Learning Mathematics in a Mobile App-Supported Math Trail Environment (eBook)

Adi Nur Cahyono is a Lecturer in the Department of Mathematics at Semarang State University in Indonesia. He completed his PhD in the Institute for Mathematics and Computer Science Education at the Johann Wolfgang von Goethe University in Frankfurt am Main, with a special research focus on the didactics of mathematics. He has presented and published several papers in his research area in national and international forums and scientific journals. 

Adi Nur Cahyono

 

 

 

 

 

 

 

 

 

About the author

Dr.rer.nat. Adi Nur Cahyono, S.Pd., M.Pd. was born on 11 March 1982 in Banjarnegara, Central Java, Indonesia. In 2000, he completed his secondary education at Senior High School in Karangkobar, Banjarnegara, and started his university studies in Mathematics Education at Universitas Negeri Semarang (UNNES), Indonesia, where he obtained a bachelor degree in 2004. He pursued a master’s degree in Mathematics Education at Universitas Negeri Semarang in 2006 and graduated in 2008. Beginning in 2005, he worked as a senior high school mathematics teacher for three years, and in 2008 he worked as a lecturer in mathematics education in an institute of teacher training and education science for one year. In 2009, he started a new job as a civil servant lecturer at the Department of Mathematics of the Faculty of Mathematics and Natural Sciences, Universitas Negeri Semarang.

He also has experience as an academic staff member at the Centre of Development of Education Media (PPMP), Universitas Negeri Semarang, and as a reviewer of Interactive Educational Multimedia Production at the Department of ICT Development for Education (BPTIKP), Central Java, Indonesia. He teaches mathematics education, especially geometry and the use of ICT in Mathematics Education, and he has participated in several research projects in these areas. He has presented and published several papers in his research area in national and international forums and scientific journals. In 2006, he was recognised as the winner of a competition of teachers with smart ideas held by Citibank Indonesia and the Indonesian Hope Foundation.

In 2013, he was awarded a PhD scholarship from the Islamic Development Bank (IDB), Jeddah, Saudi Arabia, through an IDB-UNNES PhD fellowship program. In the same year, he started his PhD study at the Institute for Mathematics and Computer Science Education, J. W. v. Goethe- University Frankfurt am Main, Germany, under the supervision of Prof. Dr. Matthias Ludwig from Goethe Universit Frankfurt and Professor Marc Schaefer, Ph.D from Rhodes University, South Africa. His PhD research focused on the MathCityMap-Project for Indonesia. In 2017 he was awarded a doctoral degree in Natural Sciences (Dr.rer.nat.) in the speciality of Didactics of Mathematics after he furnished evidence of his ability in regularly conducted doctoral proceedings through his dissertation entitled „Learning mathematics in a mobile app supported math trail environment“. In the same year, he resumed his position as a lecturer at Universitas Negeri Semarang to teach in the Undergraduate Program and Postgraduate Program of mathematics education.

 

 

Die Deutsche Nationalbibliothek verzeichnet diese Publikation

in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://d-nb.info/1137348240 abrufbar.

Learning mathematics in a mobile app-supported math trail environment

 

 

 

 

 

 

 

Springer

 

 

Adi Nur Cahyono

 

 

Learning mathematics in a mobile app-supported math trail environment

 

 

 

 

 

 

 

 

 

 

 

 

 

Springer

 

 

Adi Nur Cahyono aus Banjarnegara (Indonesien)

E-mail: cahyono@math.uni-frankfurt.de/ adinurcahyono@mail.unnes.ac.id

 

 

Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) angenommen vom Fachbereich 12 (Informatik und Mathematik) der Johann Wolfgang Goethe-Universität in Frankfurt am Main, 2017.

 

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Dekan: Prof. Dr. Andreas Bernig Erstgutachter: Prof. Dr. Matthias Ludwig

Zweitgutachter: Prof. Marc Schäfer, PhD. (Rhodes University, South Africa)

Datum der Disputation: 26. Juni 2017

 

 

 

 

 

 

 

 

 

 

 

Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://d-nb.info/1137348240 abrufbar.

 

 

Preface  

Considering mathematics as a human activity, as in Freudenthal’s educational credo, has led to the view of mathematics education as a process of doing mathematics that results in mathematics as a product. As a human activity, mathematics is also an activity of solving problems. Bringing problem-solving practice into the educational process can be useful as a tool for learning mathematics. Through this process, learners have the opportunity to reconstruct the mathematical experience and gain new mathematical knowledge. Furthermore, encouragement is needed to motivate learners to engage in this process.

However, the current public understanding of mathematics and its teaching and learning process is still not satisfactory. Many students do not enjoy engaging in mathematics activities. Mathematics is also sometimes considered to be a subject that is difficult, abstract and far from students’ lives, in terms of both daily life and occupation. This problem leads mathematicians and mathematics educators to think of ways to popularize mathematics among the public.

A variety of projects have been developed and implemented in numerous countries to raise public

 

 

awareness of mathematics. For instance, in an effort to communicate mathematics to the public, in 2008, the German “Year of Mathematics” was held successfully. One positive message from this event was Du kannst mehr Mathe, als Du denkst (You know more maths than you think). This message emphasizes the importance of working on the public’s views of what mathematics is about and what mathematicians do. The event also focused on news and challenges, as well as images and jobs.

In the same spirit, the MATIS I Team from IDMI Goethe- Universität Frankfurt, Germany has also given attention to this subject by developing and implementing a project called MathCityMap. This project comprises math trails around a city that are supported by the use of GPS-enabled mobile phone technology. It aims to share mathematics with the public (especially students), encouraging them to be more involved in mathematics. The project offers an activity that is designed to support students in constructing their own mathematical knowledge by solving the prepared mathematical tasks on the math trail and interacting with the environment, including the digital environment.

My PhD research is a part of the MathCityMap project. By following a design research paradigm, it focused on the development of a model for a mobile app-supported math trail programme and the implementation of this programme in Indonesia tailored to that country’s situation. The

 

 

implementation included a field empirical study to explore its effect on students’ motivation in mathematics, their performance in mathematics and teachers’ mathematical beliefs.

The results of the research are presented through my dissertation, which is submitted for the degree of Doctor rerum naturalium (Dr. rer. nat.) at Faculty 12 (Computer Science and Mathematics), Goethe University Frankfurt, Germany. The research described herein was conducted under the supervision of Professor Matthias Ludwig at the Institute of Mathematics and Computer Science Education, Faculty of Computer Science and Mathematics, Goethe University Frankfurt, between September 2013 and August 2016. These results have also been reviewed by Professor Marc Schäfer, SARChI Mathematics Education Chair at Rhodes University, South Africa.

Frankfurt, March 2017 Adi Nur Cahyono

 

 



 

 

Acknowledgements This thesis is the result of my PhD study at the Institute of Mathematics and Computer Science Education, Goethe University Frankfurt (GUF), Germany. It was implemented in collaboration with the Department of Mathematics, Semarang State University (UNNES), Indonesia and nine secondary schools in the city of Semarang with the agreement of the Department of Education of the city of Semarang, Indonesia. This study was funded by the Islamic Development Bank (IDB), Jeddah, Saudi Arabia through an IDB-UNNES PhD fellowship programme. It would not have been possible without the support, guidance and help of many people around me. Therefore, I would like to dedicate this section to acknowledging these people.

First, I would like to express my sincere gratitude to my Doktorvater/ supervisor, Professor Matthias Ludwig (Goethe University Frankfurt, Germany), who has supported and guided me with his impressive knowledge and expertise. It was a great opportunity for me to do my PhD research under his guidance. He guided me to see things in a comprehensive and coherent way that was not only important for my PhD work, but will also be essential for my future professional life. He not only gave me professional and academic support, but also personal support. Thank you very much for this experience, Professor Ludwig.

 

 



 

 

I am also indebted to Professor Marc Schäfer (Rhodes University, South Africa), my second supervisor, for his valuable guidance and support. He gave valuable feedback and input that stimulated me to further reflect on my own thought and works. Thank you very much for your support, enthusiasm, knowledge and friendship.

I would like to acknowledge the IDB and the PMU IDB- UNNES for the scholarship I received. I would also like to express my gratitude to UNNES, the university where I work as a lecturer, for giving me permission and encouragement to pursue my doctoral degree in Germany.

During my PhD study, I received support and help from my colleagues at the Institute of Mathematics and Computer Science Education, GUF (thanks to Phillip, Xenia, Hanna, Sam, Jorg, Iwan, Martin, and Anne) and the Department of Mathematics, Faculty of Mathematics and Natural Sciences, UNNES. I am also deeply indebted to the teachers and students in Semarang, Indonesia who participated in my studies. Completion of my PhD research would never have been possible without their help and cooperation. I would like to extend my gratitude to KJRI Frankfurt (General Consulate of Republic Indonesia Frankfurt am Main) and to all my friends during my stay in Germany.

Above all, I would like to express my immeasurable gratitude to my wife, Chatila Maharani, for her love,

 

 

understanding, unfailing encouragement and support. Without her unconditional support, I would never have completed my study. Although she was also busy with her doctoral study, she was always there whenever I needed her. To my daughter, Calya Adiyamanna Putri, you are a great kid who always cheerfully follows our journeys, and to my son, Arbiruni Neumain Cahyono, our spring child, who was born in the busy time when I prepared this dissertation submission. I dedicate this dissertation to the three of you. Special thanks to my parents (Bapak Harjono and Ibu Rusmini), my parents-in-law (Bapak Suwignyo Siswosuharjo and Ibu Nurhayati), my sisters, my brothers, my nephews and my nieces, who have always supported and encouraged me.

 

 



 

 

Zusammenfassung  

Das MathCityMap-Projekt ist ein Projekt der Arbeitsgruppe MATIS I des IDMI der Goethe-Universität Frankfurt. Es ist ein Projekt innerhalb des Math-Trail- Programms, das durch den Einsatz mobiler Technologie unterstützt wird. Die vorliegende Studie untersucht dieses Projekt speziell in Indonesien. Es wurde mittels folgendem Forschungsdesign durchgeführt und fand in einer vorbereitenden und drei nachfolgenden Phasen statt (eine Entwurfsphase, eine kleine Pilotversuchsphase und eine groß angelegte Feldversuchsphase).

Die Ziele dieser Studie konzentrierten sich auf das Design einer App und von Math-Trails rund um die Stadt, die Umsetzung des App-unterstützten Math-Trail-Programms in Indonesien und insbesondere die Untersuchung der Auswirkungen auf die Mathematik-Motivation und - Leistung der Schüler und auf die mathematische Überzeugungen der Lehrer. Die Ergebnisse dieser Studie und der Beitrag zur Entwicklung des MathCityMap- Projektes werden in dieser Arbeit vorgestellt.

Diese Studie wurde in einem theoretischen Rahmen, basierend auf der konstruktivistischen Theorie in der Mathematikausbildung, durchgeführt. Dabei wird davon ausgegangen, dass die Schüler im Mathematiklernprozess

 

 

ihr mathematisches Wissen aktiv erweitern, indem sie neu gewonnene Informationen mit ihrem bisherigen Wissen verbinden. Lehrer fungieren als Vermittler und sollen in der Lage sein, mathematische Probleme vorzubereiten und den Schülern Anleitung oder Unterstützung bei der Problemlösung zu bieten. Um diese didaktische Situation zu verwirklichen, muss eine besondere Mathematik- Lernumgebung entworfen werden.

In dieser Studie wurde eine Lernumgebung mit dem Math- Trail-Konzept als Grundidee entwickelt. Ein Math-Trail ist ein Pfad, bestehend aus einer Reihe von Stationen, an denen die Schüler Mathematik im Freien betreiben und entdecken können. Die Idee der klassischen Math-Trail- Projekte sind nicht neu, aber die Kombination dieses etablierten Outdoor-Bildungskonzepts mit dem Einsatz neuester Technologie ist es. Wir zeigen in dieser Studie dass der Einsatz von Mobiltechnologie im Freien das Potenzial hat, den Mathematik-Lernprozess zu unterstützen.

In der Designphase wurde die GPS-fähige App speziell für den Einsatz in Indonesien unter Berücksichtigung der lokalen Situationen, Bedingungen und Bedürfnisse entwickelt. Es handelt sich um eine native Android-App, welche folgendes anzeigt: die mathematischen Trailrouten/Karten einschließlich der Standortkoordinaten der mathematischen Trailaufgaben und der aktuellen

 

 

Benutzerposition, die authentischen mathematischen Probleme, die Informationen über das benötigte Werkzeug zu den Problemlösungen, die Hinweise (falls vom Benutzer angefordert) und die Rückmeldung über die eingegebenen Antworten. Die App zeigt auch die Kurzinformationen zum Objekt.

Die App wurde mit der MIT App-Inventor-Software speziell für MathCityMap. Indonesia erstellt und läuft auf Handys mit Android-Plattform. Neben der App wurden in dieser Phase auch 87 Aufgaben von Mathematiklehrinnen und Lehren die durch einen Workshop geschult wurden entworfen; die Aufgaben wurden 13 Math-Trails in verschiedenen Regionen der Stadt Semarang zugeordnet. Es gibt 6-8 Aufgaben für jeden Trail, mit verschiedenen mathematische Themen in einer sicheren Umgebung. Die entworfenen Trails führen durch über Schulgelände, Stadtparks, Märkte, aber auch zu historischen Gebäuden, Touristenattraktionen und anderen Orten.

Die Math-Trails und die Aufgaben wurden in das MathCityMap-Portal hochgeladen, so dass die Benutzer über die App darauf zugreifen können. Trail-Walker brauchen etwa zwei Stunden, um einen Trail zu erkunden (ca. 1-2 km Länge) und die Aufgaben auf diesem Trail zu lösen. Die entworfenen Math-Trails und die App bestanden zuerst den Evaluierungsprozess durch Validierung der Experten und einer Pilotphase mit einer kleinen Gruppe von Schülern und Lehrern. Einige Aufgaben wurden

 

 

aussortiert; die positiv evaluierten Aufgaben wurden für die nächste Phase verwendet.

Die groß angelegte Versuchsphase wurde in den Jahren 2015 und 2016 mit Schülern und Lehrern aus neun Sekundarschulen durchgeführt. Für jede Schule waren zwei Klassen (ca. 30 Schüler/Klasse) an dieser Studie beteiligt, eine Klasse als Versuchsgruppe und eine Klasse als Kontrollgruppe. Beide Gruppen in jeder Schule wurden von der gleichen Lehrperson mit gleichem Thema und Lehrstoff, aber mit unterschiedlichen Interventionen, unterrichtet. In der Versuchsgruppe wurden das Lehren und Lernen der Mathematik mittels App-unterstütztem Math-Trail-Programm durchgeführt. Die Schüler der Kontrollgruppe wurden auf dem herkömmlichen Weg unterrichtet.

Im App-unterstützten Math-Trail-Programm wurden Gruppen von 5-6 Schülern gebildet. Die Aktivitäten wurden während der normalen Schulzeit in etwa 2 x 45- minütige Abschnitten durchgeführt, mit einer kurzen einführenden Erklärung der Regeln und Ziele durch die Lehrer. Die Gruppen begannen dann ihre mathematische Reise von unterschiedlichen Aufgabenorten aus damit nur jeweils eine Gruppe an einer Aufgabe arbeitet. Während die Gruppen den Trail abwanderten, beobachteten und überwachten Forscher und Lehrer die Aktivitäten der Schüler, unterstützten diese aber nicht, da alle notwendigen

 

 

Informationen mittels der App verfügbar waren.

Mit Hilfe der App folgen die Schülergruppen einer geplanten Route, finden die Aufgabenorte und lösen das mathematische Problem vor Ort, und gehen dann zur nächsten Aufgaben weiter. Im Anschluss wurde mit allen Schülern der Klasse eine Nachbesprechung durchgeführt. Um Schülermotivation zu erfassen wurden Fragebögen (Situational Motivation Scale/SIMS) vor und nach den Aktivitäten verteilt, Ebenso wurden die mathematischen Überzeugungen (Beliefs) der Lehrer erfasst. Um den Lernfortschritt zu messen wurden individuelle mathematischen Tests erstellt und vor und nach den Aktivitäten durchgeführt. wurden.

Das Ergebnis der SIMS zeigt, dass die Art der Motivation der Schüler, sich zu engagieren, verschieden ist. Es hat sich gezeigt, dass alle Schüler einen positiven Self-Determined Index (SDI) besitzen. Dies bedeutet, dass ihre Motivation stark selbstbestimmt, bzw. eine verinnerlichte Motivationsform war. So waren intrinsische Motivation (IM) und identifizierte Regulierung (IR) dominannter als externe Regulierung (ER) und Amotivation (AM). Die Schüler empfanden die Aktivität interessant oder angenehm (ein Indikator für IM) sowie aussagekräftig oder wertvoll (ein Indikator für IR). Sie beteiligen sich an den Aktivitäten aus Eigeninteresse/Vergnügen/Zufriedenheit, und sie betrachteten ihre Teilnahme als ihre eigene Entscheidung.

 

 

Bei offene Anschlussfragen zum SIMS- Fragebogen berichteten die Schüler, dass sie an den mathematischen Outdoor-Aktivitäten Spaß hatten und interessiert an der Verwendung von moderner mobiler Technologie beim Lernen von Mathematik sind. Sie berichteten auch, dass das Lernen von Mathematikanwendungen in der realen Welt für sie wertvoll und sinnvoll war. Das SIMS-Ergebnis zeigt auch, dass es vor und nach der Teilnahme an diesem Programm einen signifikanten Unterschied in der Orientierung der Schülermotivation gab. Es gab eine Veränderung von extrinsisch motiviert hin zu mehr intrinsisch motiviert. Bemerkenswert ist auch, dass es diese intrinsische Motivation nicht auf den Neuigkeitseffekt zurückzuführen ist, da die Werte im zweiten Jahr der Studie sich nicht signifikant geändert haben.

Es zeigte sich ebenfalls, dass die Intervention die Leistungen der Schüler in Mathematik beeinflusst hat. Um die Wirkung  der  Intervention  auf  die Mathematikleistungen der Schüler zu bestimmen, wurden die Schülerresultate von Experimentalgruppe und Kontrollgruppe statistisch überprüft. Die Ergebnisse zeigen, dass der mittlere Leistungszuwachs der 272 Schüler in der Experimentalgruppe signifikant höher war als die der 248 Schüler in der Kontrollgruppe. Das zeigt, dass die Intervention mit diesem Programm die Leistung der Schüler in Mathematik erhöht hat.

 

 

Da der Lehr- und Lernprozess mit den Einstellungen der Lehrer verknüpft ist, beschäftigte sich diese Studie auch mit dem Einfluss des MathCityMap-Programms auf die mathematischen Überzeugungen der Lehrer im Zusammenhang mit ihren Strategien zur Förderung der Motivation der Schüler, sich in Mathematik zu engagieren. Bei den Lehrpersonen war zu erwarten dass durch die Fokusgruppen-Diskussion,         Lehrerfortbildungen (Workshop zur Aufgabengestaltung) und die Einbeziehung in den Beobachtungsprozess ein Einfluss auf die mathematischen Überzeugungen stattgefunden hat Ein Pre-Posttest-design zeugen, dass es einen Wechsel der mathematischen Überzeugungen, von traditionell orientiert hin zu mehr forschungsorientiert gab.

Lehrer, nachdem sie an diesem Programm beteiligt waren, mehr von Folgendem überzeugt wurden: Mathematik ist ein Werkzeug zum Denken, das Ziel der Schüler ist es zu verstehen, die Schüler sollten eine Autonomie haben, Mathematikfähigkeit ist offen für Änderung, und die Schüler werden sich in Mathematik engagieren wollen, wenn die Aufgaben interessant und herausfordernd sind. Es gab auch einen signifikanten positiven Effekt der mathematischen                                         Überzeugungen                   bei forschungsorientierten Lehrern auf die intrinsische Motivation der Schüler. Lehrer berichteten auch, dass durch die Nutzung der mobilen App es erleichtert wurde, Outdoor-Lehr- und Lernprozess zu gestalten.

 

 

Zusammengefasst konnte mit dieser Arbeit gezeigt werden, dass durch die für diese Studie entwickelte App und den dazu passenden Trail-Designprozess mehrere Math-Trails in der Stadt Semarang installiert werden konnten. Empirische Studien zum Einsatz der App und der Trails zeigen, dass die Studierenden waren sehr intrinsisch motiviert, sich in Mathematik zu engagieren. Sie haben durch diese Aktivitäten mathematische Erfahrungen gesammelt. Infolgedessen wurde ihre Leistung in der Mathematik verbessert. Der Einfluss dieses Programms auf den Wechsel der mathematischen Überzeugungen der Lehrer hat zu diesen Ergebnissen beigetragen. Wir empfehlen, dass Schulen und Öffentlichkeit die vorhandenen Math-Trails (einschließlich der unterstützenden Werkzeuge) nutzen oder neue Trails entwerfen, indem sie dem in dieser Studie angewendeten Entwicklungsmodell folgen. Weitere Untersuchungen sind für die Projektentwicklung und -umsetzung in anderen Orten/Städten mit anderen Situationen und auch anderen Studienaspekten und -bereichen notwendig.

 

 

Table of Contents Cover............................................................................ i
Preface........................................................................ iii
Acknowledgemens....................................................... vii
Zusammenfassung (German-Summary)........................... xi
Table of Contents....................................................... xix
Chapter 1. Introduction................................................. 1
1.1.  Background of the study.......................................... 1
1.1.1.    What is the problem?....................................... 5
1.1.2.    Can the mobile app-supported math trail programme offer a solution?        9
1.2.  Design of the study............................................... 13
1.2.1.  Topic of the study........................................... 13
1.2.2.  Aims of the study............................................ 14
1.2.3.  Research questions......................................... 16
1.2.4.  Approach of the study..................................... 19
1.2.5.  Phases of the study......................................... 21
1.3.  The book’s structure.............................................. 25
Chapter 2. Theoretical Background................................ 29
2.1.  Constructivism in mathematics education................. 30
2.2.  Students’ motivation in mathematics....................... 38
2.3.  Students’ performance in mathematics..................... 41
2.4.  Teachers’ mathematical beliefs............................... 45
2.5.  Didactical situation in mathematics.......................... 47
2.6.  Outdoor mathematics education.............................. 52
2.7.  Technology in mathematics education...................... 64
2.8.  Conceptual framework of the study......................... 71
Chapter 3. The MathCityMap Project............................ 75
3.1.  The concept and the goals of the project................... 76
3.2.  Components of the project..................................... 77
3.2.1.  Mathematical outdoor tasks............................. 77
3.2.2.  Mathematical city trips................................... 80
3.2.3.  Map-based mobile app.................................... 83
3.2.4.  MathCityMap community................................ 87
3.3.  Technical implementation of the project................... 89
3.3.1.  Steps of preparation........................................ 89
3.3.2.  Settings of the activity..................................... 90
3.3.3.  Evaluating and publishing the activity.............. 92
Chapter 4. Designing the mobile app-supported math trail environment in Indonesia        93
4.1.  Indonesian secondary school mathematics
education.................................................................... 93
4.1.1.  A brief profile of Indonesia.............................. 94
4.1.2.  Education system in Indonesia......................... 96
4.1.3.  Indonesian secondary school mathematics curriculum   98
4.2.  Design of the mobile app-supported math trail programme for Indonesia      101
4.2.1.  Design of the rules of the activity.................... 106
4.2.2.  Design of the mobile app............................... 110
4.2.3.  Design of the math trails............................... 127
Chapter 5. Evaluating the potential effects of a mobile app-supported math trail programme: An exploration study in Indonesia.................................................................. 151
5.1.  Method.............................................................. 151
5.1.1.  Approach..................................................... 152
5.1.2.  Participants, situations, activities and procedures    152
5.1.3.  Data collection and analysis.......................... 154
5.2.  Results............................................................... 158
5.2.1.  What was the initial condition of the students like?            159
5.2.2.  How has the programme been running?........... 163
5.2.3.  How did the students work?........................... 164
5.2.4.  Why do students engage in this programme?. 168
5.2.5.  What mathematical experience do they get after having engaged in this programme? 184
5.2.6.  Does the intervention promote students’ performance in mathematics?         199
5.2.7.  How does the project affect teachers’ mathematical beliefs?            202
5.3.  Discussion.......................................................... 208
Chapter 6. Conclusions and recommendations............... 219
6.1.  Conclusion......................................................... 219
6.2.  Recommendations............................................... 240
References................................................................ 243
 

 

Chapter 1 Introduction

 

This book reports a study on the development, implementation and evaluation of a mobile app-supported math trail programme in Indonesia. As an introductory chapter, this chapter contains the background of this research, including the problems that underlie the need for this research, and the solution offered through this study. The design of the study, including the topic, aims, questions, approach and phases of the study, is described in detail in this chapter. To facilitate an understanding of the flow of the study, a concept map is also presented in this part. The structure of this book is then outlined in the last section of this chapter.

1.1.  BACKGROUND OF THE STUDY

Mathematics plays a role in daily life – both individual and social – and in professional life. People, therefore, must be able to apply basic mathematics in their everyday lives, a skill that the Organisation for Economic Cooperation and Development (1999) termed “mathematics literacy”. In an educational context, mathematics activities should offer the chance for students to experience mathematics as a

 

 

meaningful subject that can be understood well (Freudenthal, 1991).

Furthermore, Ojose (2011) suggests that students should be provided with real-world situations that are relevant to their position as citizens or their concern area. Such situations can be experienced by mathematics students outside the classroom (Dubiel, 2000). This approach can show students that mathematics is all around them and not merely in textbooks. Bringing mathematics outdoors can provide an opportunity for students to experience mathematics in real-world situations.

