept in general. The
findings of this study shed light on students’ thinking and understanding. Further studies are needed to
investigate students’ thinking of other related concepts such as the area, Riemann sum and the Fundamental
Theorem of Calculus.
88
Serhan
References
Abdul-Rahman, S. (2005). Learning with examples and students’ understanding of integration. In A. Rogerson
(Ed.), Proceedings of the Eighth International Conference of Mathematics Education into the 21st
Century Project on Reform, Revolution and Paradigm Shifts in Mathematics Education. Johor Bahru:
UTM.
Ferrini-Mundy, J. & Graham, K. (1994). Research in calculus learning: Understanding of limits, derivatives, and
integrals. In J. Kaput & E. Dubinsky (Eds.), Research Issues in Undergraduate Mathematics Learning,
MAA Notes #33. Washington D.C.: Mathematical Association of America.
Grundmeier, T. A., Hansen, J., & Sousa, E. (2006). An exploration of definition and procedural fluency in
integral calculus. PRIMUS, 16(2), 178-191.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory
analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27).
Hillsdale, NJ: Lawrence Erlbaum Associates.
Mahir, N. (2009). Conceptual and procedural performance of undergraduate students integration. International
Journal of Mathematical Education in Science and Technology, 40(2), 201-211.
Orton, A. (1983). Students’ understanding of integration. Educational Studies in Mathematics. 14, 1-18.
Orton, A. (1984). Understanding rate of change. Math. School, 13(5) 23-26.
Pettersson, K., & Scheja, M. (2008). Algorithmic contexts and learning potentiality: A case study of students
understanding of calculus. International Journal of Mathematical Education in Science and Technology,
39(6), 767-784.
Rasslan, S. & Tall, D. (2002). Definitions and Images for the Definite Integral Concept, in: A. D. Cockburn &
E. Nardi (eds.) Proceedings of the 26th Conference PME, Norwich, 4, 89-96.
Sealey, V. (2006). Definite integrals, Riemann sums, and area under a curve: What is necessary and sufficient?
Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for
the Psychology of Mathematics Education, Merida, Mexico: Universidad Pedagogica Nacional, 2-46.
Serhan, D. (2009). Using concept maps to assess the effect of graphing calculators use on students' concept
images of the derivative at a point. International Journal for Mathematics Teaching and Learning.
Retrieved from http://www.cimt.plymouth.ac.uk/journal/on 09/20/2014.
Tall, D.O. (1987). Whither Calculus. Mathematics Teaching, 11, 50-54.
Tall, D.O. & Vinner, S. (1981). Concept image and concept definition in mathematics, with particular reference
to limits and continuity. Educational Studies in Mathematics, 12, 151 - 169.
International Journal of Research in Education and Science
Volume 1, Issue 1, Winter 2015
ISSN: 2148-9955
Experimental Studies on Electronic Portfolios in Turkey: A Literature
Review
1
Selahattin Alan1*, Ali Murat Sünbül2
Selçuk University, Turkey, 2 Necmettin Erbakan University, Turkey
Abstract
In this study, a literature review was conducted about an individual’s selected efforts, products stored in
electronic format, and electronic portfolios that reflect the development and capacity of multimedia systems. In
this context, relevant experimental studies performed in Turkey are collected to show e-portfolio application
forms, their development, and to provide information about current activities for the use of e-portfolios. When
the results of studies on e-portfolios are examined, it was determined achievement, attitude, creativity skills,
application skills, etc. about e-portfolio application users caused an increase in the positive direction in many
variables to reveal various features of e-portfolios. Users’ opinions have been positive.
Key words: Electronic portfolio; Alternative assessment; Technology
Introduction
In students’ evaluation in the center of the education system, the extent these individuals’ predetermined goals
and behaviors reached is measured. According to traditional evaluation approaches, whether students have
reached the determined gains or not, with a definitive statement language (successful, failed, and passed) is
expressed. In this assessment approach, the final evaluation made in the student's learning process is neglected.
For this reason, to make a more comprehensive evaluation, an assessment that includes the evaluation process is
needed to better link the student with what is learned from real life experiences and taught in the classroom for
performance-based assessment (Atılgan, Kan, & Dogan, 2009; Demirel, 2007). As a requirement of this need, in
time new searches and various changes have emerged in the field of evaluation (Altun & Olkun, 2005).
‘Alternative assessment approaches’, is a common name for the method that utilizes results and the evaluation
process together.. Self-assessment, peer assessment, performance assessment, authentic assessment, portfolio
assessment, and project-based assessment are the assessments imposed under this approach. At the same time,
alternative assessment approaches is not an approach that excludes the traditional evaluation approaches and
cannot be used instead., It should not be forgotten alternative assessment approaches is the complementary
nature to traditional evaluation (Atılgan, Kan, & Dogan, 2009; Korkmaz, 2004). As one of the alternative
assessment methods, the portfolio assessment has become synonymous with the concept of alternative
assessment. It is even used instead of performance-based assessment, the summative evaluation, and the
alternative assessment.
Portfolio can be described as a teaching and assessment tool that brings the works of students together with a
purpose to enable them to monitor their development and progress. The concept of portfolio comes from the
Latin word ‘portare’ meaning to move and ‘folio’ meaning sheets of paper (Sharp, 1997). This concept in the
educational literature is called different expressions, such as "student progress file," "product selection file," "
individual development file," and "educational development file." The Ministry of Education also used the
"student progress file" expression in an attempt to expand the portfolio with the new curriculum in 2005. Both,
in the need to use alternative assessment approaches and the need to integrate our educational technology
system in accordance with the development of the era, have come out as the need for electronic portfolios (eportfolio). E-portfolios are the multimedia systems that reflect a person’s development and capacity, selected
efforts, and products stored in electronic format. The e-portfolio has become a very effective teaching and
assessment tool as a result of unification of the advantages of computer. The web environment has
complementary evaluation approaches, in general, and advantages of the portfolio method, in particular,
integrating it into Turkey’s educational system will surely raise the quality of education (Alan, 2014). Eportfolio is also like a portfolio—an integrated, efficient approach that can be used both as an educational
method and an assessment tool.
*
Corresponding Author: Selahattin ALAN, salan@selcuk.edu.tr
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Alan & Sunbul
An e-portfolio is a collection of what students form in the learning process and choose one from those they store
in digital media. Using this system, students load the products they create onto a web site with the assistance of
the Internet. In this way, there are many advantages different from the conventional portfolio system, such as
students can access their work from anywhere and at any time. Also, students can use more multimedia
elements that appeal to other types of media, such as video, rich text, images, sounds; students do not depend
upon the paper environment and have the ability to store their work for a long time; easy sharing of work
between students through a virtual environment; and easy monitoring and evaluation by parents and educators.
Since the 1970s, various studies and applications on e-portfolios are encountered abroad (Demirli, 2007).
However, this type of assessment was introduced in Turkey after the 2000s. After 2005, with the start of new
training programs in portfolio assessment, use of the e-portfolio system in private schools was introduced to the
agenda. In state schools, because the students were unable to afford computers, e-portfolios were dropped from
the agenda. Insufficient time was provided for students to work on portfolios. Also, they were introduced only
for control by the education inspectors. Therefore, the e-portfolio system could not be attained and was dropped
from the curriculum. Some pilot studies created an infrastructure for e-portfolios in preparation for blog sites,
forum sites, and use of free and open source e-portfolio software as possible solutions. However, these types of
solutions appeared insufficient to meet the educational needs. Therefore, especially for students whose
computers skills were at a low level, such as elementary school students, a wide variety of media types revealed
products that conformed to current technology e-portfolio software (Alan, 2014).
In this study, experimental studies made on e-portfolio were combined and studied in various respects. Both
thesis and article or report formats are prepared. Only the thesis is more comprehensive. Thus, to provide
information about the application of e-portfolios in Turkey and its development, related literature was reviewed.