In some countries, interest in the development of outdoors and adventure education programmes has increased (Fägerstam, 2012). Various activities outside the classroom have been specifically designed to increase student engagement. Integrated programmes have also been developed to combine learning outside the classroom with traditional learning in the classroom. In mathematics, one of the outdoor educational programmes that have been elaborated is the math trail programme. This programme is not purely an outdoor activity, as it can also be conducted indoors, but it can be used as one alternative to encourage students to learn mathematics outside the classroom.

The math trail programme, which is a pathway to discovering mathematics, was created as a medium for experiencing mathematics in relevant applications (Shoaf,

 

 

Pollak, & Schneider, 2004), namely communication, connections, reasoning and problem solving (National Council of Teachers of Mathematics, 2000). In a math trail, students can simultaneously solve mathematical problems encountered along the path, make connections, communicate ideas and discuss them with their teammates, and use their reasoning and skills in problem solving.

Dudley Blane and his colleagues first developed a math trail by blazing trails in the centre of Melbourne as a family holiday activity (Blane & Clarke, 1984). The programme was then strengthened when some schools took advantage of the trails by integrating them into their mathematics learning programmes. The success of this idea allowed this programme to be adapted and applied in different places, and math trail projects subsequently emerged in various cities, such as Vancouver, Boston, Philadelphia and San Francisco (Richardson, 2004).

Although the math trail project is not new, the idea of an outdoor education programme supported by mobile technology appears to be new. This idea is facilitated by the fact that in recent years, developments in mobile technology and mobile phone use have improved significantly (Cisco, 2016; Lankshear & Knobel, 2006). These improvements were followed by many mobile phone applications (apps), including those intended for use in outdoor activities, such as Endomondo® for outdoor sports. In learning activities, Wijers, Jonker, and Drijvers (2010)

 

 

suggested that mobile devices could be employed to promote learning outside the classroom.

However, to date, most mobile technology apps for mathematics learning have only been employed in regular teaching settings (Trouche & Drijvers, 2010). Thus, it is necessary to explore the potential of mobile technology for teaching and learning mathematics, including for use in outdoors mathematics learning, and in so doing, engage students in meaningful mathematical activity. The combination of reality and virtual reality is expected to contribute to student engagement (Schwabe & Göth, 2005; Wijers et al., 2010).

This point is used as the basis for the development of the MathCityMap project. The project is a programme of math trails that are facilitated by the use of mobile phone technology and use special mathematical tasks (Cahyono & Ludwig, 2014; Jesberg & Ludwig, 2012). The project offers the opportunity to experience mathematics in the environment by arranging educational tasks in several locations around the city. Through a mobile phone app- supported mathematical activity, learners in groups find the task locations by tracing a planned math trail and solve mathematical problems encountered along the path.

The project aims to develop math trails and supporting tools, such as a web portal and a mobile phone app, and then share the project with the public (especially students).

 

 

This approach is expected to have the potential to encourage students to be actively involved in mathematics and promote student motivation to engage in mathematical activity. During their involvement in each activity and their interaction with the environment, students have the opportunity to construct their own mathematical knowledge. As a result, this will achieve the primary target of improving their performance in mathematics.

Referring to these basic ideas and objectives, as a part of the MathCityMap project, the model of the mobile app- supported math trail programme was then developed. The implementation of this programme in Indonesia tailored to that country’s situation was also studied scientifically. It was motivated by several problems in the field of mathematics education, especially in Indonesia.

1.1.1.   What is the problem? Public understanding of mathematics is often unsatisfactory, and only a small number of students enjoy participating in maths during the school day (Behrends, 2009, p. 1). At school, and globally, mathematics is sometimes perceived as a difficult and abstract subject. Internationally, by the eighth grade, only about a quarter of students enjoy learning mathematics (Mullis, Martin, Foy, & Arora, 2012, p. 325). This issue is a worldwide phenomenon, but given the unique aspects of each country, in this study we restrict ourselves to focusing on the issue in Indonesia.

 

 

In Indonesia, the percentage of students who report being happy at school is high in comparison with other countries (Organisation for Economic Cooperation and Development, 2013). However, Hadi (2015) reported that “most students fear mathematics, tend to skip mathematics subjects, and are happy when the mathematics teacher is not able to come to class” (p. 1).

In addition, based on the TIMSS 2011 results in mathematics, only approximately 20% of Indonesian eighth-grade students like learning mathematics and only 3% are confident in their abilities in maths (Mullis et al., 2012). In general, mathematics involves learning a lot of processes and formulae that appear to be not only unconnected with each other but also irrelevant to students’ lives (Education, Audiovisual and Culture Executive Agency, 2011).

Students appear to be less motivated to learn mathematics. This problem is serious because if we refer to the results of Hattie’s (2009) meta-analysis, students’ motivation and attitude toward mathematics and science are related to their performance in those subjects. In a synthesis of over 800 meta-analyses relating to achievement, 288 studies reported that attitudes toward mathematics and science were related to mathematics and science achievement (Hattie, 2009, p. 50).

 

 

From six meta-analyses (327 studies involving 110,373 people), Hattie concluded that motivation has a significant positive effect on achievement. Hattie’s effect size of student motivation was d = 0.48. This means that the influence was labelled as being in the “zone of desired effects”. Therefore, we assume that the above conditions have an impact on the low performance of Indonesian students.

Nationally, overall, Indonesian secondary school student performance has improved significantly. For mathematics, between 2004 and 2006, the average examination score rose from 5.0–6.2 to 6.8–7.6 (EFA Secretariat Ministry of National Education of RI, 2007). However, based on the data from the UNESCO International Bureau of Education (2011), Indonesian students still rank low in international standardized tests. Since PISA 2000, the trend has shown no significant increase or decrease and has tended to stagnate at low performance values (Baswedan, 2014).

PISA 2012 ranked Indonesian students aged 15 years 64th out of 65 countries, with a score of 375 for mathematics. PISA 2012 examines how well students can perform with the knowledge they possess (Organisation for Economic Cooperation and Development, 2013). Similarly, in TIMMS 2011, the mathematics achievement of Indonesian eighth-grade students was ranked 38th out of 42 countries, with a score of 386 in comparison to an average score of 500.

 

 

Further, PISA 2012 found that 75.7% of students were unable to reach level 2. This indicates that Indonesian students were unable to extract the relevant information for a given task or use basic algorithms to answer questions. Students also had difficulty solving realistic problems, especially in geometry. In the national exam in 2012, the average score of junior high school students in mathematics was 5.78 out of a maximum of 10, and mastery of geometry was below 50% (Kementerian Pendidikan dan Kebudayaan RI, 2013).

The World Bank (2010) reported that an ineffective learning process contributes to the low quality of graduates from Indonesian secondary schools. Previous studies indicated that several factors may affect quality, such as an increased focus on theory and routine learning, a focus on external examinations, administrative demands, textbook- based teaching and learning, and a crowded curriculum (Marsigit & Rosnawati, 2011; Sembiring, Hadi, & Dolk, 2008).

The natural next question, therefore, is how to improve Indonesian students’ motivation in mathematics in an effort to promote their performance in mathematics. One expected approach concerns the implementation of the mobile app-supported math trail programme. Conceptually, this programme is intended to address

 

 

problems similar to those encountered in Indonesia, as described above.

1.1.2.   Can    the    mobile     app-supported     math    trail programme offer a solution? Mathematics can be found everywhere and is experienced in everyday situations. The environment provides unlimited sources and ideas for teaching and learning mathematics. Students need to connect with the subject through meaningful activities, such as by implementing abstract mathematical concepts in real-life situations. The math trail programme provides opportunities in this regard. Supported by the use of mobile phone technology, in this project, math trails were designed around the city and offered opportunities to encourage student involvement in meaningful mathematical activities by utilizing the advantages of the latest technology.

This programme has the math trail concept at its core. A math trail is a planned pathway that consists of a series of stops at which students can explore mathematics in the environment (English, Humble, & Barnes, 2010; McDonald & Watson, 2010; Shoaf et al., 2004). Students find and solve real problems related to mathematics in real situations and connect mathematics with other disciplines.

Math trail tasks are created in several locations and then collected in a database. During the implementation phase, the tasks can be selected and grouped into a math trail route

 

 

based on the conditions and objectives. The tasks, their locations and directions to these sites, as well as the route and the tools required for problem solving, are provided in a trail guide, which is a scale map of the trail.

With the rapid development of technology, it is possible to collect the tasks and design a trail guide based on a digital map through a portal. Teachers can arrange the math trails in a web portal, and students can access the trails with the help of GPS-enabled mobile devices or use a paper version of the trail guide. In this project, we designed a web portal and a mobile phone application to support the math trail programme.

Using mobile devices is a familiar practice to all social and economic groups. In fact, in recent years, rapid developments have occurred in the scope, uses and convergence of mobile devices (Lankshear & Knobel, 2006) used for computing, communications and information. Within the next five years, it is estimated that the total number of smartphones will account for nearly 50 per cent of all global devices and connections (Cisco, 2016, p. 3).

As in other countries around the world, the rapid development of mobile technology has also occurred in Indonesia. Indonesia is amongst the top five countries worldwide in terms of the number of mobile phone users, with growth in 2012 reaching 60% (Prayudi & Iqbal,

 

 

2013). The same authors also reported that the percentage of market share in 2012 was 68.8% Android, 18.8% iOS, 4.5% BlackBerry OS, 3.3% Symbian, 2.5% Windows Phone, 2.0% Linux and 2.1% others. The World Bank Group (2015) reported that mobile cellular subscription in Indonesia in 2012 was 319,000,000 or 126.18 per 100 people.

Using mobile technologies is also common practice in learning activities, and these technologies offer advantages in terms of learning opportunities. The National Council of Teachers of Mathematics (2008) stated that technology has the potential to be a fundamental tool for mathematics learning. O’Malley et al. (2003) called the learning process in which learners take advantage of this approach “mobile learning”. One of the benefits of mobile learning is the opportunity to place learning outdoors in the real world. Mobile devices have the potential to integrate the characteristics of effective learning, such as situated realistic learning, motivational power and teamwork (Wijers et al., 2010). Thus, mobile learning enables learners to build knowledge and construct their understanding in unusual settings (Winter, 2007).

The math trail is the core of the activities in this project, and mobile technology is used to support these activities. This shows that this project combines real and virtual environments. According to Schwabe and Göth (2005), this combination might contribute to the improvement of

 

 

student engagement. When students are highly motivated to engage in mathematics, they are interested in, and enjoy spending more time, learning and solving mathematical tasks. Such students tend to be more persistent in solving mathematical problems (Lepper & Henderlong, 2000). Enjoyment and persistence in learning mathematics and positive views concerning the value and relevance of mathematics learning can lead to engagement with mathematics (Attard, 2012). Rigby, Deci, Patrick, and Ryan (1992) reported that student engagement in learning processes has a positive impact on their understanding of new knowledge and their flexibility in using new information.

Thus, theoretically, the concept of the mobile app- supported math trail programme offers several benefits for solving the current problems in the field of mathematics education, especially in Indonesia. We felt that if it were carefully designed and integrated into an intervention, the programme developed in this project could increase students’ motivation to engage in the mathematics learning process. As a result, in line with the results of Hattie’s (2009) meta-analysis, it will have an impact on improving their performance in mathematics. However, the design and implementation of the teaching and learning programme are also related to teachers’ attitudes toward the programme and their practices in the classroom. Thus, it is also important to address teachers’ mathematical

 

 

beliefs related to their strategies in fostering students’ motivation to engage in mathematical activity.

1.2.  DESIGN OF THE STUDY

The conceptual idea described above must be studied scientifically to develop a theory, translate the theory into a model of an educational programme, implement the programme and evaluate it through an empirical study. This need prompted us to carry out this study.

1.2.1.  Topic of the study This PhD study researches the development of the model of the mobile app-supported math trail programme and its implementation in Indonesia. The implementation in this study also includes an empirical study to explore the impact of the programme developed on student motivation in mathematics, their performance in mathematics and teachers’ mathematical beliefs. The programme was developed and implemented primarily for school activities as part of the process of teaching and learning mathematics, so the programme was developed in accordance with the school mathematics curriculum in Indonesia. Since the area of school mathematics is too broad to be investigated in the framework of this study, it must be restricted to an area that is relevant to the issues mentioned in the part of the background of this study. Therefore, the programme was implemented particularly for lower secondary school or

 

 

junior high school level (for all grades and for all mathematics topics).

As a pilot study, we implemented and studied empirically the programme for secondary schools in the city of Semarang, Indonesia from 2013 to 2016. It was carried out in cooperation between the two countries through two institutions, namely IDMI Goethe-Universität Frankfurt in Germany and FMIPA Universitas Negeri Semarang in Indonesia, and in collaboration with the Department of Education of the city of Semarang to implement the project in nine schools in the city. Semarang is not representative of Indonesia as a whole but has unique characteristics as a town and includes high-, middle- and low-level schools. Both urban and rural schools are also found in this region, which has hills and seaside areas, while the economic level of citizens varies. Thus, the implementation of the programme in this city is expected to provide a large amount of information and to serve as a model for the development and implementation of the programme in other cities in Indonesia, which is rich in diversity.

1.2.2.  Aims of the study This study consisted of two phases, namely: (1) the developmental phase and (2) the implementation and empirical study phase. These two phases were preceded by a preliminary phase. The first phase was aimed at developing a model of a mathematics educational

 

 

programme grounded in the basic idea and objectives of the MathCityMap project. The basic idea of the MathCityMap project is to set up mathematical problem-solving activities in outdoor situations following the concept of the math trail and supported by the use of mobile technology. The programme offers the opportunity for learners to construct their own mathematical knowledge during their interactions with the environment.

Therefore, the programme was developed within a framework informed by constructivism in mathematics education and built using some basic concepts, such as the concept of the math trail and the use of mobile technology in mathematics education. From this theoretical framework, the model of a mobile app-supported math trail activity was formulated. This model includes the concept, objectives, components and technical implementation of the programme and its evaluation.

The second phase was aimed at implementing the programme in Indonesia tailored to the local situation. It included the evaluation of its implementation by conducting an empirical study. Through an exploration study, the potential of the programme for eliciting engagement in meaningful mathematics teaching and learning activities was explored. Decisive factors that affect students’ motivation to engage in mathematics and their performance in mathematics were also examined in this study. Furthermore, because teaching and learning

 

 

processes are linked with teachers’ attitudes, this study also considered the effects of the project on teachers’ mathematical beliefs. Thus, teachers’ mathematical beliefs related to their strategies in fostering students’ motivation to engage in mathematics education through the implementation of the programme were also investigated in this study.

1.2.3.  Research questions Taking into consideration the aims of the research, the three central research questions of this study were clarified. The first question was:

1.    How can the mobile app-supported math trail programme promote students’ motivation to engage in mathematics?

The meaning of the keywords of this question are as follows. First, the question starts with the words “how can”. This phrasing indicates that the question focuses on identifying the conditions and approaches that optimize the benefits offered by the programme. This programme is a “mobile app-supported math trail programme”, which is a core of the MathCityMap project and was also theorized through this study.

“Engagement” leads this study to determine how students are actively engaged in this programme. The use of outdoor

 

 

activities supported by a mobile phone application is a strategy to engage students in the mathematics learning process. Because students’ engagement in an activity is related to their motivation, it is necessary to investigate the nature of “students’ motivation” to engage in such an activity. It is also important to determine the impact of this programme in “promoting” student motivation in mathematics and to investigate the differences in student motivation to engage in mathematics before and after the programme.

Furthermore, as a type of mathematics education programme, this programme focuses on “mathematics”. It was designed to motivate students to engage in an activity that helps them to construct their own mathematical knowledge. Therefore, this study was also concerned with examining the impact of the programme on student performance in mathematics by answering the question:

2.    Does an intervention with a mobile app-supported math trail programme enhance students’ performance in mathematics?

The “intervention” in this study was performed by arranging a teaching and learning mathematics in which a “mobile app-supported math trail programme” was carried out. In line with the Indonesian school mathematics curriculum contents, “students’ performance in mathematics” addressed in this study includes their mathematical ability and skill in understanding and

 

 

applying mathematics in solving problems. This question led to the findings on the effect of this intervention on student performance in mathematics. It is controlled by knowing whether the performance of the students involved in this programme is better than that of students learning in regular settings.

Furthermore, the arrangement of learning needs the support of teachers in their teaching practice. Thus, the study also intervened in teachers’ mathematical beliefs, so the teachers’ strategies in their teaching practices were in line with the concept and objectives of programmes developed through this study. The impact of the intervention was evaluated in the study by answering the question:

3.    How do teachers’ mathematical beliefs differ after being involved in the project compared with their initial beliefs?

This question addresses teachers’ view of mathematics, and its teaching and learning, before, during and after involvement in the project. It led to the findings on “teachers’ mathematical beliefs” and measured changes in those beliefs. The impact of the project on teachers’ orientation – traditional or inquiry – was also associated with this question.

 

 

1.2.4.  Approach of the study This study globally followed a design research paradigm. Design research is also known as developmental research. This approach integrates research, development, implementation and dissemination and involves all participants (researchers, developers, teachers and practitioners) in dialogue in the field (Gravemeijer & Terwel, 2000). This work produces theories and prototypes that are theoretically and empirically founded and studied. Freudenthal (1968) described the principle of developmental research as follows:

experiencing the cyclic process of development and research so consciously, and reporting on it so candidly that it justifies itself, and that this experience can be transmitted to others to become like their own experience (p. 161).

Through this principle, Freudenthal suggested that the research experiences should be transferred to outsiders, such as teachers, who can then use the experiences as the basis of decisions when they implement them in their classroom to achieve their own goals according to the actual situation. This process shows that this kind of research fits into the pedagogical tradition (Gravemeijer & Terwel, 2000).

 

 

Van den Akker (1999) stated that developmental research is based on two objectives, namely the development of prototype products and the formulation of methodological suggestions for designing and evaluating prototype products. The prototypes produced in this study were the model of the programme, the math trail tasks and the mobile phone app. The aims of design research are to develop theories or models, instructional materials and an empirically grounded understanding of the work of the learning process (NCTM Research Advisory Committee, 1996).

Design research is particularly suitable in situations for which a full theoretical framework is not yet available; thus, hypotheses must still be developed, and some new teaching materials must be designed (Drijvers, 2003). To produce prototypes, it is necessary to establish criteria for their development. Nieveen (1999) offered three criteria that can be used: validity, practicality and effectiveness.

·         The product meets the requirement of validity if the components are based on state-of-the-art knowledge and consistently linked to each other (internally consistent).

·         Practicality is fulfilled when experts and practitioners claim that the prototype can be used and when the prototype can really be employed.

 

 

·         The product meets the criterion of effectiveness when the students appreciate the learning programme and the desired learning takes place (P. 127).

Therefore, the products developed in this study must meet these requirements. The evaluation of the products was conducted through multiple phases in this study.

1.2.5.  Phases of the study According to Drijvers (2003), “Design research has a cyclic character: a design research study consists of a research cycle in which thought experiments and teaching experiments alternate” (p. 20). The development that occurs during the cycles can be characterized as ranging from qualitative formative to more quantitative summative.



 

Fig. 1.1 Layers of formative evaluation (Tessmer, 1993)



 

 

Following the layers of formative evaluation proposed by Tessmer (1993) reveals the progress from small-scale to large-scale evaluation (Fig. 1.1). This method requires mixing a qualitative formative method at the beginning of the study and a quantitative summative method afterward, using a quasi-experimental approach (Bokhove, 2011). Therefore, this study took place in one preparatory cycle and three subsequent cycles (prototypical design (CYCLE I), a small-scale field experiment (CYCLE II) and a large- scale experiment (CYCLE III)). This phase is shown in Fig. 1.2.



 

Fig. 1.2 Design research cycles within this study

 

The preparatory cycle involved the literature review, which led to the theoretical framework and the design of the prototypes. The focus of activity in this cycle was to conduct an in-depth review of the literature cross-checked with field experiences and existing data. The result is the theoretical framework for the development of the project

 

 

for implementation in Indonesia. The cross-checking process was completed through an exploration step that consisted of benchmarking, needs analysis and focus group discussion (involving experts and practitioners/educators).

Based on the results of this activity, the specification procedure was defined and the technical requirements were developed, primarily as a foundation and guideline for designing prototypes. The results of this cycle are the guidelines for the technical implementation of the project in Indonesia, a guide to designing the prototypes. In addition, the research instruments based on the theory are also reviewed in this cycle to obtain research tools that are appropriate for the theory and the chosen situations.

The first intervention cycle focused on the prototypical design, such as the model of the programme including the activity rule, the design of math trails containing varied mathematical outdoor tasks and the design of the mobile phone app. The prototypes were designed based on the specifications and requirements defined in the previous cycle. In this cycle, the trails were created by the project team and also by the task contributors (teachers, educators and others), with reference to the guidelines and characteristics that were set in the previous cycle. The math trails were created by contributors through training and a workshop session. The mobile app to support the math trail programme was also created in this cycle by the app programmer of the project team. Activities took place until

 

 

the authors uploaded the tasks into the MathCityMap website portal.

In the second cycle, the prototypes were reviewed by experts (mathematics, mathematics education and mathematics educational multimedia), and a small-scale field experiment (simulation session) was carried out to determine how the prototypes would work. In this cycle, teachers and small groups of students performed math trail activities and tested the mobile app. It is necessary to see how the math trails run and how the app works. Here, teachers and a small group of students were involved. During this session, the researchers conducted observations and interviews. The prototypes were then revised based on the results of the review and simulation in this phase.

In the final cycle, we implemented the project in a large- scale class experiment and studied its effects. This cycle was carried out in several pilot studies by involving teachers and students from nine schools. The implementation consisted of an introduction session, a math trail run and a debriefing session. In this step, we explored how the math trails ran and how the application worked in all pilot studies, how the students performed and how motivated the students were to participate in the activity. We also investigated the attitudes and beliefs of teachers related to mathematics and this project. We then

 

 

analysed the data obtained in this step to answer the research questions, drawn conclusions and recommended (and criticized) some important points for the further development of the project.

1.3. THE BOOK’S STRUCTURE

This book consists of six chapters. These six chapters represent the main content and are preceded by a preface. The concept map of this study and its link to the content of each chapter of this book are illustrated in Fig. 1.3.

Chapter 1 is an introductory chapter. In this chapter, we present some background for this study, including the problems and solutions offered through this study. The topic, objectives, questions, approach and phases of this study are also described herein. At the end of this chapter, we outline the concept map of the study and the structure of the book.

Chapter 2 focuses on the theoretical background of the study. This chapter presents the results of a literature review relating to the study. The theory of constructivism in mathematics education linked with the concepts of student motivation in mathematics, students’ performance in mathematics and teachers’ mathematical beliefs is discussed in this chapter as the theoretical basis for the framework of the study. This section is followed by a discussion on some basic concepts that build on the idea of the study, such as the concept of mathematics trails and the

 

 

use of mobile technology in math trail programmes. From this theoretical background, the framework of the study was formulated.

Chapter 3 focuses on the MathCityMap project. The concept and goals of the project are formulated and the components of the project are also described. In reference to implementing the project, we also propose the technical implementation of the project in this chapter.

Chapter 4 describes the results of the development of the model of the mobile app-supported math trail programme for implementation in Indonesia. This chapter starts with a closer look at the situation in Indonesia and contains the results of designing a mobile app and math trails in the city of Semarang, as well as the rules of the activity.

Chapter 5 is a report on an exploration study of the implementation of the programme in Indonesia. This chapter describes the methodology and the procedures of the study. The impact of the programme on Indonesian secondary school students’ motivation to engage in mathematical activity, their performance in mathematics and the teachers’ mathematical beliefs are discussed in this chapter. At the end of this chapter, the results of this study are presented.

Chapter 6 is the concluding chapter. Here, we draw conclusions from the results obtained in this study and

 

 

provide    recommendations     for   future    activities    and researches.



 

Fig. 1.3 The concept map of the study

 

 



 

 

 

 

Chapter 6 Conclusions and recommendations

 

In this final chapter, we review the detailed results of this study that were presented in the previous chapters from a more global perspective. The aim of this chapter is to draw conclusions regarding the results as the answers to the research questions stated in the first chapter. Recommendations are also given in the last section of this chapter. There are two kinds of recommendations, namely: a recommendation that the programme be implemented and recommendations for further studies on the MathCityMap project.

6.1.  CONCLUSION

Mathematics plays an important role in daily life. However, the current public understanding of mathematics remains unsatisfactory. Many students do not enjoy mathematical activities and appear unmotivated to learn mathematics, while their attitudes affect their performance in mathematics. These problems have led mathematicians and mathematics educators to think of ways to popularize mathematics. A variety of projects have been developed and implemented in numerous countries to raise public

 

 

awareness of mathematics, one of which is the MathCityMap project, a project of the working group MATIS I IDMI Goethe University Frankfurt. It is a project of a math trail programme supported by the use of mobile technology.

The research on this project was carried out in multiple places and using multiple areas of study, including this PhD study. This work is a study on this project specifically in Indonesia. It was conducted by following a design research paradigm and took place in one preparatory phase and three subsequent phases (a design phase, a small-scale field experiment phase and a large-scale experiment phase). The aims of this study were focused on designing a mobile app and math trails around the city, implementing the mobile app-supported math trail programme in Indonesia and exploring its impact on students’ motivation and performance in mathematics and on teachers’ mathematical beliefs.

Because the MathCityMap project has not been theorized yet, this study was also concerned with formulating the concept of the MathCityMap project as a guideline for the implementation of this project, both in this study and other studies in the future. Therefore, in this study, the concept of the MathCityMap project was theorized. For the implementation of the project in Indonesia, the model of the mobile app-supported math trail programme was

 

 

developed; it was followed by designing the rules of the activity, the mobile app and the math trails around the city of Semarang. Then, the programme was implemented and studied empirically in nine secondary schools in the city of Semarang.

This study was conducted within a theoretical framework informed by constructivist theory in mathematics education. This view suggested that, in the learning mathematics process, students should actively construct their own mathematical knowledge by connecting new information they have gained with their previous knowledge. However, learners’ engagement in an activity depends on their motivation. Sociocultural views state that learners need guidance as a sort of push for students to get started in their engagement in the learning process, and classroom social interaction also motivates them to engage through peer pressure.

The external factors that encourage learners to undertake the construction of knowledge should, however, be reduced and eliminated because, based on the cognitive view, students should value learning for its own sake. The learning process should prepare students to become lifelong learners that gain the confidence to construct more complex knowledge and who are better equipped to make use of their mathematical knowledge and skills throughout life. Both views affect the strategies that are used to improve student motivation: the sociocultural view leads to

 

 

extrinsic motivation, while the cognitive view leads to intrinsic motivation.

The intrinsic-extrinsic motivation axis is explained in self- determination theory (SDT). This model conceptualizes a range of regulations, from intrinsic motivation to amotivation. Between these extremes lie identified regulation and external regulation. It has been hypothesized that self-determination is associated with improved psychological performance; therefore, it would be expected that intrinsic motivation, followed by identified regulation, is mostly associated with positive outcomes. Thus, the process of learning mathematics needs to be prepared and arranged so that students can be intrinsically motivated to engage in learning mathematics.

Nonetheless, the improvement of students’ motivation is only a stimulus for students to promote their engagement in mathematics. As a serious mathematics educational programme, the main focus of this activity is to support students in learning mathematics and constructing their own mathematical knowledge. By engaging in this programme, the students got the opportunity to learn and practise solving real problems by following the stages of mathematizing. These stages are: understanding a problem situated in reality; organizing real-world problems according to mathematical concepts and identifying relevant mathematics; transforming real-world problems

 

 

into a mathematical problem that represents the problem situation; solving mathematical problems; and interpreting mathematical solutions in terms of the real situation. In the studied project, students learned and experienced using what they knew in the process of solving problems related to mathematics.

This activity helped students to construct their own mathematical knowledge by solving the mathematical tasks on the planned math trail and interacting with the environment, including the digital environment. The digital tool is used as a bridge to help teachers support students in the mathematics learning process. It is hypothesized that the intervention with this programme enhances students’ performance in mathematics. Through this activity, students were expected to increase their understanding of mathematical concepts and their relationships with real life and improve their ability to apply mathematics in solving problems.

To reach this goal, the teachers also play an important role. Learners’ confidence and performance in mathematics are significantly associated with the teachers’ mathematical beliefs and teaching practices. Belief in the value of intrinsic motivational strategies is expected to be associated with inquiry-oriented teachers’ mathematical beliefs. In their teaching and learning practices, teachers should act as facilitators and not as transmitters of information to their students. Teachers should prepare

 

 

problems that will provoke the expected learning and offer guidance for students to learn new knowledge. Students accept the problem and then speak, think and act about the problem through their own motivation. This teacher- student interaction is important in helping learners to solve problems and develop their abilities to allow them to solve other or future problems by themselves. In this environment, the interaction was facilitated by the use of mobile technology.

Furthermore, the learning environment also needs to be well organized. The environment should offer opportunities for learners to adapt and reconfigure new information to construct new knowledge. Mathematical knowledge can be constructed through the action of organizing the subject matter, both matter from reality and mathematical matter (mathematization). Students are reinventing the teaching matter prepared by teachers in such interactions and under guidance (guided reinvention).

Taking mathematics outdoors allows learners to experience these processes in an authentic situation. In this situation, students can see the connection between mathematics and their real world. It also creates greater motivation and excitement for students in the learning process. Teachers can also design a mathematical task outside the classroom, because the environment provides many ideas for the subject matter of mathematics.

 

 

Tasks about an object in the environment are designed and grouped with other tasks in one location, then linked into a route. Furthermore, students undertake the exploration of mathematics in the environment around them by conducting mathematical outdoor activities: tracking the math trail and completing these planned mathematical tasks.

By taking advantage of these benefits of mobile technology, math trail tasks can be localized with GPS coordinates and pinned onto a digital map through a web portal. The trail walkers can then access the tasks and run the math trail activity with the help of a GPS-enabled mobile application. The app helps teachers to facilitate the process of learning mathematics, from preparing the problem to directing students to find the task locations and supporting them in solving mathematical problems. Mobile devices can also be used to integrate learning activities with authentic situations in outdoor environments. The math trail increases student engagement in learning mathematics activities. The combination of the math trail and the use of mobile technology has become a basic idea of the MathCityMap project.

The MathCityMap project The MathCityMap project is a project that involves the math trail programme, which is supported by the use of mobile technology. This project was not conceived merely to design and/or use the math trails; instead, the project

 

 

includes the entire process: preparation (how to design the programme), implementation (how the programme runs) and evaluation (how the programme impacts on students’ motivation, students’ performance in mathematics and teachers’ mathematical beliefs). A mobile phone app, as a supporting tool, was also designed, created and used during this project.

As the name implies, the MathCityMap project is composed of several components. These components are: mathematical outdoor tasks, mathematical city trips, a map-based mobile app and the MathCityMap community. This project is not only related to the implementation of the activity, but also addresses how to prepare the project and its instruments and how to evaluate the project. Consequently, the technical implementation of the project is made up of three main parts – the preparation, the activity and the evaluation.

The mobile app is designed by the team. Tasks are prepared by educators or teachers in a team or individually. Then, the tasks and app are evaluated and validated to ensure that they are feasible for use. The mobile app can be downloaded and installed by the public. The tasks can be used in private or public settings by following the rules that have been prepared. Technically, with the help of the mobile app, students/users follow a planned route, discover the task locations and answer the tasks related to what they

 

 

encounter along the path; further, they continue with the subsequent tasks.

Then, the implementation of the math trail and app is evaluated for further development. The process and results of the activities are also published so that others can take the benefits of these results and be inspired to develop it in other places. However, it is important to note that the implementation of the educational programme in a particular place requires adjustment to local conditions and needs.

The mobile app-supported math trail programme in Indonesia By following the theory, the specific procedures and the technical requirements of the MathCityMap project, a mobile app-supported math trail programme was designed to be implemented in Indonesia. The designing phase started with discussions between experts and practitioners, followed by teacher training, and then the process of designing and testing the tasks and mobile phone app was performed by the team. In this phase, the GPS-enabled mobile application was designed specifically for implementation in Indonesia in relation to the local situations, conditions and needs. It is a native app that displays the math trail routes/map, including the coordinates of the math trail tasks, the user’s current position, the authentic mathematical problems, the information about the tool needed in solving problems, the

 

 

hints on demand (if needed by the users) and the feedback on the answers entered. The app also gives brief information related to the object, such as its history, its function, etc. The app was created using the MIT App Inventor program and runs on Android platform mobile phones. It can be installed by downloading the installation file available at www.mathcitymap.eu or Google PlayStore.

In addition to making the app, in this phase, 87 tasks, grouped into 13 math trail routes scattered among various areas in the city of Semarang, were also designed by educators (including teachers through training and a workshop). There were six to eight tasks for each trail with a variety of topics located in a safe and comfortable environment. The trails were designed in several school areas, city parks, historical buildings, markets, tourist attractions and other locations. The math trails and the tasks were then uploaded into the portal so that users could access them through the mobile app. Trail walkers needed about two hours to explore a trail (about 1–2 km length) and to solve the problems encountered along the path. The math trails’ and mobile app’s designs passed the evaluation process, through validation by experts, and the simulation phase involved a small group of students and teachers that were considered eligible for use in the next phase.

 

 

In the mobile app-supported math trail programme, the activities can be carried out with students from different grades and the tasks can vary in terms of level and difficulty. The activities can be conducted in the area of the school, in the school’s neighbourhood or at several locations around the city. The duration of the activity is about two to three hours during school hours (morning time is recommended), after school hours or during holidays. Students in a class are divided into groups of five to six members. The activities are conducted during normal school hours for about two 45-minute periods, beginning with the teachers giving a brief explanation of the rules and goals. Groups then start their mathematical trip, each from a task location that is different from that of the others. As the groups trek the trail, teachers observe and supervise students’ activities but are not expected to assist them, because all the necessary information is available in the app. With the help of the app, students in groups follow a planned route, discover task locations, solve the problem on-site and then move on to subsequent tasks. A debriefing session is also needed to get feedback for evaluating the activity and preparing further activities.

Implementation and empirical study of the programme in Indonesia The large-scale field experimental phase was conducted in 2015 and 2016 and involved 520 students and nine teachers from nine secondary schools. For each school, there were

 

 

two classes (about 30 students/class) involved in this study: one class was the experimental group and one class was the control group. Both groups at each school were taught by the same teacher with the same topic and subject matter but with different interventions. In the experimental group, teaching and learning mathematics was carried out by running the mobile app-supported math trail programme. Meanwhile, in the control group, students learned mathematics in the regular setting. A debriefing session was conducted after the activities. Questionnaires and evaluations regarding students’ motivation (the self- reported Situational Motivation Scale (SIMS)), teachers’ mathematical beliefs and individual mathematics performances were also given before and after the activities.

According to the situation and condition, the activities were conducted in different settings. In general, the activities were held during school hours, but some schools’ activities were also organized on a Sunday (holiday) or after school hours. On average, each activity required two to three hours to complete. Four schools carried out activities outside the school (including in some city parks and near landmarks in the city of Semarang), while five schools conducted activities in the school area. These choices were made based on the conditions and situations (such as safety and the availability of teaching and learning time) that were unique for each school.

 

 

The results of the field empirical study revealed that, in general, the activities ran smoothly, the mobile app worked well, and the rules and goals of the programme were comprehensible. Nearly the entire group attempted to work in their own way (without using hints). Most students reported no problems in participating in these activities, including the use of the app. All groups discovered the locations of the tasks, and a total of 73 tasks were completed by all groups in this activity; 14 tasks were not answered or were completed incorrectly by some groups.

The answer to the first research question: the students’ motivation to engage in mathematics The findings indicate that it was easy to involve the students in the activities. Most of the students were actively engaged in the activities, expressed positive feelings and had no problems in carrying out the activities. Students’ engagement was associated with their motivation. The SIMS results show that the students’ motivations to engage in the activity were diverse; however, all had positive Self- Determined Index (SDI) scores (ranging from 2.00 to 16.25 and M ± SD = 7.3180 ± 2.92629), which indicates that their motivations were more often the self-determined or internalized forms of motivation, namely intrinsic motivation (IM) and identified regulation (IR), than external regulation (ER) and amotivation (AM). Students perceived the activity to be interesting or enjoyable (an indicator of IM) and meaningful or valuable (an indicator

 

 

of IR). They were engaged in the activities for their own sake/pleasure/satisfaction, and their engagement was considered to be a self-choice.

Open-ended follow-up questions revealed that students enjoyed the outdoor mathematical activities (30%) and were interested in the use of advanced mobile technology to learn mathematics (23%). They also reported that learning and applying mathematics in the real world were valuable and meaningful for them (18%). Other students mentioned different reasons or did not give a reason, and slightly negative feelings were also mentioned. The SIMS results also show that there was a significant difference in the reported orientation of the students’ motivation before and after they were involved in this programme. There was a change in the orientation from being extrinsically oriented toward being more intrinsically oriented. We also found that there has not been a decrease in the intrinsic motivation of students from the first year to the second year of implementation.

Generally, the project was successful in designing and offering a programme that can promote students’ intrinsic motivation in mathematics; however, this improvement of students’ motivation is only as a stimulus for students to promote their own engagement in mathematics. Their involvement in the learning process is expected to positively affect their performance in mathematics.

 

 

The answer to the second research question: the students’ performance in mathematics As a serious mathematics educational programme, the main focus of this activity is to help students in learning mathematics and constructing their own mathematical knowledge. We found that, by engaging in this programme, the students got opportunities to learn, experience and practise solving real problems by following the stages of mathematizing. These stages are: understanding a problem situated in reality; organizing real-world problems according to mathematical concepts and identifying relevant mathematics; transforming real-world problems into a mathematical problem that represents the problem situation; solving mathematical problems; and interpreting mathematical solutions in terms of the real situation.

Through this programme, students learned and experienced using what they knew to solve problems related to mathematics. The students practised extracting information from the problems they faced and using the mathematical concepts that they knew to solve those problems. They experienced mathematics in the environment and constructed their mathematical knowledge by applying mathematics in solving the problems encountered along the path.

To determine the effect of the intervention on students’ performance in mathematics, a statistical comparison of the students’ scores between the two groups was carried out.

 

 

The results show that the mean gain in performance score of the 272 students in the experimental group (16.2684) was significantly higher (Cohen’s d = 0.31) than that of the 248 students in the control group (4.2540); in other words, the programme enhanced students’ performance in mathematics, especially their ability and skills to understand and apply mathematics to solve problems. Besides the learning arrangement, the teacher was also an important factor in these outcomes.

The answer to the third research question: the teachers’ mathematical beliefs Teachers play an important role in organizing and running the strategy of the learning process. Teaching and learning processes are linked with teachers’ attitudes; as such, this study was also concerned with the influence of the programme on the teachers’ mathematical beliefs in relation to their strategies for fostering students’ motivation to engage in mathematics. The influence was given by holding focus group discussions, teacher training and a workshop for designing tasks, and involving teachers in the processes of observing and evaluating the activity.

The findings indicated that the project changed the teachers’ mathematical beliefs in relation to their strategies for fostering students’ motivation in mathematics. The comparison test shows that there was a change in the teachers’ mathematical beliefs from traditionally oriented

 

 

beliefs toward more inquiry-oriented beliefs, i.e. after being involved in this programme, the teachers were more likely to believe that: maths is a tool for thought; the students’ goal is to understand; students should have some autonomy; maths ability is prone to change; and students will want to engage in maths if the tasks are interesting and challenging.

The regression analysis results revealed that there was a significant effect of teachers’ mathematical beliefs on students’ motivation (y = 2,239 + 0,492x) with the contribution at a rate of 77.7%. The more teachers implement inquiry-oriented mathematical beliefs in their teaching strategies and practice, the more their students have intrinsic motivation to learn mathematics. This shows that although this project aimed to improve students’ motivation by designing a learning environment and a supporting tool, it also affected teachers’ mathematical beliefs in relation to their teaching strategies and practices for fostering student motivation in mathematics.

Remarks from the answers to the three research questions Finally, the project was successful in designing and offering a programme that can intrinsically motivate students to engage in mathematics. The design and arrangement of the math trail and the mobile app, as well as a combination of both, created a pleasant situation and attractive environment that offered valuable knowledge and experience in mathematics. They embody the aspects

 

 

of enjoying or being interested in the activity and the use of advanced mobile technology for learning mathematics (an indicator of intrinsic motivation), and of the value and meaningfulness of the mathematical tasks and the activity (an indicator of identified regulation), which were generated through the implementation of this project.

The students were interested in the use of advanced mobile technology for learning mathematics and enjoyed the outdoor mathematical activity. The students also believed that the mathematical tasks and the activity were valuable and meaningful for them. The intervention with the mobile app-supported math trail programme also enhanced students’ performance in mathematics, especially their ability to understand and apply mathematics to solve problems. The change in teachers’ mathematical beliefs, from being traditionally oriented to being more inquiry- oriented, which occurred as an effect of the project implementation, also contributed to this improvement in students’ intrinsic motivation to engage in mathematics.

Some criticism of the MathCityMap project However, some feedback on the implementation of the project needs to be considered, such as: the need for the optimization of mobile phone use to motivate students; aspects of the learning mathematics outcomes; and the need for a system that helps the teachers to prepare and run the activities. Firstly, we will criticize the students’

 

 

motivation to engage in this activity. Although the SDI scores of students affected by the use of the mobile phone are the highest, the data show that the majority of students were interested in these activities because they were sited outdoors. This is important because one of the goals of this project was to develop the concept of a math trail by taking advantage of mobile technology. Therefore, the potential of mobile technology to support outdoor mathematics learning needs to be further exploited.

Currently, the use of the mobile phone is directed toward supporting the improvement of students’ intrinsic motivation. However, based on the sociocultural viewpoint, learners need encouragement to initiate and engage in a learning process, despite the fact that the encouragement of external factors needs to be reduced gradually so that in the end students must learn for its own sake, as suggested by the cognitive view.

Here, the use of a mobile phone can play a role. The possibilities of this role, for example, include the use of these tools for outdoor games or giving scores in recognition of what has been achieved by learners. These are factors that extrinsically motivate students to engage in an activity. These rewards, however, may differ from the basic concept of the math trail, but these findings provide benefits for the development of the math trail and its innovations by taking advantage of the use of mobile technology, so that the programmes developed through this

 

 

project can be optimized to increase students’ motivation to engage in mathematics.

Secondly, it is about mathematical skills. The results of the study show that once the students were involved in the activities, they had the opportunity to learn mathematics, construct mathematical knowledge and improve their performance in mathematics. The topic and subject matter used in designing tasks for this programme were adapted to the learning objectives and standards of competence expected by the school mathematics curriculum.

However, the expected learning mathematics outcomes, or more specifically the aspects that are examined in the final maths examination, are not always related to real problem- solving abilities and do not always require the problem- solving skills that are trained through the math trail activity. Even some (multiple-choice) test items can be solved by using “the quick solution”.

Therefore, this programme cannot be used as a single strategy in the process of teaching and learning because the students need to be equipped with mathematical skills in other aspects. The programme is also not the best strategy when used alone, but it can become one of several strategies used to complement each other. This programme can take a role in improving the mathematical literacy of students.

 

 

Although based on research results, teachers involved in this project experienced a change in their beliefs from traditional orientations toward being more inquiry- oriented, but they are also required to prepare students for the examination and teach the subject matter in accordance with the requirements in the curriculum, i.e. there was a conflict between their beliefs and their needs. Therefore, time efficiency and the effectiveness of the strategy in achieving the objectives need to be managed carefully. Although this programme provides many advantages, organizing the students to do outdoor activities requires extra time and attention.

Before implementing the programme, the teacher or educator also takes time for preparation, such as designing tasks and uploading them into the web portal system. In designing math trail tasks, although the environments present many ideas for mathematical tasks, safety, comfort and public order factors are also constraints. Discussion, evaluation and validation of the tasks are also required to produce good math trail tasks in accordance with the concept. These require the development of a system that helps teachers to prepare math trail tasks and activities for their students.

However, overall, it can be concluded that the project was successful in designing and offering an engaging mathematics learning environment. Students easily engaged in the activities and these activities positively

 

 

affected students’ attitudes and motivations toward learning mathematics. Students were intrinsically highly motivated to engage in these interesting and fun mathematics activities. Their involvement encouraged them to experience the process of mathematization and construct their own mathematical knowledge. As a result, their performance in mathematics improved. Teachers have a role in achieving these outcomes. Afterward, the mathematical beliefs of the teachers involved in this project became more inquiry-oriented. This supports efforts to increase the students’ intrinsic motivation in mathematics.

6.2.  RECOMMENDATIONS

Based on the conclusions of this study, some recommendations are given at the end of this report on the study for two reasons. Firstly, the aim of this part was to concentrate the findings into concrete suggestions for implementing the programme and utilizing the results of this study. Secondly, the recommendations were aimed at highlighting the issues that were not addressed in this study and require further research and development.

In this study, it was reported that math trails, supported by the use of a mobile phone app, have many potential advantages. Schools can take advantage of the results of this project by integrating the mobile app-supported math trail programme into the teaching and learning processes. The public can also participate in this programme for

 

 

leisure by walking around the city while learning mathematics. The public can freely use the mobile app and run the math trail activities that have been designed, use the existing math trails that have been designed and tested in this study, and design new math trails and upload them into the portal system by following the specification procedures and technical requirements formulated in this study.

Implementation of this programme can be strengthened through cooperation with several parties, such as city governments, schools and education authorities. Cooperation can also be established with the tourism department, museum managers, travel managers and others. With this cooperation, this educational programme can be synchronized with the programme of the parties, e.g. mathematics city tours. Financial support can also be obtained through sponsorship cooperation with several parties, such as a mobile phone service provider or mobile device manufacturers and sellers.

However, based on the results of this study, some conditions and requirements must be considered. The important role of influencing student motivation and performance in mathematics must be designated when designing the app, tasks and activity. The relevance and value of the task must be clearly identified and linked to the concept and objectives of the project. Most importantly, the students must enjoy and be attracted to the programme,

 

Adi Nur Cahyono

 

 

 

 

 

 

 

 

 

About the author

Dr.rer.nat. Adi Nur Cahyono, S.Pd., M.Pd. was born on 11 March 1982 in Banjarnegara, Central Java, Indonesia. In 2000, he completed his secondary education at Senior High School in Karangkobar, Banjarnegara, and started his university studies in Mathematics Education at Universitas Negeri Semarang (UNNES), Indonesia, where he obtained a bachelor degree in 2004. He pursued a master’s degree in Mathematics Education at Universitas Negeri Semarang in 2006 and graduated in 2008. Beginning in 2005, he worked as a senior high school mathematics teacher for three years, and in 2008 he worked as a lecturer in mathematics education in an institute of teacher training and education science for one year. In 2009, he started a new job as a civil servant lecturer at the Department of Mathematics of the Faculty of Mathematics and Natural Sciences, Universitas Negeri Semarang.

He also has experience as an academic staff member at the Centre of Development of Education Media (PPMP), Universitas Negeri Semarang, and as a reviewer of Interactive Educational Multimedia Production at the Department of ICT Development for Education (BPTIKP), Central Java, Indonesia. He teaches mathematics education, especially geometry and the use of ICT in Mathematics Education, and he has participated in several research projects in these areas. He has presented and published several papers in his research area in national and international forums and scientific journals. In 2006, he was recognised as the winner of a competition of teachers with smart ideas held by Citibank Indonesia and the Indonesian Hope Foundation.

In 2013, he was awarded a PhD scholarship from the Islamic Development Bank (IDB), Jeddah, Saudi Arabia, through an IDB-UNNES PhD fellowship program. In the same year, he started his PhD study at the Institute for Mathematics and Computer Science Education, J. W. v. Goethe- University Frankfurt am Main, Germany, under the supervision of Prof. Dr. Matthias Ludwig from Goethe Universit Frankfurt and Professor Marc Schaefer, Ph.D from Rhodes University, South Africa. His PhD research focused on the MathCityMap-Project for Indonesia. In 2017 he was awarded a doctoral degree in Natural Sciences (Dr.rer.nat.) in the speciality of Didactics of Mathematics after he furnished evidence of his ability in regularly conducted doctoral proceedings through his dissertation entitled „Learning mathematics in a mobile app supported math trail environment“. In the same year, he resumed his position as a lecturer at Universitas Negeri Semarang to teach in the Undergraduate Program and Postgraduate Program of mathematics education.

 

 

Die Deutsche Nationalbibliothek verzeichnet diese Publikation

in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://d-nb.info/1137348240 abrufbar.

Learning mathematics in a mobile app-supported math trail environment

 

 

 

 

 

 

 

Springer

 

 

Adi Nur Cahyono

 

 

Learning mathematics in a mobile app-supported math trail environment

 

 

 

 

 

 

 

 

 

 

 

 

 

Springer

 

 

Adi Nur Cahyono aus Banjarnegara (Indonesien)

E-mail: cahyono@math.uni-frankfurt.de/ adinurcahyono@mail.unnes.ac.id

 

 

Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) angenommen vom Fachbereich 12 (Informatik und Mathematik) der Johann Wolfgang Goethe-Universität in Frankfurt am Main, 2017.

 

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Dekan: Prof. Dr. Andreas Bernig Erstgutachter: Prof. Dr. Matthias Ludwig

Zweitgutachter: Prof. Marc Schäfer, PhD. (Rhodes University, South Africa)

Datum der Disputation: 26. Juni 2017

 

 

 

 

 

 

 

 

 

 

 

Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://d-nb.info/1137348240 abrufbar.

 

 

Preface  

Considering mathematics as a human activity, as in Freudenthal’s educational credo, has led to the view of mathematics education as a process of doing mathematics that results in mathematics as a product. As a human activity, mathematics is also an activity of solving problems. Bringing problem-solving practice into the educational process can be useful as a tool for learning mathematics. Through this process, learners have the opportunity to reconstruct the mathematical experience and gain new mathematical knowledge. Furthermore, encouragement is needed to motivate learners to engage in this process.

However, the current public understanding of mathematics and its teaching and learning process is still not satisfactory. Many students do not enjoy engaging in mathematics activities. Mathematics is also sometimes considered to be a subject that is difficult, abstract and far from students’ lives, in terms of both daily life and occupation. This problem leads mathematicians and mathematics educators to think of ways to popularize mathematics among the public.

A variety of projects have been developed and implemented in numerous countries to raise public

 

 

awareness of mathematics. For instance, in an effort to communicate mathematics to the public, in 2008, the German “Year of Mathematics” was held successfully. One positive message from this event was Du kannst mehr Mathe, als Du denkst (You know more maths than you think). This message emphasizes the importance of working on the public’s views of what mathematics is about and what mathematicians do. The event also focused on news and challenges, as well as images and jobs.