Related Literature
This section provides several studies and their results from 2002 to date. Various studies have been completed in
Turkey at various institutions to demonstrate the advantages and disadvantages of e-portfolios in the classroom
during this timeframe. Intended for use in an academic literacy course, Sanalan (2002) conducted a study with a
web-based assessment system at a university in the United States. The research group took certain TOEFL level
test scores from different countries for 14 undergraduate students from various departments admitted to a
university.. After a 14-week training period, nine- and three-day study summaries were collected from these
students. Microsoft Access database software for the e-portfolio software environment and a webpage creation
feature from this software were used. Researchers put forth an e-portfolio design, usage, features, and results of
students’ evaluation. The results about the use of e-portfolio in the classroom, and its advantages and
disadvantages were argued.
Baki and Birgin (2004) used a math lesson as an alternative assessment tool, which was a computer-assisted
personal development file (BDBGD) and its applicability within the education system. This study took place
during the 2002–2003 academic year in Trabzon, by two teachers who worked in two different schools in their
classes. Using e-portfolio software developed by the researcher and his team, the individual development files
were determined more efficient than conventional methods in terms of student's performance evaluations.
Additionally, students were offered the opportunity to evaluate their performances. Moreover BDBGD
improved the communications among students, parents, and teachers, which allowed parents to actively
participate in the evaluation process.
In his thesis, Özyenginer (2006) noted students, in Vocational High School in the Department of Computers
who utilized their computers / hardware lessons, related opinions and successes on electronic portfolio
preparation, writing statements that reflect the portfolio, and portfolio evaluations. Research at the Buca
Anatolian Vocational High School in the Department of Computers was conducted during the 2005–2006
academic year. There was a second class of 28 students studying on courses in computer hardware. The eportfolio software environment used was Microsoft Office PowerPoint. As a result of his research, it was
observed that students made research, learned new concepts, self-esteem, self-assessment of learning, use of
time, feel responsible for their work, and developed skills related to creativity. In addition, e-portfolio motivated
students to interact with teachers and felt a need for their ideas to be evaluated within the course, Özyenginer
concluded the students were completing courses more willingly.
International Journal of Research in Education and Science (IJRES)
91
Demirli (2007) studied the effects of e-portfolio learners' attitudes and perceptions of the teaching process. His
research was conducted on 33 in the Technical Education Faculty. E-portfolio software was used as a web-based
environment developed by the researcher. At the end of the study, students from many perspectives found the
process of e-portfolio teaching interesting and developed positive attitudes towards the process. Moreover, the
courses taught using e-portfolios observed students have a learner-centered attitude, and have made in-depth,
accurate concepts.
Çayırcı (2007) completed a project to determine the effects of a web-based, portfolio site on 7th grade students’
verbal and numerical courses. It was conducted on 67 students from the Marmara Region on social studies and
science courses. For the e-portfolio software environment, Çayırcı used the website he developed. At the end of
the study, the results showed positive effects for both verbal and numerical courses, and students’ attitudes
towards these courses showed positive effects as well.
Arap (2008) completed a project to discover the contributions of electronic portfolio ( e-portfolio) application
for English teachers and to determine if it has any effects on achievement scores. The project was conducted on
44 English teacher candidates, who are interns in state schools, studying in Mersin University, Department of
Foreign Language Education during the 2006–2007 academic year. In an e-portfolio software environment, the
free, easy website creation tool on Google Page Creator Google Sites is used. At the end of the study, the
practice of e-portfolios for English teachers produced positive effects, including positive achievement scores.
Erice (2008) used an electronic portfolio to determine the effects of English language skills for students with an
intermediate level on writing skills. Students in the electronic portfolio group showed students who keep
portfolios are more successful. An e-portfolio software environment, Dokeos, was the web-based e-learning
platform utilized. At the end of Erice’s study, the digital environment for second language writing skills
contributed positively to the computing experience and using a computer affected the user's attitudes towards
computers, especially towards reducing anxiety levels.
Döşlü’s (2009) research included 77 tenth grade students from four different classes of an Information and
Communication Technologies course in the district of Adana Pozantı, who studied at Martyrs Victory Sabancı
High School. This study was to determine attitudes toward teaching students the e-portfolio process, and also to
observe and evaluate whether students can prepare a web-based portfolio. The e-portfolio software environment
has a free blog site to provide web services (blogcu.com). At the end of the study, students’ attitudes were
determined positively affected by the e-portfolio. Furthermore, by utilizing the web-based portfolio method, the
students’ success was favorably influenced. Also, students achieved a high rate of positive findings from the
students' views on the subject.