In the same spirit, the MATIS I Team from IDMI Goethe- Universität Frankfurt, Germany has also given attention to this subject by developing and implementing a project called MathCityMap. This project comprises math trails around a city that are supported by the use of GPS-enabled mobile phone technology. It aims to share mathematics with the public (especially students), encouraging them to be more involved in mathematics. The project offers an activity that is designed to support students in constructing their own mathematical knowledge by solving the prepared mathematical tasks on the math trail and interacting with the environment, including the digital environment.

My PhD research is a part of the MathCityMap project. By following a design research paradigm, it focused on the development of a model for a mobile app-supported math trail programme and the implementation of this programme in Indonesia tailored to that country’s situation. The

 

 

implementation included a field empirical study to explore its effect on students’ motivation in mathematics, their performance in mathematics and teachers’ mathematical beliefs.

The results of the research are presented through my dissertation, which is submitted for the degree of Doctor rerum naturalium (Dr. rer. nat.) at Faculty 12 (Computer Science and Mathematics), Goethe University Frankfurt, Germany. The research described herein was conducted under the supervision of Professor Matthias Ludwig at the Institute of Mathematics and Computer Science Education, Faculty of Computer Science and Mathematics, Goethe University Frankfurt, between September 2013 and August 2016. These results have also been reviewed by Professor Marc Schäfer, SARChI Mathematics Education Chair at Rhodes University, South Africa.

Frankfurt, March 2017 Adi Nur Cahyono

 

 



 

 

Acknowledgements This thesis is the result of my PhD study at the Institute of Mathematics and Computer Science Education, Goethe University Frankfurt (GUF), Germany. It was implemented in collaboration with the Department of Mathematics, Semarang State University (UNNES), Indonesia and nine secondary schools in the city of Semarang with the agreement of the Department of Education of the city of Semarang, Indonesia. This study was funded by the Islamic Development Bank (IDB), Jeddah, Saudi Arabia through an IDB-UNNES PhD fellowship programme. It would not have been possible without the support, guidance and help of many people around me. Therefore, I would like to dedicate this section to acknowledging these people.

First, I would like to express my sincere gratitude to my Doktorvater/ supervisor, Professor Matthias Ludwig (Goethe University Frankfurt, Germany), who has supported and guided me with his impressive knowledge and expertise. It was a great opportunity for me to do my PhD research under his guidance. He guided me to see things in a comprehensive and coherent way that was not only important for my PhD work, but will also be essential for my future professional life. He not only gave me professional and academic support, but also personal support. Thank you very much for this experience, Professor Ludwig.

 

 



 

 

I am also indebted to Professor Marc Schäfer (Rhodes University, South Africa), my second supervisor, for his valuable guidance and support. He gave valuable feedback and input that stimulated me to further reflect on my own thought and works. Thank you very much for your support, enthusiasm, knowledge and friendship.

I would like to acknowledge the IDB and the PMU IDB- UNNES for the scholarship I received. I would also like to express my gratitude to UNNES, the university where I work as a lecturer, for giving me permission and encouragement to pursue my doctoral degree in Germany.

During my PhD study, I received support and help from my colleagues at the Institute of Mathematics and Computer Science Education, GUF (thanks to Phillip, Xenia, Hanna, Sam, Jorg, Iwan, Martin, and Anne) and the Department of Mathematics, Faculty of Mathematics and Natural Sciences, UNNES. I am also deeply indebted to the teachers and students in Semarang, Indonesia who participated in my studies. Completion of my PhD research would never have been possible without their help and cooperation. I would like to extend my gratitude to KJRI Frankfurt (General Consulate of Republic Indonesia Frankfurt am Main) and to all my friends during my stay in Germany.

Above all, I would like to express my immeasurable gratitude to my wife, Chatila Maharani, for her love,

 

 

understanding, unfailing encouragement and support. Without her unconditional support, I would never have completed my study. Although she was also busy with her doctoral study, she was always there whenever I needed her. To my daughter, Calya Adiyamanna Putri, you are a great kid who always cheerfully follows our journeys, and to my son, Arbiruni Neumain Cahyono, our spring child, who was born in the busy time when I prepared this dissertation submission. I dedicate this dissertation to the three of you. Special thanks to my parents (Bapak Harjono and Ibu Rusmini), my parents-in-law (Bapak Suwignyo Siswosuharjo and Ibu Nurhayati), my sisters, my brothers, my nephews and my nieces, who have always supported and encouraged me.

 

 



 

 

Zusammenfassung  

Das MathCityMap-Projekt ist ein Projekt der Arbeitsgruppe MATIS I des IDMI der Goethe-Universität Frankfurt. Es ist ein Projekt innerhalb des Math-Trail- Programms, das durch den Einsatz mobiler Technologie unterstützt wird. Die vorliegende Studie untersucht dieses Projekt speziell in Indonesien. Es wurde mittels folgendem Forschungsdesign durchgeführt und fand in einer vorbereitenden und drei nachfolgenden Phasen statt (eine Entwurfsphase, eine kleine Pilotversuchsphase und eine groß angelegte Feldversuchsphase).

Die Ziele dieser Studie konzentrierten sich auf das Design einer App und von Math-Trails rund um die Stadt, die Umsetzung des App-unterstützten Math-Trail-Programms in Indonesien und insbesondere die Untersuchung der Auswirkungen auf die Mathematik-Motivation und - Leistung der Schüler und auf die mathematische Überzeugungen der Lehrer. Die Ergebnisse dieser Studie und der Beitrag zur Entwicklung des MathCityMap- Projektes werden in dieser Arbeit vorgestellt.

Diese Studie wurde in einem theoretischen Rahmen, basierend auf der konstruktivistischen Theorie in der Mathematikausbildung, durchgeführt. Dabei wird davon ausgegangen, dass die Schüler im Mathematiklernprozess

 

 

ihr mathematisches Wissen aktiv erweitern, indem sie neu gewonnene Informationen mit ihrem bisherigen Wissen verbinden. Lehrer fungieren als Vermittler und sollen in der Lage sein, mathematische Probleme vorzubereiten und den Schülern Anleitung oder Unterstützung bei der Problemlösung zu bieten. Um diese didaktische Situation zu verwirklichen, muss eine besondere Mathematik- Lernumgebung entworfen werden.

In dieser Studie wurde eine Lernumgebung mit dem Math- Trail-Konzept als Grundidee entwickelt. Ein Math-Trail ist ein Pfad, bestehend aus einer Reihe von Stationen, an denen die Schüler Mathematik im Freien betreiben und entdecken können. Die Idee der klassischen Math-Trail- Projekte sind nicht neu, aber die Kombination dieses etablierten Outdoor-Bildungskonzepts mit dem Einsatz neuester Technologie ist es. Wir zeigen in dieser Studie dass der Einsatz von Mobiltechnologie im Freien das Potenzial hat, den Mathematik-Lernprozess zu unterstützen.

In der Designphase wurde die GPS-fähige App speziell für den Einsatz in Indonesien unter Berücksichtigung der lokalen Situationen, Bedingungen und Bedürfnisse entwickelt. Es handelt sich um eine native Android-App, welche folgendes anzeigt: die mathematischen Trailrouten/Karten einschließlich der Standortkoordinaten der mathematischen Trailaufgaben und der aktuellen

 

 

Benutzerposition, die authentischen mathematischen Probleme, die Informationen über das benötigte Werkzeug zu den Problemlösungen, die Hinweise (falls vom Benutzer angefordert) und die Rückmeldung über die eingegebenen Antworten. Die App zeigt auch die Kurzinformationen zum Objekt.

Die App wurde mit der MIT App-Inventor-Software speziell für MathCityMap. Indonesia erstellt und läuft auf Handys mit Android-Plattform. Neben der App wurden in dieser Phase auch 87 Aufgaben von Mathematiklehrinnen und Lehren die durch einen Workshop geschult wurden entworfen; die Aufgaben wurden 13 Math-Trails in verschiedenen Regionen der Stadt Semarang zugeordnet. Es gibt 6-8 Aufgaben für jeden Trail, mit verschiedenen mathematische Themen in einer sicheren Umgebung. Die entworfenen Trails führen durch über Schulgelände, Stadtparks, Märkte, aber auch zu historischen Gebäuden, Touristenattraktionen und anderen Orten.

Die Math-Trails und die Aufgaben wurden in das MathCityMap-Portal hochgeladen, so dass die Benutzer über die App darauf zugreifen können. Trail-Walker brauchen etwa zwei Stunden, um einen Trail zu erkunden (ca. 1-2 km Länge) und die Aufgaben auf diesem Trail zu lösen. Die entworfenen Math-Trails und die App bestanden zuerst den Evaluierungsprozess durch Validierung der Experten und einer Pilotphase mit einer kleinen Gruppe von Schülern und Lehrern. Einige Aufgaben wurden

 

 

aussortiert; die positiv evaluierten Aufgaben wurden für die nächste Phase verwendet.

Die groß angelegte Versuchsphase wurde in den Jahren 2015 und 2016 mit Schülern und Lehrern aus neun Sekundarschulen durchgeführt. Für jede Schule waren zwei Klassen (ca. 30 Schüler/Klasse) an dieser Studie beteiligt, eine Klasse als Versuchsgruppe und eine Klasse als Kontrollgruppe. Beide Gruppen in jeder Schule wurden von der gleichen Lehrperson mit gleichem Thema und Lehrstoff, aber mit unterschiedlichen Interventionen, unterrichtet. In der Versuchsgruppe wurden das Lehren und Lernen der Mathematik mittels App-unterstütztem Math-Trail-Programm durchgeführt. Die Schüler der Kontrollgruppe wurden auf dem herkömmlichen Weg unterrichtet.

Im App-unterstützten Math-Trail-Programm wurden Gruppen von 5-6 Schülern gebildet. Die Aktivitäten wurden während der normalen Schulzeit in etwa 2 x 45- minütige Abschnitten durchgeführt, mit einer kurzen einführenden Erklärung der Regeln und Ziele durch die Lehrer. Die Gruppen begannen dann ihre mathematische Reise von unterschiedlichen Aufgabenorten aus damit nur jeweils eine Gruppe an einer Aufgabe arbeitet. Während die Gruppen den Trail abwanderten, beobachteten und überwachten Forscher und Lehrer die Aktivitäten der Schüler, unterstützten diese aber nicht, da alle notwendigen

 

 

Informationen mittels der App verfügbar waren.

Mit Hilfe der App folgen die Schülergruppen einer geplanten Route, finden die Aufgabenorte und lösen das mathematische Problem vor Ort, und gehen dann zur nächsten Aufgaben weiter. Im Anschluss wurde mit allen Schülern der Klasse eine Nachbesprechung durchgeführt. Um Schülermotivation zu erfassen wurden Fragebögen (Situational Motivation Scale/SIMS) vor und nach den Aktivitäten verteilt, Ebenso wurden die mathematischen Überzeugungen (Beliefs) der Lehrer erfasst. Um den Lernfortschritt zu messen wurden individuelle mathematischen Tests erstellt und vor und nach den Aktivitäten durchgeführt. wurden.

Das Ergebnis der SIMS zeigt, dass die Art der Motivation der Schüler, sich zu engagieren, verschieden ist. Es hat sich gezeigt, dass alle Schüler einen positiven Self-Determined Index (SDI) besitzen. Dies bedeutet, dass ihre Motivation stark selbstbestimmt, bzw. eine verinnerlichte Motivationsform war. So waren intrinsische Motivation (IM) und identifizierte Regulierung (IR) dominannter als externe Regulierung (ER) und Amotivation (AM). Die Schüler empfanden die Aktivität interessant oder angenehm (ein Indikator für IM) sowie aussagekräftig oder wertvoll (ein Indikator für IR). Sie beteiligen sich an den Aktivitäten aus Eigeninteresse/Vergnügen/Zufriedenheit, und sie betrachteten ihre Teilnahme als ihre eigene Entscheidung.

 

 

Bei offene Anschlussfragen zum SIMS- Fragebogen berichteten die Schüler, dass sie an den mathematischen Outdoor-Aktivitäten Spaß hatten und interessiert an der Verwendung von moderner mobiler Technologie beim Lernen von Mathematik sind. Sie berichteten auch, dass das Lernen von Mathematikanwendungen in der realen Welt für sie wertvoll und sinnvoll war. Das SIMS-Ergebnis zeigt auch, dass es vor und nach der Teilnahme an diesem Programm einen signifikanten Unterschied in der Orientierung der Schülermotivation gab. Es gab eine Veränderung von extrinsisch motiviert hin zu mehr intrinsisch motiviert. Bemerkenswert ist auch, dass es diese intrinsische Motivation nicht auf den Neuigkeitseffekt zurückzuführen ist, da die Werte im zweiten Jahr der Studie sich nicht signifikant geändert haben.

Es zeigte sich ebenfalls, dass die Intervention die Leistungen der Schüler in Mathematik beeinflusst hat. Um die Wirkung  der  Intervention  auf  die Mathematikleistungen der Schüler zu bestimmen, wurden die Schülerresultate von Experimentalgruppe und Kontrollgruppe statistisch überprüft. Die Ergebnisse zeigen, dass der mittlere Leistungszuwachs der 272 Schüler in der Experimentalgruppe signifikant höher war als die der 248 Schüler in der Kontrollgruppe. Das zeigt, dass die Intervention mit diesem Programm die Leistung der Schüler in Mathematik erhöht hat.

 

 

Da der Lehr- und Lernprozess mit den Einstellungen der Lehrer verknüpft ist, beschäftigte sich diese Studie auch mit dem Einfluss des MathCityMap-Programms auf die mathematischen Überzeugungen der Lehrer im Zusammenhang mit ihren Strategien zur Förderung der Motivation der Schüler, sich in Mathematik zu engagieren. Bei den Lehrpersonen war zu erwarten dass durch die Fokusgruppen-Diskussion,         Lehrerfortbildungen (Workshop zur Aufgabengestaltung) und die Einbeziehung in den Beobachtungsprozess ein Einfluss auf die mathematischen Überzeugungen stattgefunden hat Ein Pre-Posttest-design zeugen, dass es einen Wechsel der mathematischen Überzeugungen, von traditionell orientiert hin zu mehr forschungsorientiert gab.

Lehrer, nachdem sie an diesem Programm beteiligt waren, mehr von Folgendem überzeugt wurden: Mathematik ist ein Werkzeug zum Denken, das Ziel der Schüler ist es zu verstehen, die Schüler sollten eine Autonomie haben, Mathematikfähigkeit ist offen für Änderung, und die Schüler werden sich in Mathematik engagieren wollen, wenn die Aufgaben interessant und herausfordernd sind. Es gab auch einen signifikanten positiven Effekt der mathematischen                                         Überzeugungen                   bei forschungsorientierten Lehrern auf die intrinsische Motivation der Schüler. Lehrer berichteten auch, dass durch die Nutzung der mobilen App es erleichtert wurde, Outdoor-Lehr- und Lernprozess zu gestalten.

 

 

Zusammengefasst konnte mit dieser Arbeit gezeigt werden, dass durch die für diese Studie entwickelte App und den dazu passenden Trail-Designprozess mehrere Math-Trails in der Stadt Semarang installiert werden konnten. Empirische Studien zum Einsatz der App und der Trails zeigen, dass die Studierenden waren sehr intrinsisch motiviert, sich in Mathematik zu engagieren. Sie haben durch diese Aktivitäten mathematische Erfahrungen gesammelt. Infolgedessen wurde ihre Leistung in der Mathematik verbessert. Der Einfluss dieses Programms auf den Wechsel der mathematischen Überzeugungen der Lehrer hat zu diesen Ergebnissen beigetragen. Wir empfehlen, dass Schulen und Öffentlichkeit die vorhandenen Math-Trails (einschließlich der unterstützenden Werkzeuge) nutzen oder neue Trails entwerfen, indem sie dem in dieser Studie angewendeten Entwicklungsmodell folgen. Weitere Untersuchungen sind für die Projektentwicklung und -umsetzung in anderen Orten/Städten mit anderen Situationen und auch anderen Studienaspekten und -bereichen notwendig.

 

 

Table of Contents Cover............................................................................ i
Preface........................................................................ iii
Acknowledgemens....................................................... vii
Zusammenfassung (German-Summary)........................... xi
Table of Contents....................................................... xix
Chapter 1. Introduction................................................. 1
1.1.  Background of the study.......................................... 1
1.1.1.    What is the problem?....................................... 5
1.1.2.    Can the mobile app-supported math trail programme offer a solution?        9
1.2.  Design of the study............................................... 13
1.2.1.  Topic of the study........................................... 13
1.2.2.  Aims of the study............................................ 14
1.2.3.  Research questions......................................... 16
1.2.4.  Approach of the study..................................... 19
1.2.5.  Phases of the study......................................... 21
1.3.  The book’s structure.............................................. 25
Chapter 2. Theoretical Background................................ 29
2.1.  Constructivism in mathematics education................. 30
2.2.  Students’ motivation in mathematics....................... 38
2.3.  Students’ performance in mathematics..................... 41
2.4.  Teachers’ mathematical beliefs............................... 45
2.5.  Didactical situation in mathematics.......................... 47
2.6.  Outdoor mathematics education.............................. 52
2.7.  Technology in mathematics education...................... 64
2.8.  Conceptual framework of the study......................... 71
Chapter 3. The MathCityMap Project............................ 75
3.1.  The concept and the goals of the project................... 76
3.2.  Components of the project..................................... 77
3.2.1.  Mathematical outdoor tasks............................. 77
3.2.2.  Mathematical city trips................................... 80
3.2.3.  Map-based mobile app.................................... 83
3.2.4.  MathCityMap community................................ 87
3.3.  Technical implementation of the project................... 89
3.3.1.  Steps of preparation........................................ 89
3.3.2.  Settings of the activity..................................... 90
3.3.3.  Evaluating and publishing the activity.............. 92
Chapter 4. Designing the mobile app-supported math trail environment in Indonesia        93
4.1.  Indonesian secondary school mathematics
education.................................................................... 93
4.1.1.  A brief profile of Indonesia.............................. 94
4.1.2.  Education system in Indonesia......................... 96
4.1.3.  Indonesian secondary school mathematics curriculum   98
4.2.  Design of the mobile app-supported math trail programme for Indonesia      101
4.2.1.  Design of the rules of the activity.................... 106
4.2.2.  Design of the mobile app............................... 110
4.2.3.  Design of the math trails............................... 127
Chapter 5. Evaluating the potential effects of a mobile app-supported math trail programme: An exploration study in Indonesia.................................................................. 151
5.1.  Method.............................................................. 151
5.1.1.  Approach..................................................... 152
5.1.2.  Participants, situations, activities and procedures    152
5.1.3.  Data collection and analysis.......................... 154
5.2.  Results............................................................... 158
5.2.1.  What was the initial condition of the students like?            159
5.2.2.  How has the programme been running?........... 163
5.2.3.  How did the students work?........................... 164
5.2.4.  Why do students engage in this programme?. 168
5.2.5.  What mathematical experience do they get after having engaged in this programme? 184
5.2.6.  Does the intervention promote students’ performance in mathematics?         199
5.2.7.  How does the project affect teachers’ mathematical beliefs?            202
5.3.  Discussion.......................................................... 208
Chapter 6. Conclusions and recommendations............... 219
6.1.  Conclusion......................................................... 219
6.2.  Recommendations............................................... 240
References................................................................ 243
 

 

Chapter 1 Introduction

 

This book reports a study on the development, implementation and evaluation of a mobile app-supported math trail programme in Indonesia. As an introductory chapter, this chapter contains the background of this research, including the problems that underlie the need for this research, and the solution offered through this study. The design of the study, including the topic, aims, questions, approach and phases of the study, is described in detail in this chapter. To facilitate an understanding of the flow of the study, a concept map is also presented in this part. The structure of this book is then outlined in the last section of this chapter.

1.1.  BACKGROUND OF THE STUDY

Mathematics plays a role in daily life – both individual and social – and in professional life. People, therefore, must be able to apply basic mathematics in their everyday lives, a skill that the Organisation for Economic Cooperation and Development (1999) termed “mathematics literacy”. In an educational context, mathematics activities should offer the chance for students to experience mathematics as a

 

 

meaningful subject that can be understood well (Freudenthal, 1991).

Furthermore, Ojose (2011) suggests that students should be provided with real-world situations that are relevant to their position as citizens or their concern area. Such situations can be experienced by mathematics students outside the classroom (Dubiel, 2000). This approach can show students that mathematics is all around them and not merely in textbooks. Bringing mathematics outdoors can provide an opportunity for students to experience mathematics in real-world situations.

In some countries, interest in the development of outdoors and adventure education programmes has increased (Fägerstam, 2012). Various activities outside the classroom have been specifically designed to increase student engagement. Integrated programmes have also been developed to combine learning outside the classroom with traditional learning in the classroom. In mathematics, one of the outdoor educational programmes that have been elaborated is the math trail programme. This programme is not purely an outdoor activity, as it can also be conducted indoors, but it can be used as one alternative to encourage students to learn mathematics outside the classroom.

The math trail programme, which is a pathway to discovering mathematics, was created as a medium for experiencing mathematics in relevant applications (Shoaf,

 

 

Pollak, & Schneider, 2004), namely communication, connections, reasoning and problem solving (National Council of Teachers of Mathematics, 2000). In a math trail, students can simultaneously solve mathematical problems encountered along the path, make connections, communicate ideas and discuss them with their teammates, and use their reasoning and skills in problem solving.

Dudley Blane and his colleagues first developed a math trail by blazing trails in the centre of Melbourne as a family holiday activity (Blane & Clarke, 1984). The programme was then strengthened when some schools took advantage of the trails by integrating them into their mathematics learning programmes. The success of this idea allowed this programme to be adapted and applied in different places, and math trail projects subsequently emerged in various cities, such as Vancouver, Boston, Philadelphia and San Francisco (Richardson, 2004).

Although the math trail project is not new, the idea of an outdoor education programme supported by mobile technology appears to be new. This idea is facilitated by the fact that in recent years, developments in mobile technology and mobile phone use have improved significantly (Cisco, 2016; Lankshear & Knobel, 2006). These improvements were followed by many mobile phone applications (apps), including those intended for use in outdoor activities, such as Endomondo® for outdoor sports. In learning activities, Wijers, Jonker, and Drijvers (2010)

 

 

suggested that mobile devices could be employed to promote learning outside the classroom.

However, to date, most mobile technology apps for mathematics learning have only been employed in regular teaching settings (Trouche & Drijvers, 2010). Thus, it is necessary to explore the potential of mobile technology for teaching and learning mathematics, including for use in outdoors mathematics learning, and in so doing, engage students in meaningful mathematical activity. The combination of reality and virtual reality is expected to contribute to student engagement (Schwabe & Göth, 2005; Wijers et al., 2010).

This point is used as the basis for the development of the MathCityMap project. The project is a programme of math trails that are facilitated by the use of mobile phone technology and use special mathematical tasks (Cahyono & Ludwig, 2014; Jesberg & Ludwig, 2012). The project offers the opportunity to experience mathematics in the environment by arranging educational tasks in several locations around the city. Through a mobile phone app- supported mathematical activity, learners in groups find the task locations by tracing a planned math trail and solve mathematical problems encountered along the path.

The project aims to develop math trails and supporting tools, such as a web portal and a mobile phone app, and then share the project with the public (especially students).

 

 

This approach is expected to have the potential to encourage students to be actively involved in mathematics and promote student motivation to engage in mathematical activity. During their involvement in each activity and their interaction with the environment, students have the opportunity to construct their own mathematical knowledge. As a result, this will achieve the primary target of improving their performance in mathematics.

Referring to these basic ideas and objectives, as a part of the MathCityMap project, the model of the mobile app- supported math trail programme was then developed. The implementation of this programme in Indonesia tailored to that country’s situation was also studied scientifically. It was motivated by several problems in the field of mathematics education, especially in Indonesia.

1.1.1.   What is the problem? Public understanding of mathematics is often unsatisfactory, and only a small number of students enjoy participating in maths during the school day (Behrends, 2009, p. 1). At school, and globally, mathematics is sometimes perceived as a difficult and abstract subject. Internationally, by the eighth grade, only about a quarter of students enjoy learning mathematics (Mullis, Martin, Foy, & Arora, 2012, p. 325). This issue is a worldwide phenomenon, but given the unique aspects of each country, in this study we restrict ourselves to focusing on the issue in Indonesia.

 

 

In Indonesia, the percentage of students who report being happy at school is high in comparison with other countries (Organisation for Economic Cooperation and Development, 2013). However, Hadi (2015) reported that “most students fear mathematics, tend to skip mathematics subjects, and are happy when the mathematics teacher is not able to come to class” (p. 1).

In addition, based on the TIMSS 2011 results in mathematics, only approximately 20% of Indonesian eighth-grade students like learning mathematics and only 3% are confident in their abilities in maths (Mullis et al., 2012). In general, mathematics involves learning a lot of processes and formulae that appear to be not only unconnected with each other but also irrelevant to students’ lives (Education, Audiovisual and Culture Executive Agency, 2011).

Students appear to be less motivated to learn mathematics. This problem is serious because if we refer to the results of Hattie’s (2009) meta-analysis, students’ motivation and attitude toward mathematics and science are related to their performance in those subjects. In a synthesis of over 800 meta-analyses relating to achievement, 288 studies reported that attitudes toward mathematics and science were related to mathematics and science achievement (Hattie, 2009, p. 50).

 

 

From six meta-analyses (327 studies involving 110,373 people), Hattie concluded that motivation has a significant positive effect on achievement. Hattie’s effect size of student motivation was d = 0.48. This means that the influence was labelled as being in the “zone of desired effects”. Therefore, we assume that the above conditions have an impact on the low performance of Indonesian students.

Nationally, overall, Indonesian secondary school student performance has improved significantly. For mathematics, between 2004 and 2006, the average examination score rose from 5.0–6.2 to 6.8–7.6 (EFA Secretariat Ministry of National Education of RI, 2007). However, based on the data from the UNESCO International Bureau of Education (2011), Indonesian students still rank low in international standardized tests. Since PISA 2000, the trend has shown no significant increase or decrease and has tended to stagnate at low performance values (Baswedan, 2014).

PISA 2012 ranked Indonesian students aged 15 years 64th out of 65 countries, with a score of 375 for mathematics. PISA 2012 examines how well students can perform with the knowledge they possess (Organisation for Economic Cooperation and Development, 2013). Similarly, in TIMMS 2011, the mathematics achievement of Indonesian eighth-grade students was ranked 38th out of 42 countries, with a score of 386 in comparison to an average score of 500.

 

 

Further, PISA 2012 found that 75.7% of students were unable to reach level 2. This indicates that Indonesian students were unable to extract the relevant information for a given task or use basic algorithms to answer questions. Students also had difficulty solving realistic problems, especially in geometry. In the national exam in 2012, the average score of junior high school students in mathematics was 5.78 out of a maximum of 10, and mastery of geometry was below 50% (Kementerian Pendidikan dan Kebudayaan RI, 2013).

The World Bank (2010) reported that an ineffective learning process contributes to the low quality of graduates from Indonesian secondary schools. Previous studies indicated that several factors may affect quality, such as an increased focus on theory and routine learning, a focus on external examinations, administrative demands, textbook- based teaching and learning, and a crowded curriculum (Marsigit & Rosnawati, 2011; Sembiring, Hadi, & Dolk, 2008).

The natural next question, therefore, is how to improve Indonesian students’ motivation in mathematics in an effort to promote their performance in mathematics. One expected approach concerns the implementation of the mobile app-supported math trail programme. Conceptually, this programme is intended to address

 

 

problems similar to those encountered in Indonesia, as described above.

1.1.2.   Can    the    mobile     app-supported     math    trail programme offer a solution? Mathematics can be found everywhere and is experienced in everyday situations. The environment provides unlimited sources and ideas for teaching and learning mathematics. Students need to connect with the subject through meaningful activities, such as by implementing abstract mathematical concepts in real-life situations. The math trail programme provides opportunities in this regard. Supported by the use of mobile phone technology, in this project, math trails were designed around the city and offered opportunities to encourage student involvement in meaningful mathematical activities by utilizing the advantages of the latest technology.

This programme has the math trail concept at its core. A math trail is a planned pathway that consists of a series of stops at which students can explore mathematics in the environment (English, Humble, & Barnes, 2010; McDonald & Watson, 2010; Shoaf et al., 2004). Students find and solve real problems related to mathematics in real situations and connect mathematics with other disciplines.

Math trail tasks are created in several locations and then collected in a database. During the implementation phase, the tasks can be selected and grouped into a math trail route

 

 

based on the conditions and objectives. The tasks, their locations and directions to these sites, as well as the route and the tools required for problem solving, are provided in a trail guide, which is a scale map of the trail.

With the rapid development of technology, it is possible to collect the tasks and design a trail guide based on a digital map through a portal. Teachers can arrange the math trails in a web portal, and students can access the trails with the help of GPS-enabled mobile devices or use a paper version of the trail guide. In this project, we designed a web portal and a mobile phone application to support the math trail programme.

Using mobile devices is a familiar practice to all social and economic groups. In fact, in recent years, rapid developments have occurred in the scope, uses and convergence of mobile devices (Lankshear & Knobel, 2006) used for computing, communications and information. Within the next five years, it is estimated that the total number of smartphones will account for nearly 50 per cent of all global devices and connections (Cisco, 2016, p. 3).

As in other countries around the world, the rapid development of mobile technology has also occurred in Indonesia. Indonesia is amongst the top five countries worldwide in terms of the number of mobile phone users, with growth in 2012 reaching 60% (Prayudi & Iqbal,

 

 

2013). The same authors also reported that the percentage of market share in 2012 was 68.8% Android, 18.8% iOS, 4.5% BlackBerry OS, 3.3% Symbian, 2.5% Windows Phone, 2.0% Linux and 2.1% others. The World Bank Group (2015) reported that mobile cellular subscription in Indonesia in 2012 was 319,000,000 or 126.18 per 100 people.

Using mobile technologies is also common practice in learning activities, and these technologies offer advantages in terms of learning opportunities. The National Council of Teachers of Mathematics (2008) stated that technology has the potential to be a fundamental tool for mathematics learning. O’Malley et al. (2003) called the learning process in which learners take advantage of this approach “mobile learning”. One of the benefits of mobile learning is the opportunity to place learning outdoors in the real world. Mobile devices have the potential to integrate the characteristics of effective learning, such as situated realistic learning, motivational power and teamwork (Wijers et al., 2010). Thus, mobile learning enables learners to build knowledge and construct their understanding in unusual settings (Winter, 2007).

The math trail is the core of the activities in this project, and mobile technology is used to support these activities. This shows that this project combines real and virtual environments. According to Schwabe and Göth (2005), this combination might contribute to the improvement of

 

 

student engagement. When students are highly motivated to engage in mathematics, they are interested in, and enjoy spending more time, learning and solving mathematical tasks. Such students tend to be more persistent in solving mathematical problems (Lepper & Henderlong, 2000). Enjoyment and persistence in learning mathematics and positive views concerning the value and relevance of mathematics learning can lead to engagement with mathematics (Attard, 2012). Rigby, Deci, Patrick, and Ryan (1992) reported that student engagement in learning processes has a positive impact on their understanding of new knowledge and their flexibility in using new information.

Thus, theoretically, the concept of the mobile app- supported math trail programme offers several benefits for solving the current problems in the field of mathematics education, especially in Indonesia. We felt that if it were carefully designed and integrated into an intervention, the programme developed in this project could increase students’ motivation to engage in the mathematics learning process. As a result, in line with the results of Hattie’s (2009) meta-analysis, it will have an impact on improving their performance in mathematics. However, the design and implementation of the teaching and learning programme are also related to teachers’ attitudes toward the programme and their practices in the classroom. Thus, it is also important to address teachers’ mathematical

 

 

beliefs related to their strategies in fostering students’ motivation to engage in mathematical activity.

1.2.  DESIGN OF THE STUDY

The conceptual idea described above must be studied scientifically to develop a theory, translate the theory into a model of an educational programme, implement the programme and evaluate it through an empirical study. This need prompted us to carry out this study.

1.2.1.  Topic of the study This PhD study researches the development of the model of the mobile app-supported math trail programme and its implementation in Indonesia. The implementation in this study also includes an empirical study to explore the impact of the programme developed on student motivation in mathematics, their performance in mathematics and teachers’ mathematical beliefs. The programme was developed and implemented primarily for school activities as part of the process of teaching and learning mathematics, so the programme was developed in accordance with the school mathematics curriculum in Indonesia. Since the area of school mathematics is too broad to be investigated in the framework of this study, it must be restricted to an area that is relevant to the issues mentioned in the part of the background of this study. Therefore, the programme was implemented particularly for lower secondary school or

 

 

junior high school level (for all grades and for all mathematics topics).

As a pilot study, we implemented and studied empirically the programme for secondary schools in the city of Semarang, Indonesia from 2013 to 2016. It was carried out in cooperation between the two countries through two institutions, namely IDMI Goethe-Universität Frankfurt in Germany and FMIPA Universitas Negeri Semarang in Indonesia, and in collaboration with the Department of Education of the city of Semarang to implement the project in nine schools in the city. Semarang is not representative of Indonesia as a whole but has unique characteristics as a town and includes high-, middle- and low-level schools. Both urban and rural schools are also found in this region, which has hills and seaside areas, while the economic level of citizens varies. Thus, the implementation of the programme in this city is expected to provide a large amount of information and to serve as a model for the development and implementation of the programme in other cities in Indonesia, which is rich in diversity.

1.2.2.  Aims of the study This study consisted of two phases, namely: (1) the developmental phase and (2) the implementation and empirical study phase. These two phases were preceded by a preliminary phase. The first phase was aimed at developing a model of a mathematics educational

 

 

programme grounded in the basic idea and objectives of the MathCityMap project. The basic idea of the MathCityMap project is to set up mathematical problem-solving activities in outdoor situations following the concept of the math trail and supported by the use of mobile technology. The programme offers the opportunity for learners to construct their own mathematical knowledge during their interactions with the environment.

Therefore, the programme was developed within a framework informed by constructivism in mathematics education and built using some basic concepts, such as the concept of the math trail and the use of mobile technology in mathematics education. From this theoretical framework, the model of a mobile app-supported math trail activity was formulated. This model includes the concept, objectives, components and technical implementation of the programme and its evaluation.

The second phase was aimed at implementing the programme in Indonesia tailored to the local situation. It included the evaluation of its implementation by conducting an empirical study. Through an exploration study, the potential of the programme for eliciting engagement in meaningful mathematics teaching and learning activities was explored. Decisive factors that affect students’ motivation to engage in mathematics and their performance in mathematics were also examined in this study. Furthermore, because teaching and learning

 

 

processes are linked with teachers’ attitudes, this study also considered the effects of the project on teachers’ mathematical beliefs. Thus, teachers’ mathematical beliefs related to their strategies in fostering students’ motivation to engage in mathematics education through the implementation of the programme were also investigated in this study.

1.2.3.  Research questions Taking into consideration the aims of the research, the three central research questions of this study were clarified. The first question was:

1.    How can the mobile app-supported math trail programme promote students’ motivation to engage in mathematics?

The meaning of the keywords of this question are as follows. First, the question starts with the words “how can”. This phrasing indicates that the question focuses on identifying the conditions and approaches that optimize the benefits offered by the programme. This programme is a “mobile app-supported math trail programme”, which is a core of the MathCityMap project and was also theorized through this study.

“Engagement” leads this study to determine how students are actively engaged in this programme. The use of outdoor

 

 

activities supported by a mobile phone application is a strategy to engage students in the mathematics learning process. Because students’ engagement in an activity is related to their motivation, it is necessary to investigate the nature of “students’ motivation” to engage in such an activity. It is also important to determine the impact of this programme in “promoting” student motivation in mathematics and to investigate the differences in student motivation to engage in mathematics before and after the programme.

Furthermore, as a type of mathematics education programme, this programme focuses on “mathematics”. It was designed to motivate students to engage in an activity that helps them to construct their own mathematical knowledge. Therefore, this study was also concerned with examining the impact of the programme on student performance in mathematics by answering the question:

2.    Does an intervention with a mobile app-supported math trail programme enhance students’ performance in mathematics?

The “intervention” in this study was performed by arranging a teaching and learning mathematics in which a “mobile app-supported math trail programme” was carried out. In line with the Indonesian school mathematics curriculum contents, “students’ performance in mathematics” addressed in this study includes their mathematical ability and skill in understanding and

 

 

applying mathematics in solving problems. This question led to the findings on the effect of this intervention on student performance in mathematics. It is controlled by knowing whether the performance of the students involved in this programme is better than that of students learning in regular settings.

Furthermore, the arrangement of learning needs the support of teachers in their teaching practice. Thus, the study also intervened in teachers’ mathematical beliefs, so the teachers’ strategies in their teaching practices were in line with the concept and objectives of programmes developed through this study. The impact of the intervention was evaluated in the study by answering the question:

3.    How do teachers’ mathematical beliefs differ after being involved in the project compared with their initial beliefs?

This question addresses teachers’ view of mathematics, and its teaching and learning, before, during and after involvement in the project. It led to the findings on “teachers’ mathematical beliefs” and measured changes in those beliefs. The impact of the project on teachers’ orientation – traditional or inquiry – was also associated with this question.

 

 

1.2.4.  Approach of the study This study globally followed a design research paradigm. Design research is also known as developmental research. This approach integrates research, development, implementation and dissemination and involves all participants (researchers, developers, teachers and practitioners) in dialogue in the field (Gravemeijer & Terwel, 2000). This work produces theories and prototypes that are theoretically and empirically founded and studied. Freudenthal (1968) described the principle of developmental research as follows:

experiencing the cyclic process of development and research so consciously, and reporting on it so candidly that it justifies itself, and that this experience can be transmitted to others to become like their own experience (p. 161).

Through this principle, Freudenthal suggested that the research experiences should be transferred to outsiders, such as teachers, who can then use the experiences as the basis of decisions when they implement them in their classroom to achieve their own goals according to the actual situation. This process shows that this kind of research fits into the pedagogical tradition (Gravemeijer & Terwel, 2000).

 

 

Van den Akker (1999) stated that developmental research is based on two objectives, namely the development of prototype products and the formulation of methodological suggestions for designing and evaluating prototype products. The prototypes produced in this study were the model of the programme, the math trail tasks and the mobile phone app. The aims of design research are to develop theories or models, instructional materials and an empirically grounded understanding of the work of the learning process (NCTM Research Advisory Committee, 1996).

Design research is particularly suitable in situations for which a full theoretical framework is not yet available; thus, hypotheses must still be developed, and some new teaching materials must be designed (Drijvers, 2003). To produce prototypes, it is necessary to establish criteria for their development. Nieveen (1999) offered three criteria that can be used: validity, practicality and effectiveness.

·         The product meets the requirement of validity if the components are based on state-of-the-art knowledge and consistently linked to each other (internally consistent).

·         Practicality is fulfilled when experts and practitioners claim that the prototype can be used and when the prototype can really be employed.

 

 

·         The product meets the criterion of effectiveness when the students appreciate the learning programme and the desired learning takes place (P. 127).

Therefore, the products developed in this study must meet these requirements. The evaluation of the products was conducted through multiple phases in this study.

1.2.5.  Phases of the study According to Drijvers (2003), “Design research has a cyclic character: a design research study consists of a research cycle in which thought experiments and teaching experiments alternate” (p. 20). The development that occurs during the cycles can be characterized as ranging from qualitative formative to more quantitative summative.



 

Fig. 1.1 Layers of formative evaluation (Tessmer, 1993)



 

 

Following the layers of formative evaluation proposed by Tessmer (1993) reveals the progress from small-scale to large-scale evaluation (Fig. 1.1). This method requires mixing a qualitative formative method at the beginning of the study and a quantitative summative method afterward, using a quasi-experimental approach (Bokhove, 2011). Therefore, this study took place in one preparatory cycle and three subsequent cycles (prototypical design (CYCLE I), a small-scale field experiment (CYCLE II) and a large- scale experiment (CYCLE III)). This phase is shown in Fig. 1.2.



 

Fig. 1.2 Design research cycles within this study

 

The preparatory cycle involved the literature review, which led to the theoretical framework and the design of the prototypes. The focus of activity in this cycle was to conduct an in-depth review of the literature cross-checked with field experiences and existing data. The result is the theoretical framework for the development of the project

 

 

for implementation in Indonesia. The cross-checking process was completed through an exploration step that consisted of benchmarking, needs analysis and focus group discussion (involving experts and practitioners/educators).

Based on the results of this activity, the specification procedure was defined and the technical requirements were developed, primarily as a foundation and guideline for designing prototypes. The results of this cycle are the guidelines for the technical implementation of the project in Indonesia, a guide to designing the prototypes. In addition, the research instruments based on the theory are also reviewed in this cycle to obtain research tools that are appropriate for the theory and the chosen situations.

The first intervention cycle focused on the prototypical design, such as the model of the programme including the activity rule, the design of math trails containing varied mathematical outdoor tasks and the design of the mobile phone app. The prototypes were designed based on the specifications and requirements defined in the previous cycle. In this cycle, the trails were created by the project team and also by the task contributors (teachers, educators and others), with reference to the guidelines and characteristics that were set in the previous cycle. The math trails were created by contributors through training and a workshop session. The mobile app to support the math trail programme was also created in this cycle by the app programmer of the project team. Activities took place until

 

 

the authors uploaded the tasks into the MathCityMap website portal.

In the second cycle, the prototypes were reviewed by experts (mathematics, mathematics education and mathematics educational multimedia), and a small-scale field experiment (simulation session) was carried out to determine how the prototypes would work. In this cycle, teachers and small groups of students performed math trail activities and tested the mobile app. It is necessary to see how the math trails run and how the app works. Here, teachers and a small group of students were involved. During this session, the researchers conducted observations and interviews. The prototypes were then revised based on the results of the review and simulation in this phase.

In the final cycle, we implemented the project in a large- scale class experiment and studied its effects. This cycle was carried out in several pilot studies by involving teachers and students from nine schools. The implementation consisted of an introduction session, a math trail run and a debriefing session. In this step, we explored how the math trails ran and how the application worked in all pilot studies, how the students performed and how motivated the students were to participate in the activity. We also investigated the attitudes and beliefs of teachers related to mathematics and this project. We then

 

 

analysed the data obtained in this step to answer the research questions, drawn conclusions and recommended (and criticized) some important points for the further development of the project.

1.3. THE BOOK’S STRUCTURE

This book consists of six chapters. These six chapters represent the main content and are preceded by a preface. The concept map of this study and its link to the content of each chapter of this book are illustrated in Fig. 1.3.

Chapter 1 is an introductory chapter. In this chapter, we present some background for this study, including the problems and solutions offered through this study. The topic, objectives, questions, approach and phases of this study are also described herein. At the end of this chapter, we outline the concept map of the study and the structure of the book.

Chapter 2 focuses on the theoretical background of the study. This chapter presents the results of a literature review relating to the study. The theory of constructivism in mathematics education linked with the concepts of student motivation in mathematics, students’ performance in mathematics and teachers’ mathematical beliefs is discussed in this chapter as the theoretical basis for the framework of the study. This section is followed by a discussion on some basic concepts that build on the idea of the study, such as the concept of mathematics trails and the

 

 

use of mobile technology in math trail programmes. From this theoretical background, the framework of the study was formulated.

Chapter 3 focuses on the MathCityMap project. The concept and goals of the project are formulated and the components of the project are also described. In reference to implementing the project, we also propose the technical implementation of the project in this chapter.

Chapter 4 describes the results of the development of the model of the mobile app-supported math trail programme for implementation in Indonesia. This chapter starts with a closer look at the situation in Indonesia and contains the results of designing a mobile app and math trails in the city of Semarang, as well as the rules of the activity.

Chapter 5 is a report on an exploration study of the implementation of the programme in Indonesia. This chapter describes the methodology and the procedures of the study. The impact of the programme on Indonesian secondary school students’ motivation to engage in mathematical activity, their performance in mathematics and the teachers’ mathematical beliefs are discussed in this chapter. At the end of this chapter, the results of this study are presented.

Chapter 6 is the concluding chapter. Here, we draw conclusions from the results obtained in this study and

 

 

provide    recommendations     for   future    activities    and researches.



 

Fig. 1.3 The concept map of the study

 

 



 

 

 

 

Chapter 6 Conclusions and recommendations

 

In this final chapter, we review the detailed results of this study that were presented in the previous chapters from a more global perspective. The aim of this chapter is to draw conclusions regarding the results as the answers to the research questions stated in the first chapter. Recommendations are also given in the last section of this chapter. There are two kinds of recommendations, namely: a recommendation that the programme be implemented and recommendations for further studies on the MathCityMap project.

6.1.  CONCLUSION

Mathematics plays an important role in daily life. However, the current public understanding of mathematics remains unsatisfactory. Many students do not enjoy mathematical activities and appear unmotivated to learn mathematics, while their attitudes affect their performance in mathematics. These problems have led mathematicians and mathematics educators to think of ways to popularize mathematics. A variety of projects have been developed and implemented in numerous countries to raise public

 

 

awareness of mathematics, one of which is the MathCityMap project, a project of the working group MATIS I IDMI Goethe University Frankfurt. It is a project of a math trail programme supported by the use of mobile technology.

The research on this project was carried out in multiple places and using multiple areas of study, including this PhD study. This work is a study on this project specifically in Indonesia. It was conducted by following a design research paradigm and took place in one preparatory phase and three subsequent phases (a design phase, a small-scale field experiment phase and a large-scale experiment phase). The aims of this study were focused on designing a mobile app and math trails around the city, implementing the mobile app-supported math trail programme in Indonesia and exploring its impact on students’ motivation and performance in mathematics and on teachers’ mathematical beliefs.

Because the MathCityMap project has not been theorized yet, this study was also concerned with formulating the concept of the MathCityMap project as a guideline for the implementation of this project, both in this study and other studies in the future. Therefore, in this study, the concept of the MathCityMap project was theorized. For the implementation of the project in Indonesia, the model of the mobile app-supported math trail programme was

 

 

developed; it was followed by designing the rules of the activity, the mobile app and the math trails around the city of Semarang. Then, the programme was implemented and studied empirically in nine secondary schools in the city of Semarang.

This study was conducted within a theoretical framework informed by constructivist theory in mathematics education. This view suggested that, in the learning mathematics process, students should actively construct their own mathematical knowledge by connecting new information they have gained with their previous knowledge. However, learners’ engagement in an activity depends on their motivation. Sociocultural views state that learners need guidance as a sort of push for students to get started in their engagement in the learning process, and classroom social interaction also motivates them to engage through peer pressure.

The external factors that encourage learners to undertake the construction of knowledge should, however, be reduced and eliminated because, based on the cognitive view, students should value learning for its own sake. The learning process should prepare students to become lifelong learners that gain the confidence to construct more complex knowledge and who are better equipped to make use of their mathematical knowledge and skills throughout life. Both views affect the strategies that are used to improve student motivation: the sociocultural view leads to

 

 

extrinsic motivation, while the cognitive view leads to intrinsic motivation.

The intrinsic-extrinsic motivation axis is explained in self- determination theory (SDT). This model conceptualizes a range of regulations, from intrinsic motivation to amotivation. Between these extremes lie identified regulation and external regulation. It has been hypothesized that self-determination is associated with improved psychological performance; therefore, it would be expected that intrinsic motivation, followed by identified regulation, is mostly associated with positive outcomes. Thus, the process of learning mathematics needs to be prepared and arranged so that students can be intrinsically motivated to engage in learning mathematics.

Nonetheless, the improvement of students’ motivation is only a stimulus for students to promote their engagement in mathematics. As a serious mathematics educational programme, the main focus of this activity is to support students in learning mathematics and constructing their own mathematical knowledge. By engaging in this programme, the students got the opportunity to learn and practise solving real problems by following the stages of mathematizing. These stages are: understanding a problem situated in reality; organizing real-world problems according to mathematical concepts and identifying relevant mathematics; transforming real-world problems

 

 

into a mathematical problem that represents the problem situation; solving mathematical problems; and interpreting mathematical solutions in terms of the real situation. In the studied project, students learned and experienced using what they knew in the process of solving problems related to mathematics.

This activity helped students to construct their own mathematical knowledge by solving the mathematical tasks on the planned math trail and interacting with the environment, including the digital environment. The digital tool is used as a bridge to help teachers support students in the mathematics learning process. It is hypothesized that the intervention with this programme enhances students’ performance in mathematics. Through this activity, students were expected to increase their understanding of mathematical concepts and their relationships with real life and improve their ability to apply mathematics in solving problems.

To reach this goal, the teachers also play an important role. Learners’ confidence and performance in mathematics are significantly associated with the teachers’ mathematical beliefs and teaching practices. Belief in the value of intrinsic motivational strategies is expected to be associated with inquiry-oriented teachers’ mathematical beliefs. In their teaching and learning practices, teachers should act as facilitators and not as transmitters of information to their students. Teachers should prepare

 

 

problems that will provoke the expected learning and offer guidance for students to learn new knowledge. Students accept the problem and then speak, think and act about the problem through their own motivation. This teacher- student interaction is important in helping learners to solve problems and develop their abilities to allow them to solve other or future problems by themselves. In this environment, the interaction was facilitated by the use of mobile technology.

Furthermore, the learning environment also needs to be well organized. The environment should offer opportunities for learners to adapt and reconfigure new information to construct new knowledge. Mathematical knowledge can be constructed through the action of organizing the subject matter, both matter from reality and mathematical matter (mathematization). Students are reinventing the teaching matter prepared by teachers in such interactions and under guidance (guided reinvention).

Taking mathematics outdoors allows learners to experience these processes in an authentic situation. In this situation, students can see the connection between mathematics and their real world. It also creates greater motivation and excitement for students in the learning process. Teachers can also design a mathematical task outside the classroom, because the environment provides many ideas for the subject matter of mathematics.

 

 

Tasks about an object in the environment are designed and grouped with other tasks in one location, then linked into a route. Furthermore, students undertake the exploration of mathematics in the environment around them by conducting mathematical outdoor activities: tracking the math trail and completing these planned mathematical tasks.

By taking advantage of these benefits of mobile technology, math trail tasks can be localized with GPS coordinates and pinned onto a digital map through a web portal. The trail walkers can then access the tasks and run the math trail activity with the help of a GPS-enabled mobile application. The app helps teachers to facilitate the process of learning mathematics, from preparing the problem to directing students to find the task locations and supporting them in solving mathematical problems. Mobile devices can also be used to integrate learning activities with authentic situations in outdoor environments. The math trail increases student engagement in learning mathematics activities. The combination of the math trail and the use of mobile technology has become a basic idea of the MathCityMap project.