Tonbul (2009), in his project entitled, "An E-Portfolio Model for Students of Department of English Language
Teaching, Gazi University," studied the perceptions and attitudes of the electronic portfolio as a learning and
assessment tool for students of English Language Teaching. His study sought to determine the experiences
regarding the electronic portfolio development and to propose an electronic portfolio model for the educators
and students. The designs of the electronic portfolio and its practice in the classroom are the basic topics for the
project with 26 students attending the English Teaching Department of Gazi University. For Tonbul’s study,
participants spent two months to develop an electronic portfolio application. E-portfolio software environment
for personal blog sites from Microsoft Corporation (MSN Spaces) was used. At the end of the research, students
liked e-portfolio practice, and student-teacher and student-student interactions using their e-portfolios were
effective. In the process of preparing the e-portfolio, it was observed that writing skills developed. Moreover, it
was difficult to provide good Internet access to create good e-portfolio that took longer to prepare and student’s
evaluations of each other might have created some problems.
Koç (2010), in the acquisition of computer literacy skills, studied the e-portfolio process on learner’s
performances and its effects towards attitudes. This study was conducted with 69 first year students registered in
a Computer-I course from Erzincan University, Faculty of Education, and the Department of Primary School
Education. In an e-portfolio software environment, a European web-based portal named "E-Portfolio Process in
Vocational Education" was developed under the project and utilized. At the end of the research, e-portfolio in
the usage of computer literacy teaching did not show any effect on students’ theoretical knowledge, but was
effective in the development of practical skills. Also, the study revealed students' self-assessment was positive.
Moreover, the process provided students with a sense of responsibility and contributed towards organization.
Taking into consideration the interest of youths towards the computer and Internet, Ogmen (2011) developed a
more current vocabulary learning tool to help students improve their vocabulary learning strategies. Ogmen
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Alan & Sunbul
aimed to raise the level of learner autonomy by requesting 89 students from 9th grade high school to utilize a
word e-portfolio for 24 weeks. Before and after use of participatory strategies to determine if there was a change
in the levels of learner autonomy, a poll was provided to participants before and after the study. Finally, the
students were asked to register in a distance education platform. They utilized the e-portfolio software
environment, Dokeos, a web-based e-learning. To follow the process, research logs were retained and interviews
were conducted with the active participants. At the end of the research period, 67% of the participants were
interested in e-portfolios. For assignments based on computer use, the participants paid more attention to using
the words they learned during the course. Also, participants obtained some new strategies, which means new
words from the applied learning e-portfolio. This made them develop more word strategies and had positive
effects for creating learner autonomy.
Özgür and Kaya (2011), the name of the project is “The Management Aspect of the E-Portfolio as an
Assessment Tool: Sample of Anatolian University”, in 2008-2009 academic year, a distance education program
serving approximately 12,000 students, Eskisehir Anatolian University Open Education Faculty, the design of
the e-portfolio system which was developed for the students of Pre-School Education Department, the students,
have conducted a study on the management, implementation and evaluation. As the e-portfolio software
environment a web-based developed environment is used in the Anatolian University. According to the result of
the research and university staff, e-portfolio's dynamism, ease of planning and organization has put forth that it
has a good working environment to share the recommendations, ideas and criticisms for the participants of the
application.
Akdoğan Yeşilova (2011), in his doctoral thesis, provided observations regarding the process of preparing
electronic portfolios for 35 7th grade students’ learning styles, attending Ülkü Primary School. Using their eportfolios, students delivered electronic presentations prepared with Microsoft PowerPoint software. The study
found, even though students were a bit lazy in the beginning, their attitudes changed during the process. These
students described the e-portfolio as exciting and fun, and added it made them learn new things in a fun way.
Looking at the students’ performances at the conclusion of the study, the students exhibited various learning
styles. The group with the highest performance was the largest group of visual learners, followed by kinesthetic
and visual learners.