The MathCityMap project The MathCityMap project is a project that involves the math trail programme, which is supported by the use of mobile technology. This project was not conceived merely to design and/or use the math trails; instead, the project

 

 

includes the entire process: preparation (how to design the programme), implementation (how the programme runs) and evaluation (how the programme impacts on students’ motivation, students’ performance in mathematics and teachers’ mathematical beliefs). A mobile phone app, as a supporting tool, was also designed, created and used during this project.

As the name implies, the MathCityMap project is composed of several components. These components are: mathematical outdoor tasks, mathematical city trips, a map-based mobile app and the MathCityMap community. This project is not only related to the implementation of the activity, but also addresses how to prepare the project and its instruments and how to evaluate the project. Consequently, the technical implementation of the project is made up of three main parts – the preparation, the activity and the evaluation.

The mobile app is designed by the team. Tasks are prepared by educators or teachers in a team or individually. Then, the tasks and app are evaluated and validated to ensure that they are feasible for use. The mobile app can be downloaded and installed by the public. The tasks can be used in private or public settings by following the rules that have been prepared. Technically, with the help of the mobile app, students/users follow a planned route, discover the task locations and answer the tasks related to what they

 

 

encounter along the path; further, they continue with the subsequent tasks.

Then, the implementation of the math trail and app is evaluated for further development. The process and results of the activities are also published so that others can take the benefits of these results and be inspired to develop it in other places. However, it is important to note that the implementation of the educational programme in a particular place requires adjustment to local conditions and needs.

The mobile app-supported math trail programme in Indonesia By following the theory, the specific procedures and the technical requirements of the MathCityMap project, a mobile app-supported math trail programme was designed to be implemented in Indonesia. The designing phase started with discussions between experts and practitioners, followed by teacher training, and then the process of designing and testing the tasks and mobile phone app was performed by the team. In this phase, the GPS-enabled mobile application was designed specifically for implementation in Indonesia in relation to the local situations, conditions and needs. It is a native app that displays the math trail routes/map, including the coordinates of the math trail tasks, the user’s current position, the authentic mathematical problems, the information about the tool needed in solving problems, the

 

 

hints on demand (if needed by the users) and the feedback on the answers entered. The app also gives brief information related to the object, such as its history, its function, etc. The app was created using the MIT App Inventor program and runs on Android platform mobile phones. It can be installed by downloading the installation file available at www.mathcitymap.eu or Google PlayStore.

In addition to making the app, in this phase, 87 tasks, grouped into 13 math trail routes scattered among various areas in the city of Semarang, were also designed by educators (including teachers through training and a workshop). There were six to eight tasks for each trail with a variety of topics located in a safe and comfortable environment. The trails were designed in several school areas, city parks, historical buildings, markets, tourist attractions and other locations. The math trails and the tasks were then uploaded into the portal so that users could access them through the mobile app. Trail walkers needed about two hours to explore a trail (about 1–2 km length) and to solve the problems encountered along the path. The math trails’ and mobile app’s designs passed the evaluation process, through validation by experts, and the simulation phase involved a small group of students and teachers that were considered eligible for use in the next phase.

 

 

In the mobile app-supported math trail programme, the activities can be carried out with students from different grades and the tasks can vary in terms of level and difficulty. The activities can be conducted in the area of the school, in the school’s neighbourhood or at several locations around the city. The duration of the activity is about two to three hours during school hours (morning time is recommended), after school hours or during holidays. Students in a class are divided into groups of five to six members. The activities are conducted during normal school hours for about two 45-minute periods, beginning with the teachers giving a brief explanation of the rules and goals. Groups then start their mathematical trip, each from a task location that is different from that of the others. As the groups trek the trail, teachers observe and supervise students’ activities but are not expected to assist them, because all the necessary information is available in the app. With the help of the app, students in groups follow a planned route, discover task locations, solve the problem on-site and then move on to subsequent tasks. A debriefing session is also needed to get feedback for evaluating the activity and preparing further activities.

Implementation and empirical study of the programme in Indonesia The large-scale field experimental phase was conducted in 2015 and 2016 and involved 520 students and nine teachers from nine secondary schools. For each school, there were

 

 

two classes (about 30 students/class) involved in this study: one class was the experimental group and one class was the control group. Both groups at each school were taught by the same teacher with the same topic and subject matter but with different interventions. In the experimental group, teaching and learning mathematics was carried out by running the mobile app-supported math trail programme. Meanwhile, in the control group, students learned mathematics in the regular setting. A debriefing session was conducted after the activities. Questionnaires and evaluations regarding students’ motivation (the self- reported Situational Motivation Scale (SIMS)), teachers’ mathematical beliefs and individual mathematics performances were also given before and after the activities.

According to the situation and condition, the activities were conducted in different settings. In general, the activities were held during school hours, but some schools’ activities were also organized on a Sunday (holiday) or after school hours. On average, each activity required two to three hours to complete. Four schools carried out activities outside the school (including in some city parks and near landmarks in the city of Semarang), while five schools conducted activities in the school area. These choices were made based on the conditions and situations (such as safety and the availability of teaching and learning time) that were unique for each school.

 

 

The results of the field empirical study revealed that, in general, the activities ran smoothly, the mobile app worked well, and the rules and goals of the programme were comprehensible. Nearly the entire group attempted to work in their own way (without using hints). Most students reported no problems in participating in these activities, including the use of the app. All groups discovered the locations of the tasks, and a total of 73 tasks were completed by all groups in this activity; 14 tasks were not answered or were completed incorrectly by some groups.

The answer to the first research question: the students’ motivation to engage in mathematics The findings indicate that it was easy to involve the students in the activities. Most of the students were actively engaged in the activities, expressed positive feelings and had no problems in carrying out the activities. Students’ engagement was associated with their motivation. The SIMS results show that the students’ motivations to engage in the activity were diverse; however, all had positive Self- Determined Index (SDI) scores (ranging from 2.00 to 16.25 and M ± SD = 7.3180 ± 2.92629), which indicates that their motivations were more often the self-determined or internalized forms of motivation, namely intrinsic motivation (IM) and identified regulation (IR), than external regulation (ER) and amotivation (AM). Students perceived the activity to be interesting or enjoyable (an indicator of IM) and meaningful or valuable (an indicator

 

 

of IR). They were engaged in the activities for their own sake/pleasure/satisfaction, and their engagement was considered to be a self-choice.

Open-ended follow-up questions revealed that students enjoyed the outdoor mathematical activities (30%) and were interested in the use of advanced mobile technology to learn mathematics (23%). They also reported that learning and applying mathematics in the real world were valuable and meaningful for them (18%). Other students mentioned different reasons or did not give a reason, and slightly negative feelings were also mentioned. The SIMS results also show that there was a significant difference in the reported orientation of the students’ motivation before and after they were involved in this programme. There was a change in the orientation from being extrinsically oriented toward being more intrinsically oriented. We also found that there has not been a decrease in the intrinsic motivation of students from the first year to the second year of implementation.

Generally, the project was successful in designing and offering a programme that can promote students’ intrinsic motivation in mathematics; however, this improvement of students’ motivation is only as a stimulus for students to promote their own engagement in mathematics. Their involvement in the learning process is expected to positively affect their performance in mathematics.

 

 

The answer to the second research question: the students’ performance in mathematics As a serious mathematics educational programme, the main focus of this activity is to help students in learning mathematics and constructing their own mathematical knowledge. We found that, by engaging in this programme, the students got opportunities to learn, experience and practise solving real problems by following the stages of mathematizing. These stages are: understanding a problem situated in reality; organizing real-world problems according to mathematical concepts and identifying relevant mathematics; transforming real-world problems into a mathematical problem that represents the problem situation; solving mathematical problems; and interpreting mathematical solutions in terms of the real situation.

Through this programme, students learned and experienced using what they knew to solve problems related to mathematics. The students practised extracting information from the problems they faced and using the mathematical concepts that they knew to solve those problems. They experienced mathematics in the environment and constructed their mathematical knowledge by applying mathematics in solving the problems encountered along the path.

To determine the effect of the intervention on students’ performance in mathematics, a statistical comparison of the students’ scores between the two groups was carried out.

 

 

The results show that the mean gain in performance score of the 272 students in the experimental group (16.2684) was significantly higher (Cohen’s d = 0.31) than that of the 248 students in the control group (4.2540); in other words, the programme enhanced students’ performance in mathematics, especially their ability and skills to understand and apply mathematics to solve problems. Besides the learning arrangement, the teacher was also an important factor in these outcomes.

The answer to the third research question: the teachers’ mathematical beliefs Teachers play an important role in organizing and running the strategy of the learning process. Teaching and learning processes are linked with teachers’ attitudes; as such, this study was also concerned with the influence of the programme on the teachers’ mathematical beliefs in relation to their strategies for fostering students’ motivation to engage in mathematics. The influence was given by holding focus group discussions, teacher training and a workshop for designing tasks, and involving teachers in the processes of observing and evaluating the activity.

The findings indicated that the project changed the teachers’ mathematical beliefs in relation to their strategies for fostering students’ motivation in mathematics. The comparison test shows that there was a change in the teachers’ mathematical beliefs from traditionally oriented

 

 

beliefs toward more inquiry-oriented beliefs, i.e. after being involved in this programme, the teachers were more likely to believe that: maths is a tool for thought; the students’ goal is to understand; students should have some autonomy; maths ability is prone to change; and students will want to engage in maths if the tasks are interesting and challenging.

The regression analysis results revealed that there was a significant effect of teachers’ mathematical beliefs on students’ motivation (y = 2,239 + 0,492x) with the contribution at a rate of 77.7%. The more teachers implement inquiry-oriented mathematical beliefs in their teaching strategies and practice, the more their students have intrinsic motivation to learn mathematics. This shows that although this project aimed to improve students’ motivation by designing a learning environment and a supporting tool, it also affected teachers’ mathematical beliefs in relation to their teaching strategies and practices for fostering student motivation in mathematics.

Remarks from the answers to the three research questions Finally, the project was successful in designing and offering a programme that can intrinsically motivate students to engage in mathematics. The design and arrangement of the math trail and the mobile app, as well as a combination of both, created a pleasant situation and attractive environment that offered valuable knowledge and experience in mathematics. They embody the aspects

 

 

of enjoying or being interested in the activity and the use of advanced mobile technology for learning mathematics (an indicator of intrinsic motivation), and of the value and meaningfulness of the mathematical tasks and the activity (an indicator of identified regulation), which were generated through the implementation of this project.

The students were interested in the use of advanced mobile technology for learning mathematics and enjoyed the outdoor mathematical activity. The students also believed that the mathematical tasks and the activity were valuable and meaningful for them. The intervention with the mobile app-supported math trail programme also enhanced students’ performance in mathematics, especially their ability to understand and apply mathematics to solve problems. The change in teachers’ mathematical beliefs, from being traditionally oriented to being more inquiry- oriented, which occurred as an effect of the project implementation, also contributed to this improvement in students’ intrinsic motivation to engage in mathematics.

Some criticism of the MathCityMap project However, some feedback on the implementation of the project needs to be considered, such as: the need for the optimization of mobile phone use to motivate students; aspects of the learning mathematics outcomes; and the need for a system that helps the teachers to prepare and run the activities. Firstly, we will criticize the students’

 

 

motivation to engage in this activity. Although the SDI scores of students affected by the use of the mobile phone are the highest, the data show that the majority of students were interested in these activities because they were sited outdoors. This is important because one of the goals of this project was to develop the concept of a math trail by taking advantage of mobile technology. Therefore, the potential of mobile technology to support outdoor mathematics learning needs to be further exploited.

Currently, the use of the mobile phone is directed toward supporting the improvement of students’ intrinsic motivation. However, based on the sociocultural viewpoint, learners need encouragement to initiate and engage in a learning process, despite the fact that the encouragement of external factors needs to be reduced gradually so that in the end students must learn for its own sake, as suggested by the cognitive view.

Here, the use of a mobile phone can play a role. The possibilities of this role, for example, include the use of these tools for outdoor games or giving scores in recognition of what has been achieved by learners. These are factors that extrinsically motivate students to engage in an activity. These rewards, however, may differ from the basic concept of the math trail, but these findings provide benefits for the development of the math trail and its innovations by taking advantage of the use of mobile technology, so that the programmes developed through this

 

 

project can be optimized to increase students’ motivation to engage in mathematics.

Secondly, it is about mathematical skills. The results of the study show that once the students were involved in the activities, they had the opportunity to learn mathematics, construct mathematical knowledge and improve their performance in mathematics. The topic and subject matter used in designing tasks for this programme were adapted to the learning objectives and standards of competence expected by the school mathematics curriculum.

However, the expected learning mathematics outcomes, or more specifically the aspects that are examined in the final maths examination, are not always related to real problem- solving abilities and do not always require the problem- solving skills that are trained through the math trail activity. Even some (multiple-choice) test items can be solved by using “the quick solution”.

Therefore, this programme cannot be used as a single strategy in the process of teaching and learning because the students need to be equipped with mathematical skills in other aspects. The programme is also not the best strategy when used alone, but it can become one of several strategies used to complement each other. This programme can take a role in improving the mathematical literacy of students.

 

 

Although based on research results, teachers involved in this project experienced a change in their beliefs from traditional orientations toward being more inquiry- oriented, but they are also required to prepare students for the examination and teach the subject matter in accordance with the requirements in the curriculum, i.e. there was a conflict between their beliefs and their needs. Therefore, time efficiency and the effectiveness of the strategy in achieving the objectives need to be managed carefully. Although this programme provides many advantages, organizing the students to do outdoor activities requires extra time and attention.

Before implementing the programme, the teacher or educator also takes time for preparation, such as designing tasks and uploading them into the web portal system. In designing math trail tasks, although the environments present many ideas for mathematical tasks, safety, comfort and public order factors are also constraints. Discussion, evaluation and validation of the tasks are also required to produce good math trail tasks in accordance with the concept. These require the development of a system that helps teachers to prepare math trail tasks and activities for their students.

However, overall, it can be concluded that the project was successful in designing and offering an engaging mathematics learning environment. Students easily engaged in the activities and these activities positively

 

 

affected students’ attitudes and motivations toward learning mathematics. Students were intrinsically highly motivated to engage in these interesting and fun mathematics activities. Their involvement encouraged them to experience the process of mathematization and construct their own mathematical knowledge. As a result, their performance in mathematics improved. Teachers have a role in achieving these outcomes. Afterward, the mathematical beliefs of the teachers involved in this project became more inquiry-oriented. This supports efforts to increase the students’ intrinsic motivation in mathematics.

6.2.  RECOMMENDATIONS

Based on the conclusions of this study, some recommendations are given at the end of this report on the study for two reasons. Firstly, the aim of this part was to concentrate the findings into concrete suggestions for implementing the programme and utilizing the results of this study. Secondly, the recommendations were aimed at highlighting the issues that were not addressed in this study and require further research and development.

In this study, it was reported that math trails, supported by the use of a mobile phone app, have many potential advantages. Schools can take advantage of the results of this project by integrating the mobile app-supported math trail programme into the teaching and learning processes. The public can also participate in this programme for

 

 

leisure by walking around the city while learning mathematics. The public can freely use the mobile app and run the math trail activities that have been designed, use the existing math trails that have been designed and tested in this study, and design new math trails and upload them into the portal system by following the specification procedures and technical requirements formulated in this study.

Implementation of this programme can be strengthened through cooperation with several parties, such as city governments, schools and education authorities. Cooperation can also be established with the tourism department, museum managers, travel managers and others. With this cooperation, this educational programme can be synchronized with the programme of the parties, e.g. mathematics city tours. Financial support can also be obtained through sponsorship cooperation with several parties, such as a mobile phone service provider or mobile device manufacturers and sellers.

However, based on the results of this study, some conditions and requirements must be considered. The important role of influencing student motivation and performance in mathematics must be designated when designing the app, tasks and activity. The relevance and value of the task must be clearly identified and linked to the concept and objectives of the project. Most importantly, the students must enjoy and be attracted to the programme,

 

 

both in terms of completing the mathematical tasks and doing activities outdoors, and in using the mobile app.

As a pilot study, the research was conducted in a limited manner in one place. Although this pilot study provided information about the overall project implementation and provided suggestions for models of the programme, further development needs to be carried out, both in the development of the concept of the programme and its supporting tools. Further research needs to be conducted to inform the development of this project in other places/cities with different conditions, which will enrich the information available and improve the model of the programme. It will also be possible to dig deeper and explore more broadly the implementation of the programme and its impact on students’ motivation and their performance in mathematics, teachers’ mathematical beliefs, and other aspects and areas of study.

Adi Nur Cahyono

 

 

 

 

 

 

 

 

 

About the author

Dr.rer.nat. Adi Nur Cahyono, S.Pd., M.Pd. was born on 11 March 1982 in Banjarnegara, Central Java, Indonesia. In 2000, he completed his secondary education at Senior High School in Karangkobar, Banjarnegara, and started his university studies in Mathematics Education at Universitas Negeri Semarang (UNNES), Indonesia, where he obtained a bachelor degree in 2004. He pursued a master’s degree in Mathematics Education at Universitas Negeri Semarang in 2006 and graduated in 2008. Beginning in 2005, he worked as a senior high school mathematics teacher for three years, and in 2008 he worked as a lecturer in mathematics education in an institute of teacher training and education science for one year. In 2009, he started a new job as a civil servant lecturer at the Department of Mathematics of the Faculty of Mathematics and Natural Sciences, Universitas Negeri Semarang.

He also has experience as an academic staff member at the Centre of Development of Education Media (PPMP), Universitas Negeri Semarang, and as a reviewer of Interactive Educational Multimedia Production at the Department of ICT Development for Education (BPTIKP), Central Java, Indonesia. He teaches mathematics education, especially geometry and the use of ICT in Mathematics Education, and he has participated in several research projects in these areas. He has presented and published several papers in his research area in national and international forums and scientific journals. In 2006, he was recognised as the winner of a competition of teachers with smart ideas held by Citibank Indonesia and the Indonesian Hope Foundation.

In 2013, he was awarded a PhD scholarship from the Islamic Development Bank (IDB), Jeddah, Saudi Arabia, through an IDB-UNNES PhD fellowship program. In the same year, he started his PhD study at the Institute for Mathematics and Computer Science Education, J. W. v. Goethe- University Frankfurt am Main, Germany, under the supervision of Prof. Dr. Matthias Ludwig from Goethe Universit Frankfurt and Professor Marc Schaefer, Ph.D from Rhodes University, South Africa. His PhD research focused on the MathCityMap-Project for Indonesia. In 2017 he was awarded a doctoral degree in Natural Sciences (Dr.rer.nat.) in the speciality of Didactics of Mathematics after he furnished evidence of his ability in regularly conducted doctoral proceedings through his dissertation entitled „Learning mathematics in a mobile app supported math trail environment“. In the same year, he resumed his position as a lecturer at Universitas Negeri Semarang to teach in the Undergraduate Program and Postgraduate Program of mathematics education.

 

 

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Learning mathematics in a mobile app-supported math trail environment

 

 

 

 

 

 

 

Springer

 

 

Adi Nur Cahyono

 

 

Learning mathematics in a mobile app-supported math trail environment

 

 

 

 

 

 

 

 

 

 

 

 

 

Springer

 

 

Adi Nur Cahyono aus Banjarnegara (Indonesien)

E-mail: cahyono@math.uni-frankfurt.de/ adinurcahyono@mail.unnes.ac.id

 

 

Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) angenommen vom Fachbereich 12 (Informatik und Mathematik) der Johann Wolfgang Goethe-Universität in Frankfurt am Main, 2017.

 

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Dekan: Prof. Dr. Andreas Bernig Erstgutachter: Prof. Dr. Matthias Ludwig

Zweitgutachter: Prof. Marc Schäfer, PhD. (Rhodes University, South Africa)

Datum der Disputation: 26. Juni 2017

 

 

 

 

 

 

 

 

 

 

 

Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://d-nb.info/1137348240 abrufbar.

 

 

Preface  

Considering mathematics as a human activity, as in Freudenthal’s educational credo, has led to the view of mathematics education as a process of doing mathematics that results in mathematics as a product. As a human activity, mathematics is also an activity of solving problems. Bringing problem-solving practice into the educational process can be useful as a tool for learning mathematics. Through this process, learners have the opportunity to reconstruct the mathematical experience and gain new mathematical knowledge. Furthermore, encouragement is needed to motivate learners to engage in this process.

However, the current public understanding of mathematics and its teaching and learning process is still not satisfactory. Many students do not enjoy engaging in mathematics activities. Mathematics is also sometimes considered to be a subject that is difficult, abstract and far from students’ lives, in terms of both daily life and occupation. This problem leads mathematicians and mathematics educators to think of ways to popularize mathematics among the public.

A variety of projects have been developed and implemented in numerous countries to raise public

 

 

awareness of mathematics. For instance, in an effort to communicate mathematics to the public, in 2008, the German “Year of Mathematics” was held successfully. One positive message from this event was Du kannst mehr Mathe, als Du denkst (You know more maths than you think). This message emphasizes the importance of working on the public’s views of what mathematics is about and what mathematicians do. The event also focused on news and challenges, as well as images and jobs.

In the same spirit, the MATIS I Team from IDMI Goethe- Universität Frankfurt, Germany has also given attention to this subject by developing and implementing a project called MathCityMap. This project comprises math trails around a city that are supported by the use of GPS-enabled mobile phone technology. It aims to share mathematics with the public (especially students), encouraging them to be more involved in mathematics. The project offers an activity that is designed to support students in constructing their own mathematical knowledge by solving the prepared mathematical tasks on the math trail and interacting with the environment, including the digital environment.

My PhD research is a part of the MathCityMap project. By following a design research paradigm, it focused on the development of a model for a mobile app-supported math trail programme and the implementation of this programme in Indonesia tailored to that country’s situation. The

 

 

implementation included a field empirical study to explore its effect on students’ motivation in mathematics, their performance in mathematics and teachers’ mathematical beliefs.

The results of the research are presented through my dissertation, which is submitted for the degree of Doctor rerum naturalium (Dr. rer. nat.) at Faculty 12 (Computer Science and Mathematics), Goethe University Frankfurt, Germany. The research described herein was conducted under the supervision of Professor Matthias Ludwig at the Institute of Mathematics and Computer Science Education, Faculty of Computer Science and Mathematics, Goethe University Frankfurt, between September 2013 and August 2016. These results have also been reviewed by Professor Marc Schäfer, SARChI Mathematics Education Chair at Rhodes University, South Africa.

Frankfurt, March 2017 Adi Nur Cahyono

 

 



 

 

Acknowledgements This thesis is the result of my PhD study at the Institute of Mathematics and Computer Science Education, Goethe University Frankfurt (GUF), Germany. It was implemented in collaboration with the Department of Mathematics, Semarang State University (UNNES), Indonesia and nine secondary schools in the city of Semarang with the agreement of the Department of Education of the city of Semarang, Indonesia. This study was funded by the Islamic Development Bank (IDB), Jeddah, Saudi Arabia through an IDB-UNNES PhD fellowship programme. It would not have been possible without the support, guidance and help of many people around me. Therefore, I would like to dedicate this section to acknowledging these people.

First, I would like to express my sincere gratitude to my Doktorvater/ supervisor, Professor Matthias Ludwig (Goethe University Frankfurt, Germany), who has supported and guided me with his impressive knowledge and expertise. It was a great opportunity for me to do my PhD research under his guidance. He guided me to see things in a comprehensive and coherent way that was not only important for my PhD work, but will also be essential for my future professional life. He not only gave me professional and academic support, but also personal support. Thank you very much for this experience, Professor Ludwig.

 

 



 

 

I am also indebted to Professor Marc Schäfer (Rhodes University, South Africa), my second supervisor, for his valuable guidance and support. He gave valuable feedback and input that stimulated me to further reflect on my own thought and works. Thank you very much for your support, enthusiasm, knowledge and friendship.

I would like to acknowledge the IDB and the PMU IDB- UNNES for the scholarship I received. I would also like to express my gratitude to UNNES, the university where I work as a lecturer, for giving me permission and encouragement to pursue my doctoral degree in Germany.

During my PhD study, I received support and help from my colleagues at the Institute of Mathematics and Computer Science Education, GUF (thanks to Phillip, Xenia, Hanna, Sam, Jorg, Iwan, Martin, and Anne) and the Department of Mathematics, Faculty of Mathematics and Natural Sciences, UNNES. I am also deeply indebted to the teachers and students in Semarang, Indonesia who participated in my studies. Completion of my PhD research would never have been possible without their help and cooperation. I would like to extend my gratitude to KJRI Frankfurt (General Consulate of Republic Indonesia Frankfurt am Main) and to all my friends during my stay in Germany.

Above all, I would like to express my immeasurable gratitude to my wife, Chatila Maharani, for her love,

 

 

understanding, unfailing encouragement and support. Without her unconditional support, I would never have completed my study. Although she was also busy with her doctoral study, she was always there whenever I needed her. To my daughter, Calya Adiyamanna Putri, you are a great kid who always cheerfully follows our journeys, and to my son, Arbiruni Neumain Cahyono, our spring child, who was born in the busy time when I prepared this dissertation submission. I dedicate this dissertation to the three of you. Special thanks to my parents (Bapak Harjono and Ibu Rusmini), my parents-in-law (Bapak Suwignyo Siswosuharjo and Ibu Nurhayati), my sisters, my brothers, my nephews and my nieces, who have always supported and encouraged me.

 

 



 

 

Zusammenfassung  

Das MathCityMap-Projekt ist ein Projekt der Arbeitsgruppe MATIS I des IDMI der Goethe-Universität Frankfurt. Es ist ein Projekt innerhalb des Math-Trail- Programms, das durch den Einsatz mobiler Technologie unterstützt wird. Die vorliegende Studie untersucht dieses Projekt speziell in Indonesien. Es wurde mittels folgendem Forschungsdesign durchgeführt und fand in einer vorbereitenden und drei nachfolgenden Phasen statt (eine Entwurfsphase, eine kleine Pilotversuchsphase und eine groß angelegte Feldversuchsphase).

Die Ziele dieser Studie konzentrierten sich auf das Design einer App und von Math-Trails rund um die Stadt, die Umsetzung des App-unterstützten Math-Trail-Programms in Indonesien und insbesondere die Untersuchung der Auswirkungen auf die Mathematik-Motivation und - Leistung der Schüler und auf die mathematische Überzeugungen der Lehrer. Die Ergebnisse dieser Studie und der Beitrag zur Entwicklung des MathCityMap- Projektes werden in dieser Arbeit vorgestellt.

Diese Studie wurde in einem theoretischen Rahmen, basierend auf der konstruktivistischen Theorie in der Mathematikausbildung, durchgeführt. Dabei wird davon ausgegangen, dass die Schüler im Mathematiklernprozess

 

 

ihr mathematisches Wissen aktiv erweitern, indem sie neu gewonnene Informationen mit ihrem bisherigen Wissen verbinden. Lehrer fungieren als Vermittler und sollen in der Lage sein, mathematische Probleme vorzubereiten und den Schülern Anleitung oder Unterstützung bei der Problemlösung zu bieten. Um diese didaktische Situation zu verwirklichen, muss eine besondere Mathematik- Lernumgebung entworfen werden.

In dieser Studie wurde eine Lernumgebung mit dem Math- Trail-Konzept als Grundidee entwickelt. Ein Math-Trail ist ein Pfad, bestehend aus einer Reihe von Stationen, an denen die Schüler Mathematik im Freien betreiben und entdecken können. Die Idee der klassischen Math-Trail- Projekte sind nicht neu, aber die Kombination dieses etablierten Outdoor-Bildungskonzepts mit dem Einsatz neuester Technologie ist es. Wir zeigen in dieser Studie dass der Einsatz von Mobiltechnologie im Freien das Potenzial hat, den Mathematik-Lernprozess zu unterstützen.

In der Designphase wurde die GPS-fähige App speziell für den Einsatz in Indonesien unter Berücksichtigung der lokalen Situationen, Bedingungen und Bedürfnisse entwickelt. Es handelt sich um eine native Android-App, welche folgendes anzeigt: die mathematischen Trailrouten/Karten einschließlich der Standortkoordinaten der mathematischen Trailaufgaben und der aktuellen

 

 

Benutzerposition, die authentischen mathematischen Probleme, die Informationen über das benötigte Werkzeug zu den Problemlösungen, die Hinweise (falls vom Benutzer angefordert) und die Rückmeldung über die eingegebenen Antworten. Die App zeigt auch die Kurzinformationen zum Objekt.

Die App wurde mit der MIT App-Inventor-Software speziell für MathCityMap. Indonesia erstellt und läuft auf Handys mit Android-Plattform. Neben der App wurden in dieser Phase auch 87 Aufgaben von Mathematiklehrinnen und Lehren die durch einen Workshop geschult wurden entworfen; die Aufgaben wurden 13 Math-Trails in verschiedenen Regionen der Stadt Semarang zugeordnet. Es gibt 6-8 Aufgaben für jeden Trail, mit verschiedenen mathematische Themen in einer sicheren Umgebung. Die entworfenen Trails führen durch über Schulgelände, Stadtparks, Märkte, aber auch zu historischen Gebäuden, Touristenattraktionen und anderen Orten.

Die Math-Trails und die Aufgaben wurden in das MathCityMap-Portal hochgeladen, so dass die Benutzer über die App darauf zugreifen können. Trail-Walker brauchen etwa zwei Stunden, um einen Trail zu erkunden (ca. 1-2 km Länge) und die Aufgaben auf diesem Trail zu lösen. Die entworfenen Math-Trails und die App bestanden zuerst den Evaluierungsprozess durch Validierung der Experten und einer Pilotphase mit einer kleinen Gruppe von Schülern und Lehrern. Einige Aufgaben wurden

 

 

aussortiert; die positiv evaluierten Aufgaben wurden für die nächste Phase verwendet.

Die groß angelegte Versuchsphase wurde in den Jahren 2015 und 2016 mit Schülern und Lehrern aus neun Sekundarschulen durchgeführt. Für jede Schule waren zwei Klassen (ca. 30 Schüler/Klasse) an dieser Studie beteiligt, eine Klasse als Versuchsgruppe und eine Klasse als Kontrollgruppe. Beide Gruppen in jeder Schule wurden von der gleichen Lehrperson mit gleichem Thema und Lehrstoff, aber mit unterschiedlichen Interventionen, unterrichtet. In der Versuchsgruppe wurden das Lehren und Lernen der Mathematik mittels App-unterstütztem Math-Trail-Programm durchgeführt. Die Schüler der Kontrollgruppe wurden auf dem herkömmlichen Weg unterrichtet.

Im App-unterstützten Math-Trail-Programm wurden Gruppen von 5-6 Schülern gebildet. Die Aktivitäten wurden während der normalen Schulzeit in etwa 2 x 45- minütige Abschnitten durchgeführt, mit einer kurzen einführenden Erklärung der Regeln und Ziele durch die Lehrer. Die Gruppen begannen dann ihre mathematische Reise von unterschiedlichen Aufgabenorten aus damit nur jeweils eine Gruppe an einer Aufgabe arbeitet. Während die Gruppen den Trail abwanderten, beobachteten und überwachten Forscher und Lehrer die Aktivitäten der Schüler, unterstützten diese aber nicht, da alle notwendigen

 

 

Informationen mittels der App verfügbar waren.

Mit Hilfe der App folgen die Schülergruppen einer geplanten Route, finden die Aufgabenorte und lösen das mathematische Problem vor Ort, und gehen dann zur nächsten Aufgaben weiter. Im Anschluss wurde mit allen Schülern der Klasse eine Nachbesprechung durchgeführt. Um Schülermotivation zu erfassen wurden Fragebögen (Situational Motivation Scale/SIMS) vor und nach den Aktivitäten verteilt, Ebenso wurden die mathematischen Überzeugungen (Beliefs) der Lehrer erfasst. Um den Lernfortschritt zu messen wurden individuelle mathematischen Tests erstellt und vor und nach den Aktivitäten durchgeführt. wurden.

Das Ergebnis der SIMS zeigt, dass die Art der Motivation der Schüler, sich zu engagieren, verschieden ist. Es hat sich gezeigt, dass alle Schüler einen positiven Self-Determined Index (SDI) besitzen. Dies bedeutet, dass ihre Motivation stark selbstbestimmt, bzw. eine verinnerlichte Motivationsform war. So waren intrinsische Motivation (IM) und identifizierte Regulierung (IR) dominannter als externe Regulierung (ER) und Amotivation (AM). Die Schüler empfanden die Aktivität interessant oder angenehm (ein Indikator für IM) sowie aussagekräftig oder wertvoll (ein Indikator für IR). Sie beteiligen sich an den Aktivitäten aus Eigeninteresse/Vergnügen/Zufriedenheit, und sie betrachteten ihre Teilnahme als ihre eigene Entscheidung.

 

 

Bei offene Anschlussfragen zum SIMS- Fragebogen berichteten die Schüler, dass sie an den mathematischen Outdoor-Aktivitäten Spaß hatten und interessiert an der Verwendung von moderner mobiler Technologie beim Lernen von Mathematik sind. Sie berichteten auch, dass das Lernen von Mathematikanwendungen in der realen Welt für sie wertvoll und sinnvoll war. Das SIMS-Ergebnis zeigt auch, dass es vor und nach der Teilnahme an diesem Programm einen signifikanten Unterschied in der Orientierung der Schülermotivation gab. Es gab eine Veränderung von extrinsisch motiviert hin zu mehr intrinsisch motiviert. Bemerkenswert ist auch, dass es diese intrinsische Motivation nicht auf den Neuigkeitseffekt zurückzuführen ist, da die Werte im zweiten Jahr der Studie sich nicht signifikant geändert haben.

Es zeigte sich ebenfalls, dass die Intervention die Leistungen der Schüler in Mathematik beeinflusst hat. Um die Wirkung  der  Intervention  auf  die Mathematikleistungen der Schüler zu bestimmen, wurden die Schülerresultate von Experimentalgruppe und Kontrollgruppe statistisch überprüft. Die Ergebnisse zeigen, dass der mittlere Leistungszuwachs der 272 Schüler in der Experimentalgruppe signifikant höher war als die der 248 Schüler in der Kontrollgruppe. Das zeigt, dass die Intervention mit diesem Programm die Leistung der Schüler in Mathematik erhöht hat.

 

 

Da der Lehr- und Lernprozess mit den Einstellungen der Lehrer verknüpft ist, beschäftigte sich diese Studie auch mit dem Einfluss des MathCityMap-Programms auf die mathematischen Überzeugungen der Lehrer im Zusammenhang mit ihren Strategien zur Förderung der Motivation der Schüler, sich in Mathematik zu engagieren. Bei den Lehrpersonen war zu erwarten dass durch die Fokusgruppen-Diskussion,         Lehrerfortbildungen (Workshop zur Aufgabengestaltung) und die Einbeziehung in den Beobachtungsprozess ein Einfluss auf die mathematischen Überzeugungen stattgefunden hat Ein Pre-Posttest-design zeugen, dass es einen Wechsel der mathematischen Überzeugungen, von traditionell orientiert hin zu mehr forschungsorientiert gab.

Lehrer, nachdem sie an diesem Programm beteiligt waren, mehr von Folgendem überzeugt wurden: Mathematik ist ein Werkzeug zum Denken, das Ziel der Schüler ist es zu verstehen, die Schüler sollten eine Autonomie haben, Mathematikfähigkeit ist offen für Änderung, und die Schüler werden sich in Mathematik engagieren wollen, wenn die Aufgaben interessant und herausfordernd sind. Es gab auch einen signifikanten positiven Effekt der mathematischen                                         Überzeugungen                   bei forschungsorientierten Lehrern auf die intrinsische Motivation der Schüler. Lehrer berichteten auch, dass durch die Nutzung der mobilen App es erleichtert wurde, Outdoor-Lehr- und Lernprozess zu gestalten.

 

 

Zusammengefasst konnte mit dieser Arbeit gezeigt werden, dass durch die für diese Studie entwickelte App und den dazu passenden Trail-Designprozess mehrere Math-Trails in der Stadt Semarang installiert werden konnten. Empirische Studien zum Einsatz der App und der Trails zeigen, dass die Studierenden waren sehr intrinsisch motiviert, sich in Mathematik zu engagieren. Sie haben durch diese Aktivitäten mathematische Erfahrungen gesammelt. Infolgedessen wurde ihre Leistung in der Mathematik verbessert. Der Einfluss dieses Programms auf den Wechsel der mathematischen Überzeugungen der Lehrer hat zu diesen Ergebnissen beigetragen. Wir empfehlen, dass Schulen und Öffentlichkeit die vorhandenen Math-Trails (einschließlich der unterstützenden Werkzeuge) nutzen oder neue Trails entwerfen, indem sie dem in dieser Studie angewendeten Entwicklungsmodell folgen. Weitere Untersuchungen sind für die Projektentwicklung und -umsetzung in anderen Orten/Städten mit anderen Situationen und auch anderen Studienaspekten und -bereichen notwendig.

 

 

Table of Contents Cover............................................................................ i
Preface........................................................................ iii
Acknowledgemens....................................................... vii
Zusammenfassung (German-Summary)........................... xi
Table of Contents....................................................... xix
Chapter 1. Introduction................................................. 1
1.1.  Background of the study.......................................... 1
1.1.1.    What is the problem?....................................... 5
1.1.2.    Can the mobile app-supported math trail programme offer a solution?        9
1.2.  Design of the study............................................... 13
1.2.1.  Topic of the study........................................... 13
1.2.2.  Aims of the study............................................ 14
1.2.3.  Research questions......................................... 16
1.2.4.  Approach of the study..................................... 19
1.2.5.  Phases of the study......................................... 21
1.3.  The book’s structure.............................................. 25
Chapter 2. Theoretical Background................................ 29
2.1.  Constructivism in mathematics education................. 30
2.2.  Students’ motivation in mathematics....................... 38
2.3.  Students’ performance in mathematics..................... 41
2.4.  Teachers’ mathematical beliefs............................... 45
2.5.  Didactical situation in mathematics.......................... 47
2.6.  Outdoor mathematics education.............................. 52
2.7.  Technology in mathematics education...................... 64
2.8.  Conceptual framework of the study......................... 71
Chapter 3. The MathCityMap Project............................ 75
3.1.  The concept and the goals of the project................... 76
3.2.  Components of the project..................................... 77
3.2.1.  Mathematical outdoor tasks............................. 77
3.2.2.  Mathematical city trips................................... 80
3.2.3.  Map-based mobile app.................................... 83
3.2.4.  MathCityMap community................................ 87
3.3.  Technical implementation of the project................... 89
3.3.1.  Steps of preparation........................................ 89
3.3.2.  Settings of the activity..................................... 90
3.3.3.  Evaluating and publishing the activity.............. 92
Chapter 4. Designing the mobile app-supported math trail environment in Indonesia        93
4.1.  Indonesian secondary school mathematics
education.................................................................... 93
4.1.1.  A brief profile of Indonesia.............................. 94
4.1.2.  Education system in Indonesia......................... 96
4.1.3.  Indonesian secondary school mathematics curriculum   98
4.2.  Design of the mobile app-supported math trail programme for Indonesia      101
4.2.1.  Design of the rules of the activity.................... 106
4.2.2.  Design of the mobile app............................... 110
4.2.3.  Design of the math trails............................... 127
Chapter 5. Evaluating the potential effects of a mobile app-supported math trail programme: An exploration study in Indonesia.................................................................. 151
5.1.  Method.............................................................. 151
5.1.1.  Approach..................................................... 152
5.1.2.  Participants, situations, activities and procedures    152
5.1.3.  Data collection and analysis.......................... 154
5.2.  Results............................................................... 158
5.2.1.  What was the initial condition of the students like?            159
5.2.2.  How has the programme been running?........... 163
5.2.3.  How did the students work?........................... 164
5.2.4.  Why do students engage in this programme?. 168
5.2.5.  What mathematical experience do they get after having engaged in this programme? 184
5.2.6.  Does the intervention promote students’ performance in mathematics?         199
5.2.7.  How does the project affect teachers’ mathematical beliefs?            202
5.3.  Discussion.......................................................... 208
Chapter 6. Conclusions and recommendations............... 219
6.1.  Conclusion......................................................... 219
6.2.  Recommendations............................................... 240
References................................................................ 243
 

 

Chapter 1 Introduction

 

This book reports a study on the development, implementation and evaluation of a mobile app-supported math trail programme in Indonesia. As an introductory chapter, this chapter contains the background of this research, including the problems that underlie the need for this research, and the solution offered through this study. The design of the study, including the topic, aims, questions, approach and phases of the study, is described in detail in this chapter. To facilitate an understanding of the flow of the study, a concept map is also presented in this part. The structure of this book is then outlined in the last section of this chapter.

1.1.  BACKGROUND OF THE STUDY

Mathematics plays a role in daily life – both individual and social – and in professional life. People, therefore, must be able to apply basic mathematics in their everyday lives, a skill that the Organisation for Economic Cooperation and Development (1999) termed “mathematics literacy”. In an educational context, mathematics activities should offer the chance for students to experience mathematics as a

 

 

meaningful subject that can be understood well (Freudenthal, 1991).

Furthermore, Ojose (2011) suggests that students should be provided with real-world situations that are relevant to their position as citizens or their concern area. Such situations can be experienced by mathematics students outside the classroom (Dubiel, 2000). This approach can show students that mathematics is all around them and not merely in textbooks. Bringing mathematics outdoors can provide an opportunity for students to experience mathematics in real-world situations.

In some countries, interest in the development of outdoors and adventure education programmes has increased (Fägerstam, 2012). Various activities outside the classroom have been specifically designed to increase student engagement. Integrated programmes have also been developed to combine learning outside the classroom with traditional learning in the classroom. In mathematics, one of the outdoor educational programmes that have been elaborated is the math trail programme. This programme is not purely an outdoor activity, as it can also be conducted indoors, but it can be used as one alternative to encourage students to learn mathematics outside the classroom.

The math trail programme, which is a pathway to discovering mathematics, was created as a medium for experiencing mathematics in relevant applications (Shoaf,

 

 

Pollak, & Schneider, 2004), namely communication, connections, reasoning and problem solving (National Council of Teachers of Mathematics, 2000). In a math trail, students can simultaneously solve mathematical problems encountered along the path, make connections, communicate ideas and discuss them with their teammates, and use their reasoning and skills in problem solving.

Dudley Blane and his colleagues first developed a math trail by blazing trails in the centre of Melbourne as a family holiday activity (Blane & Clarke, 1984). The programme was then strengthened when some schools took advantage of the trails by integrating them into their mathematics learning programmes. The success of this idea allowed this programme to be adapted and applied in different places, and math trail projects subsequently emerged in various cities, such as Vancouver, Boston, Philadelphia and San Francisco (Richardson, 2004).

Although the math trail project is not new, the idea of an outdoor education programme supported by mobile technology appears to be new. This idea is facilitated by the fact that in recent years, developments in mobile technology and mobile phone use have improved significantly (Cisco, 2016; Lankshear & Knobel, 2006). These improvements were followed by many mobile phone applications (apps), including those intended for use in outdoor activities, such as Endomondo® for outdoor sports. In learning activities, Wijers, Jonker, and Drijvers (2010)

 

 

suggested that mobile devices could be employed to promote learning outside the classroom.

However, to date, most mobile technology apps for mathematics learning have only been employed in regular teaching settings (Trouche & Drijvers, 2010). Thus, it is necessary to explore the potential of mobile technology for teaching and learning mathematics, including for use in outdoors mathematics learning, and in so doing, engage students in meaningful mathematical activity. The combination of reality and virtual reality is expected to contribute to student engagement (Schwabe & Göth, 2005; Wijers et al., 2010).

This point is used as the basis for the development of the MathCityMap project. The project is a programme of math trails that are facilitated by the use of mobile phone technology and use special mathematical tasks (Cahyono & Ludwig, 2014; Jesberg & Ludwig, 2012). The project offers the opportunity to experience mathematics in the environment by arranging educational tasks in several locations around the city. Through a mobile phone app- supported mathematical activity, learners in groups find the task locations by tracing a planned math trail and solve mathematical problems encountered along the path.

The project aims to develop math trails and supporting tools, such as a web portal and a mobile phone app, and then share the project with the public (especially students).

 

 

This approach is expected to have the potential to encourage students to be actively involved in mathematics and promote student motivation to engage in mathematical activity. During their involvement in each activity and their interaction with the environment, students have the opportunity to construct their own mathematical knowledge. As a result, this will achieve the primary target of improving their performance in mathematics.

Referring to these basic ideas and objectives, as a part of the MathCityMap project, the model of the mobile app- supported math trail programme was then developed. The implementation of this programme in Indonesia tailored to that country’s situation was also studied scientifically. It was motivated by several problems in the field of mathematics education, especially in Indonesia.

1.1.1.   What is the problem? Public understanding of mathematics is often unsatisfactory, and only a small number of students enjoy participating in maths during the school day (Behrends, 2009, p. 1). At school, and globally, mathematics is sometimes perceived as a difficult and abstract subject. Internationally, by the eighth grade, only about a quarter of students enjoy learning mathematics (Mullis, Martin, Foy, & Arora, 2012, p. 325). This issue is a worldwide phenomenon, but given the unique aspects of each country, in this study we restrict ourselves to focusing on the issue in Indonesia.

 

 

In Indonesia, the percentage of students who report being happy at school is high in comparison with other countries (Organisation for Economic Cooperation and Development, 2013). However, Hadi (2015) reported that “most students fear mathematics, tend to skip mathematics subjects, and are happy when the mathematics teacher is not able to come to class” (p. 1).

In addition, based on the TIMSS 2011 results in mathematics, only approximately 20% of Indonesian eighth-grade students like learning mathematics and only 3% are confident in their abilities in maths (Mullis et al., 2012). In general, mathematics involves learning a lot of processes and formulae that appear to be not only unconnected with each other but also irrelevant to students’ lives (Education, Audiovisual and Culture Executive Agency, 2011).

Students appear to be less motivated to learn mathematics. This problem is serious because if we refer to the results of Hattie’s (2009) meta-analysis, students’ motivation and attitude toward mathematics and science are related to their performance in those subjects. In a synthesis of over 800 meta-analyses relating to achievement, 288 studies reported that attitudes toward mathematics and science were related to mathematics and science achievement (Hattie, 2009, p. 50).

 

 

From six meta-analyses (327 studies involving 110,373 people), Hattie concluded that motivation has a significant positive effect on achievement. Hattie’s effect size of student motivation was d = 0.48. This means that the influence was labelled as being in the “zone of desired effects”. Therefore, we assume that the above conditions have an impact on the low performance of Indonesian students.

Nationally, overall, Indonesian secondary school student performance has improved significantly. For mathematics, between 2004 and 2006, the average examination score rose from 5.0–6.2 to 6.8–7.6 (EFA Secretariat Ministry of National Education of RI, 2007). However, based on the data from the UNESCO International Bureau of Education (2011), Indonesian students still rank low in international standardized tests. Since PISA 2000, the trend has shown no significant increase or decrease and has tended to stagnate at low performance values (Baswedan, 2014).

PISA 2012 ranked Indonesian students aged 15 years 64th out of 65 countries, with a score of 375 for mathematics. PISA 2012 examines how well students can perform with the knowledge they possess (Organisation for Economic Cooperation and Development, 2013). Similarly, in TIMMS 2011, the mathematics achievement of Indonesian eighth-grade students was ranked 38th out of 42 countries, with a score of 386 in comparison to an average score of 500.

 

 

Further, PISA 2012 found that 75.7% of students were unable to reach level 2. This indicates that Indonesian students were unable to extract the relevant information for a given task or use basic algorithms to answer questions. Students also had difficulty solving realistic problems, especially in geometry. In the national exam in 2012, the average score of junior high school students in mathematics was 5.78 out of a maximum of 10, and mastery of geometry was below 50% (Kementerian Pendidikan dan Kebudayaan RI, 2013).

The World Bank (2010) reported that an ineffective learning process contributes to the low quality of graduates from Indonesian secondary schools. Previous studies indicated that several factors may affect quality, such as an increased focus on theory and routine learning, a focus on external examinations, administrative demands, textbook- based teaching and learning, and a crowded curriculum (Marsigit & Rosnawati, 2011; Sembiring, Hadi, & Dolk, 2008).

The natural next question, therefore, is how to improve Indonesian students’ motivation in mathematics in an effort to promote their performance in mathematics. One expected approach concerns the implementation of the mobile app-supported math trail programme. Conceptually, this programme is intended to address

 

 

problems similar to those encountered in Indonesia, as described above.

1.1.2.   Can    the    mobile     app-supported     math    trail programme offer a solution? Mathematics can be found everywhere and is experienced in everyday situations. The environment provides unlimited sources and ideas for teaching and learning mathematics. Students need to connect with the subject through meaningful activities, such as by implementing abstract mathematical concepts in real-life situations. The math trail programme provides opportunities in this regard. Supported by the use of mobile phone technology, in this project, math trails were designed around the city and offered opportunities to encourage student involvement in meaningful mathematical activities by utilizing the advantages of the latest technology.

This programme has the math trail concept at its core. A math trail is a planned pathway that consists of a series of stops at which students can explore mathematics in the environment (English, Humble, & Barnes, 2010; McDonald & Watson, 2010; Shoaf et al., 2004). Students find and solve real problems related to mathematics in real situations and connect mathematics with other disciplines.

Math trail tasks are created in several locations and then collected in a database. During the implementation phase, the tasks can be selected and grouped into a math trail route

 

 

based on the conditions and objectives. The tasks, their locations and directions to these sites, as well as the route and the tools required for problem solving, are provided in a trail guide, which is a scale map of the trail.

With the rapid development of technology, it is possible to collect the tasks and design a trail guide based on a digital map through a portal. Teachers can arrange the math trails in a web portal, and students can access the trails with the help of GPS-enabled mobile devices or use a paper version of the trail guide. In this project, we designed a web portal and a mobile phone application to support the math trail programme.

Using mobile devices is a familiar practice to all social and economic groups. In fact, in recent years, rapid developments have occurred in the scope, uses and convergence of mobile devices (Lankshear & Knobel, 2006) used for computing, communications and information. Within the next five years, it is estimated that the total number of smartphones will account for nearly 50 per cent of all global devices and connections (Cisco, 2016, p. 3).

As in other countries around the world, the rapid development of mobile technology has also occurred in Indonesia. Indonesia is amongst the top five countries worldwide in terms of the number of mobile phone users, with growth in 2012 reaching 60% (Prayudi & Iqbal,

 

 

2013). The same authors also reported that the percentage of market share in 2012 was 68.8% Android, 18.8% iOS, 4.5% BlackBerry OS, 3.3% Symbian, 2.5% Windows Phone, 2.0% Linux and 2.1% others. The World Bank Group (2015) reported that mobile cellular subscription in Indonesia in 2012 was 319,000,000 or 126.18 per 100 people.