Aktay’s (2011) project, "Web-Based Primary Portfolio (Webfolio) Application," aimed to analyze the
functionality of e-portfolio application completed in primary schools with a group of 18 students from the 4/C
class of MAT-FKB attending Eskisehir Provincial Directorate of Education Special Developmental School. The
application was conducted in Science and Technology, Mathematics, Social Studies, and Turkish courses. Using
e-portfolio software in a web-based environment developed by Aktay, it was observed the students thought the
web folio system for sharing communications, and peer assessment provided convenience in handling issues.
With regard to students' e-portfolio processes, it was both safety and fun, and increased cooperation between
students. Aktay’s study determined the web folio system was more effective from the traditional portfolio.
Barış’ (2011) thesis investigated the availability of e-portfolio activity on a social networking site and analyzed
how the e-portfolio evaluation process affected students’ success. Participants were 202 students in the 10th
grade, studying at the Technical High and Vocational High schools in Tekirdağ. The e-portfolio software
environment on a social networking platform, Facebook, was utilized, using software developed by Barış. This
study determined the e-portfolio assisted education to positively affect students' successes. Also, students'
attitudes changed in a positive direction. Moreover, the use of social networking in education developed
communications between teacher-student and student-student, improved learning responsibilities, and in terms
of changing the intended use of social networking initiated positive consequences.
Dağ (2011) completed a study in a primary school in the province of Trabzon to determine the effects of using
the web environment in fifth grade students’ math lessons. By creating a homework site, assignments were
provided to students in the on-line environment.. Thirty-three fifth grade students shared their homework using
the web environment and the teachers provided feedback using the web environment. Its impact on increasing
motivation towards mathematics was observed. Dağ developed the e-portfolio software environment, a webbased dynamic assessment system. According to results, in addition to traditional training to use the web
environment, the fifth grade students learning mathematics were affected positively and increased their interests
in the course. In addition, feedback provided by the teacher noted the e-portfolio had a positive impact on
students. Thus, the e-portfolio encouraged students to learn the course better, especially when reviewing the
topics and requesting further information to understand the topics.
International Journal of Research in Education and Science (IJRES)
93
Yastıbaş (2013) utilized an electronic portfolio with 17 students from English Preparatory Department at Zirve
University to assess speaking skills and whether it can be used for writing skills. This application determined if
it was beneficial to students' attitudes towards speaking skills and if beneficial, how much? Yastıbaş utilized an
e-portfolio software environment web-based e-learning and course method tools (lore.com) for eight weeks. At
the conclusion of the study, data were analyzed using content analysis method. This study determined the
electronic portfolio can be used for speaking skills’ purposes in evaluation. Also, the attitudes towards the
students' speaking skills were affected positively and showed improvement. According to study results, the
students improved their speaking, grammar, pronunciation, and vocabulary, and now they feel more efficient.
Conclusions
In these studies, the e-portfolio was developed from a variety of office software as a software environment, such
as e-learning portals, blogs, websites, social networking sites, desktop software by the researchers. Various
websites features were used in many environments. The diversity of these environments, while enriching the eportfolio areas, also provided convenience for interactions and communications in the use of web-based portals.
Finally, e-portfolios also raise the suitability of current technology methods.
When the conclusions of these studies are analyzed, the users of the e-portfolio environment had many positive
remarks, such as academic achievement, attitudes towards the course, sense of responsibility, creativity skills,
and practical skills. Moreover, with regard to user applications, it was observed e-portfolios put forth positive
opinions to highlight various features of the e-portfolio.
As can be seen through these various studies, e-portfolio applications have many positive contributions both in
teaching and evaluation. For this reason, to fully integrate this system into the Turkish education system, it is
necessary to attain a better quality of construction in our education system.
Notes
This paper is resulted from the doctoral (PhD) thesis study of the first author.
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Education. Mersin: Mersin Üniversitesi, Sosyal Bilimler Enstitüsü.
Atılgan, H., Kan, A., & Doğan, N. (2009). Eğitimde Ölçme ve Değerlendirme. Ankara: Anı Yayıncılık.
Baki, A., & Birgin, O. (2004). Alternatif Değerlendirme Aracı Olarak Bilgisayar Destekli Bireysel Gelişim
Dosyası Uygulamasından Yansımalar: Bir Özel Durum Çalışması. The Turkish Online Journal Of
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