Using mobile technologies is also common practice in learning activities, and these technologies offer advantages in terms of learning opportunities. The National Council of Teachers of Mathematics (2008) stated that technology has the potential to be a fundamental tool for mathematics learning. O’Malley et al. (2003) called the learning process in which learners take advantage of this approach “mobile learning”. One of the benefits of mobile learning is the opportunity to place learning outdoors in the real world. Mobile devices have the potential to integrate the characteristics of effective learning, such as situated realistic learning, motivational power and teamwork (Wijers et al., 2010). Thus, mobile learning enables learners to build knowledge and construct their understanding in unusual settings (Winter, 2007).

The math trail is the core of the activities in this project, and mobile technology is used to support these activities. This shows that this project combines real and virtual environments. According to Schwabe and Göth (2005), this combination might contribute to the improvement of

 

 

student engagement. When students are highly motivated to engage in mathematics, they are interested in, and enjoy spending more time, learning and solving mathematical tasks. Such students tend to be more persistent in solving mathematical problems (Lepper & Henderlong, 2000). Enjoyment and persistence in learning mathematics and positive views concerning the value and relevance of mathematics learning can lead to engagement with mathematics (Attard, 2012). Rigby, Deci, Patrick, and Ryan (1992) reported that student engagement in learning processes has a positive impact on their understanding of new knowledge and their flexibility in using new information.

Thus, theoretically, the concept of the mobile app- supported math trail programme offers several benefits for solving the current problems in the field of mathematics education, especially in Indonesia. We felt that if it were carefully designed and integrated into an intervention, the programme developed in this project could increase students’ motivation to engage in the mathematics learning process. As a result, in line with the results of Hattie’s (2009) meta-analysis, it will have an impact on improving their performance in mathematics. However, the design and implementation of the teaching and learning programme are also related to teachers’ attitudes toward the programme and their practices in the classroom. Thus, it is also important to address teachers’ mathematical

 

 

beliefs related to their strategies in fostering students’ motivation to engage in mathematical activity.

1.2.  DESIGN OF THE STUDY

The conceptual idea described above must be studied scientifically to develop a theory, translate the theory into a model of an educational programme, implement the programme and evaluate it through an empirical study. This need prompted us to carry out this study.

1.2.1.  Topic of the study This PhD study researches the development of the model of the mobile app-supported math trail programme and its implementation in Indonesia. The implementation in this study also includes an empirical study to explore the impact of the programme developed on student motivation in mathematics, their performance in mathematics and teachers’ mathematical beliefs. The programme was developed and implemented primarily for school activities as part of the process of teaching and learning mathematics, so the programme was developed in accordance with the school mathematics curriculum in Indonesia. Since the area of school mathematics is too broad to be investigated in the framework of this study, it must be restricted to an area that is relevant to the issues mentioned in the part of the background of this study. Therefore, the programme was implemented particularly for lower secondary school or

 

 

junior high school level (for all grades and for all mathematics topics).

As a pilot study, we implemented and studied empirically the programme for secondary schools in the city of Semarang, Indonesia from 2013 to 2016. It was carried out in cooperation between the two countries through two institutions, namely IDMI Goethe-Universität Frankfurt in Germany and FMIPA Universitas Negeri Semarang in Indonesia, and in collaboration with the Department of Education of the city of Semarang to implement the project in nine schools in the city. Semarang is not representative of Indonesia as a whole but has unique characteristics as a town and includes high-, middle- and low-level schools. Both urban and rural schools are also found in this region, which has hills and seaside areas, while the economic level of citizens varies. Thus, the implementation of the programme in this city is expected to provide a large amount of information and to serve as a model for the development and implementation of the programme in other cities in Indonesia, which is rich in diversity.

1.2.2.  Aims of the study This study consisted of two phases, namely: (1) the developmental phase and (2) the implementation and empirical study phase. These two phases were preceded by a preliminary phase. The first phase was aimed at developing a model of a mathematics educational

 

 

programme grounded in the basic idea and objectives of the MathCityMap project. The basic idea of the MathCityMap project is to set up mathematical problem-solving activities in outdoor situations following the concept of the math trail and supported by the use of mobile technology. The programme offers the opportunity for learners to construct their own mathematical knowledge during their interactions with the environment.

Therefore, the programme was developed within a framework informed by constructivism in mathematics education and built using some basic concepts, such as the concept of the math trail and the use of mobile technology in mathematics education. From this theoretical framework, the model of a mobile app-supported math trail activity was formulated. This model includes the concept, objectives, components and technical implementation of the programme and its evaluation.

The second phase was aimed at implementing the programme in Indonesia tailored to the local situation. It included the evaluation of its implementation by conducting an empirical study. Through an exploration study, the potential of the programme for eliciting engagement in meaningful mathematics teaching and learning activities was explored. Decisive factors that affect students’ motivation to engage in mathematics and their performance in mathematics were also examined in this study. Furthermore, because teaching and learning

 

 

processes are linked with teachers’ attitudes, this study also considered the effects of the project on teachers’ mathematical beliefs. Thus, teachers’ mathematical beliefs related to their strategies in fostering students’ motivation to engage in mathematics education through the implementation of the programme were also investigated in this study.

1.2.3.  Research questions Taking into consideration the aims of the research, the three central research questions of this study were clarified. The first question was:

1.    How can the mobile app-supported math trail programme promote students’ motivation to engage in mathematics?

The meaning of the keywords of this question are as follows. First, the question starts with the words “how can”. This phrasing indicates that the question focuses on identifying the conditions and approaches that optimize the benefits offered by the programme. This programme is a “mobile app-supported math trail programme”, which is a core of the MathCityMap project and was also theorized through this study.

“Engagement” leads this study to determine how students are actively engaged in this programme. The use of outdoor

 

 

activities supported by a mobile phone application is a strategy to engage students in the mathematics learning process. Because students’ engagement in an activity is related to their motivation, it is necessary to investigate the nature of “students’ motivation” to engage in such an activity. It is also important to determine the impact of this programme in “promoting” student motivation in mathematics and to investigate the differences in student motivation to engage in mathematics before and after the programme.

Furthermore, as a type of mathematics education programme, this programme focuses on “mathematics”. It was designed to motivate students to engage in an activity that helps them to construct their own mathematical knowledge. Therefore, this study was also concerned with examining the impact of the programme on student performance in mathematics by answering the question:

2.    Does an intervention with a mobile app-supported math trail programme enhance students’ performance in mathematics?

The “intervention” in this study was performed by arranging a teaching and learning mathematics in which a “mobile app-supported math trail programme” was carried out. In line with the Indonesian school mathematics curriculum contents, “students’ performance in mathematics” addressed in this study includes their mathematical ability and skill in understanding and

 

 

applying mathematics in solving problems. This question led to the findings on the effect of this intervention on student performance in mathematics. It is controlled by knowing whether the performance of the students involved in this programme is better than that of students learning in regular settings.

Furthermore, the arrangement of learning needs the support of teachers in their teaching practice. Thus, the study also intervened in teachers’ mathematical beliefs, so the teachers’ strategies in their teaching practices were in line with the concept and objectives of programmes developed through this study. The impact of the intervention was evaluated in the study by answering the question:

3.    How do teachers’ mathematical beliefs differ after being involved in the project compared with their initial beliefs?

This question addresses teachers’ view of mathematics, and its teaching and learning, before, during and after involvement in the project. It led to the findings on “teachers’ mathematical beliefs” and measured changes in those beliefs. The impact of the project on teachers’ orientation – traditional or inquiry – was also associated with this question.

 

 

1.2.4.  Approach of the study This study globally followed a design research paradigm. Design research is also known as developmental research. This approach integrates research, development, implementation and dissemination and involves all participants (researchers, developers, teachers and practitioners) in dialogue in the field (Gravemeijer & Terwel, 2000). This work produces theories and prototypes that are theoretically and empirically founded and studied. Freudenthal (1968) described the principle of developmental research as follows:

experiencing the cyclic process of development and research so consciously, and reporting on it so candidly that it justifies itself, and that this experience can be transmitted to others to become like their own experience (p. 161).

Through this principle, Freudenthal suggested that the research experiences should be transferred to outsiders, such as teachers, who can then use the experiences as the basis of decisions when they implement them in their classroom to achieve their own goals according to the actual situation. This process shows that this kind of research fits into the pedagogical tradition (Gravemeijer & Terwel, 2000).

 

 

Van den Akker (1999) stated that developmental research is based on two objectives, namely the development of prototype products and the formulation of methodological suggestions for designing and evaluating prototype products. The prototypes produced in this study were the model of the programme, the math trail tasks and the mobile phone app. The aims of design research are to develop theories or models, instructional materials and an empirically grounded understanding of the work of the learning process (NCTM Research Advisory Committee, 1996).

Design research is particularly suitable in situations for which a full theoretical framework is not yet available; thus, hypotheses must still be developed, and some new teaching materials must be designed (Drijvers, 2003). To produce prototypes, it is necessary to establish criteria for their development. Nieveen (1999) offered three criteria that can be used: validity, practicality and effectiveness.

·         The product meets the requirement of validity if the components are based on state-of-the-art knowledge and consistently linked to each other (internally consistent).

·         Practicality is fulfilled when experts and practitioners claim that the prototype can be used and when the prototype can really be employed.

 

 

·         The product meets the criterion of effectiveness when the students appreciate the learning programme and the desired learning takes place (P. 127).

Therefore, the products developed in this study must meet these requirements. The evaluation of the products was conducted through multiple phases in this study.

1.2.5.  Phases of the study According to Drijvers (2003), “Design research has a cyclic character: a design research study consists of a research cycle in which thought experiments and teaching experiments alternate” (p. 20). The development that occurs during the cycles can be characterized as ranging from qualitative formative to more quantitative summative.



 

Fig. 1.1 Layers of formative evaluation (Tessmer, 1993)



 

 

Following the layers of formative evaluation proposed by Tessmer (1993) reveals the progress from small-scale to large-scale evaluation (Fig. 1.1). This method requires mixing a qualitative formative method at the beginning of the study and a quantitative summative method afterward, using a quasi-experimental approach (Bokhove, 2011). Therefore, this study took place in one preparatory cycle and three subsequent cycles (prototypical design (CYCLE I), a small-scale field experiment (CYCLE II) and a large- scale experiment (CYCLE III)). This phase is shown in Fig. 1.2.



 

Fig. 1.2 Design research cycles within this study

 

The preparatory cycle involved the literature review, which led to the theoretical framework and the design of the prototypes. The focus of activity in this cycle was to conduct an in-depth review of the literature cross-checked with field experiences and existing data. The result is the theoretical framework for the development of the project

 

 

for implementation in Indonesia. The cross-checking process was completed through an exploration step that consisted of benchmarking, needs analysis and focus group discussion (involving experts and practitioners/educators).

Based on the results of this activity, the specification procedure was defined and the technical requirements were developed, primarily as a foundation and guideline for designing prototypes. The results of this cycle are the guidelines for the technical implementation of the project in Indonesia, a guide to designing the prototypes. In addition, the research instruments based on the theory are also reviewed in this cycle to obtain research tools that are appropriate for the theory and the chosen situations.

The first intervention cycle focused on the prototypical design, such as the model of the programme including the activity rule, the design of math trails containing varied mathematical outdoor tasks and the design of the mobile phone app. The prototypes were designed based on the specifications and requirements defined in the previous cycle. In this cycle, the trails were created by the project team and also by the task contributors (teachers, educators and others), with reference to the guidelines and characteristics that were set in the previous cycle. The math trails were created by contributors through training and a workshop session. The mobile app to support the math trail programme was also created in this cycle by the app programmer of the project team. Activities took place until

 

 

the authors uploaded the tasks into the MathCityMap website portal.

In the second cycle, the prototypes were reviewed by experts (mathematics, mathematics education and mathematics educational multimedia), and a small-scale field experiment (simulation session) was carried out to determine how the prototypes would work. In this cycle, teachers and small groups of students performed math trail activities and tested the mobile app. It is necessary to see how the math trails run and how the app works. Here, teachers and a small group of students were involved. During this session, the researchers conducted observations and interviews. The prototypes were then revised based on the results of the review and simulation in this phase.

In the final cycle, we implemented the project in a large- scale class experiment and studied its effects. This cycle was carried out in several pilot studies by involving teachers and students from nine schools. The implementation consisted of an introduction session, a math trail run and a debriefing session. In this step, we explored how the math trails ran and how the application worked in all pilot studies, how the students performed and how motivated the students were to participate in the activity. We also investigated the attitudes and beliefs of teachers related to mathematics and this project. We then

 

 

analysed the data obtained in this step to answer the research questions, drawn conclusions and recommended (and criticized) some important points for the further development of the project.

1.3. THE BOOK’S STRUCTURE

This book consists of six chapters. These six chapters represent the main content and are preceded by a preface. The concept map of this study and its link to the content of each chapter of this book are illustrated in Fig. 1.3.

Chapter 1 is an introductory chapter. In this chapter, we present some background for this study, including the problems and solutions offered through this study. The topic, objectives, questions, approach and phases of this study are also described herein. At the end of this chapter, we outline the concept map of the study and the structure of the book.

Chapter 2 focuses on the theoretical background of the study. This chapter presents the results of a literature review relating to the study. The theory of constructivism in mathematics education linked with the concepts of student motivation in mathematics, students’ performance in mathematics and teachers’ mathematical beliefs is discussed in this chapter as the theoretical basis for the framework of the study. This section is followed by a discussion on some basic concepts that build on the idea of the study, such as the concept of mathematics trails and the

 

 

use of mobile technology in math trail programmes. From this theoretical background, the framework of the study was formulated.

Chapter 3 focuses on the MathCityMap project. The concept and goals of the project are formulated and the components of the project are also described. In reference to implementing the project, we also propose the technical implementation of the project in this chapter.

Chapter 4 describes the results of the development of the model of the mobile app-supported math trail programme for implementation in Indonesia. This chapter starts with a closer look at the situation in Indonesia and contains the results of designing a mobile app and math trails in the city of Semarang, as well as the rules of the activity.

Chapter 5 is a report on an exploration study of the implementation of the programme in Indonesia. This chapter describes the methodology and the procedures of the study. The impact of the programme on Indonesian secondary school students’ motivation to engage in mathematical activity, their performance in mathematics and the teachers’ mathematical beliefs are discussed in this chapter. At the end of this chapter, the results of this study are presented.

Chapter 6 is the concluding chapter. Here, we draw conclusions from the results obtained in this study and

 

 

provide    recommendations     for   future    activities    and researches.



 

Fig. 1.3 The concept map of the study

 

 



 

 

 

 

Chapter 6 Conclusions and recommendations

 

In this final chapter, we review the detailed results of this study that were presented in the previous chapters from a more global perspective. The aim of this chapter is to draw conclusions regarding the results as the answers to the research questions stated in the first chapter. Recommendations are also given in the last section of this chapter. There are two kinds of recommendations, namely: a recommendation that the programme be implemented and recommendations for further studies on the MathCityMap project.

6.1.  CONCLUSION

Mathematics plays an important role in daily life. However, the current public understanding of mathematics remains unsatisfactory. Many students do not enjoy mathematical activities and appear unmotivated to learn mathematics, while their attitudes affect their performance in mathematics. These problems have led mathematicians and mathematics educators to think of ways to popularize mathematics. A variety of projects have been developed and implemented in numerous countries to raise public

 

 

awareness of mathematics, one of which is the MathCityMap project, a project of the working group MATIS I IDMI Goethe University Frankfurt. It is a project of a math trail programme supported by the use of mobile technology.

The research on this project was carried out in multiple places and using multiple areas of study, including this PhD study. This work is a study on this project specifically in Indonesia. It was conducted by following a design research paradigm and took place in one preparatory phase and three subsequent phases (a design phase, a small-scale field experiment phase and a large-scale experiment phase). The aims of this study were focused on designing a mobile app and math trails around the city, implementing the mobile app-supported math trail programme in Indonesia and exploring its impact on students’ motivation and performance in mathematics and on teachers’ mathematical beliefs.

Because the MathCityMap project has not been theorized yet, this study was also concerned with formulating the concept of the MathCityMap project as a guideline for the implementation of this project, both in this study and other studies in the future. Therefore, in this study, the concept of the MathCityMap project was theorized. For the implementation of the project in Indonesia, the model of the mobile app-supported math trail programme was

 

 

developed; it was followed by designing the rules of the activity, the mobile app and the math trails around the city of Semarang. Then, the programme was implemented and studied empirically in nine secondary schools in the city of Semarang.

This study was conducted within a theoretical framework informed by constructivist theory in mathematics education. This view suggested that, in the learning mathematics process, students should actively construct their own mathematical knowledge by connecting new information they have gained with their previous knowledge. However, learners’ engagement in an activity depends on their motivation. Sociocultural views state that learners need guidance as a sort of push for students to get started in their engagement in the learning process, and classroom social interaction also motivates them to engage through peer pressure.

The external factors that encourage learners to undertake the construction of knowledge should, however, be reduced and eliminated because, based on the cognitive view, students should value learning for its own sake. The learning process should prepare students to become lifelong learners that gain the confidence to construct more complex knowledge and who are better equipped to make use of their mathematical knowledge and skills throughout life. Both views affect the strategies that are used to improve student motivation: the sociocultural view leads to

 

 

extrinsic motivation, while the cognitive view leads to intrinsic motivation.

The intrinsic-extrinsic motivation axis is explained in self- determination theory (SDT). This model conceptualizes a range of regulations, from intrinsic motivation to amotivation. Between these extremes lie identified regulation and external regulation. It has been hypothesized that self-determination is associated with improved psychological performance; therefore, it would be expected that intrinsic motivation, followed by identified regulation, is mostly associated with positive outcomes. Thus, the process of learning mathematics needs to be prepared and arranged so that students can be intrinsically motivated to engage in learning mathematics.

Nonetheless, the improvement of students’ motivation is only a stimulus for students to promote their engagement in mathematics. As a serious mathematics educational programme, the main focus of this activity is to support students in learning mathematics and constructing their own mathematical knowledge. By engaging in this programme, the students got the opportunity to learn and practise solving real problems by following the stages of mathematizing. These stages are: understanding a problem situated in reality; organizing real-world problems according to mathematical concepts and identifying relevant mathematics; transforming real-world problems

 

 

into a mathematical problem that represents the problem situation; solving mathematical problems; and interpreting mathematical solutions in terms of the real situation. In the studied project, students learned and experienced using what they knew in the process of solving problems related to mathematics.

This activity helped students to construct their own mathematical knowledge by solving the mathematical tasks on the planned math trail and interacting with the environment, including the digital environment. The digital tool is used as a bridge to help teachers support students in the mathematics learning process. It is hypothesized that the intervention with this programme enhances students’ performance in mathematics. Through this activity, students were expected to increase their understanding of mathematical concepts and their relationships with real life and improve their ability to apply mathematics in solving problems.

To reach this goal, the teachers also play an important role. Learners’ confidence and performance in mathematics are significantly associated with the teachers’ mathematical beliefs and teaching practices. Belief in the value of intrinsic motivational strategies is expected to be associated with inquiry-oriented teachers’ mathematical beliefs. In their teaching and learning practices, teachers should act as facilitators and not as transmitters of information to their students. Teachers should prepare

 

 

problems that will provoke the expected learning and offer guidance for students to learn new knowledge. Students accept the problem and then speak, think and act about the problem through their own motivation. This teacher- student interaction is important in helping learners to solve problems and develop their abilities to allow them to solve other or future problems by themselves. In this environment, the interaction was facilitated by the use of mobile technology.

Furthermore, the learning environment also needs to be well organized. The environment should offer opportunities for learners to adapt and reconfigure new information to construct new knowledge. Mathematical knowledge can be constructed through the action of organizing the subject matter, both matter from reality and mathematical matter (mathematization). Students are reinventing the teaching matter prepared by teachers in such interactions and under guidance (guided reinvention).

Taking mathematics outdoors allows learners to experience these processes in an authentic situation. In this situation, students can see the connection between mathematics and their real world. It also creates greater motivation and excitement for students in the learning process. Teachers can also design a mathematical task outside the classroom, because the environment provides many ideas for the subject matter of mathematics.

 

 

Tasks about an object in the environment are designed and grouped with other tasks in one location, then linked into a route. Furthermore, students undertake the exploration of mathematics in the environment around them by conducting mathematical outdoor activities: tracking the math trail and completing these planned mathematical tasks.

By taking advantage of these benefits of mobile technology, math trail tasks can be localized with GPS coordinates and pinned onto a digital map through a web portal. The trail walkers can then access the tasks and run the math trail activity with the help of a GPS-enabled mobile application. The app helps teachers to facilitate the process of learning mathematics, from preparing the problem to directing students to find the task locations and supporting them in solving mathematical problems. Mobile devices can also be used to integrate learning activities with authentic situations in outdoor environments. The math trail increases student engagement in learning mathematics activities. The combination of the math trail and the use of mobile technology has become a basic idea of the MathCityMap project.

The MathCityMap project The MathCityMap project is a project that involves the math trail programme, which is supported by the use of mobile technology. This project was not conceived merely to design and/or use the math trails; instead, the project

 

 

includes the entire process: preparation (how to design the programme), implementation (how the programme runs) and evaluation (how the programme impacts on students’ motivation, students’ performance in mathematics and teachers’ mathematical beliefs). A mobile phone app, as a supporting tool, was also designed, created and used during this project.

As the name implies, the MathCityMap project is composed of several components. These components are: mathematical outdoor tasks, mathematical city trips, a map-based mobile app and the MathCityMap community. This project is not only related to the implementation of the activity, but also addresses how to prepare the project and its instruments and how to evaluate the project. Consequently, the technical implementation of the project is made up of three main parts – the preparation, the activity and the evaluation.

The mobile app is designed by the team. Tasks are prepared by educators or teachers in a team or individually. Then, the tasks and app are evaluated and validated to ensure that they are feasible for use. The mobile app can be downloaded and installed by the public. The tasks can be used in private or public settings by following the rules that have been prepared. Technically, with the help of the mobile app, students/users follow a planned route, discover the task locations and answer the tasks related to what they

 

 

encounter along the path; further, they continue with the subsequent tasks.

Then, the implementation of the math trail and app is evaluated for further development. The process and results of the activities are also published so that others can take the benefits of these results and be inspired to develop it in other places. However, it is important to note that the implementation of the educational programme in a particular place requires adjustment to local conditions and needs.

The mobile app-supported math trail programme in Indonesia By following the theory, the specific procedures and the technical requirements of the MathCityMap project, a mobile app-supported math trail programme was designed to be implemented in Indonesia. The designing phase started with discussions between experts and practitioners, followed by teacher training, and then the process of designing and testing the tasks and mobile phone app was performed by the team. In this phase, the GPS-enabled mobile application was designed specifically for implementation in Indonesia in relation to the local situations, conditions and needs. It is a native app that displays the math trail routes/map, including the coordinates of the math trail tasks, the user’s current position, the authentic mathematical problems, the information about the tool needed in solving problems, the

 

 

hints on demand (if needed by the users) and the feedback on the answers entered. The app also gives brief information related to the object, such as its history, its function, etc. The app was created using the MIT App Inventor program and runs on Android platform mobile phones. It can be installed by downloading the installation file available at www.mathcitymap.eu or Google PlayStore.

In addition to making the app, in this phase, 87 tasks, grouped into 13 math trail routes scattered among various areas in the city of Semarang, were also designed by educators (including teachers through training and a workshop). There were six to eight tasks for each trail with a variety of topics located in a safe and comfortable environment. The trails were designed in several school areas, city parks, historical buildings, markets, tourist attractions and other locations. The math trails and the tasks were then uploaded into the portal so that users could access them through the mobile app. Trail walkers needed about two hours to explore a trail (about 1–2 km length) and to solve the problems encountered along the path. The math trails’ and mobile app’s designs passed the evaluation process, through validation by experts, and the simulation phase involved a small group of students and teachers that were considered eligible for use in the next phase.

 

 

In the mobile app-supported math trail programme, the activities can be carried out with students from different grades and the tasks can vary in terms of level and difficulty. The activities can be conducted in the area of the school, in the school’s neighbourhood or at several locations around the city. The duration of the activity is about two to three hours during school hours (morning time is recommended), after school hours or during holidays. Students in a class are divided into groups of five to six members. The activities are conducted during normal school hours for about two 45-minute periods, beginning with the teachers giving a brief explanation of the rules and goals. Groups then start their mathematical trip, each from a task location that is different from that of the others. As the groups trek the trail, teachers observe and supervise students’ activities but are not expected to assist them, because all the necessary information is available in the app. With the help of the app, students in groups follow a planned route, discover task locations, solve the problem on-site and then move on to subsequent tasks. A debriefing session is also needed to get feedback for evaluating the activity and preparing further activities.

Implementation and empirical study of the programme in Indonesia The large-scale field experimental phase was conducted in 2015 and 2016 and involved 520 students and nine teachers from nine secondary schools. For each school, there were

 

 

two classes (about 30 students/class) involved in this study: one class was the experimental group and one class was the control group. Both groups at each school were taught by the same teacher with the same topic and subject matter but with different interventions. In the experimental group, teaching and learning mathematics was carried out by running the mobile app-supported math trail programme. Meanwhile, in the control group, students learned mathematics in the regular setting. A debriefing session was conducted after the activities. Questionnaires and evaluations regarding students’ motivation (the self- reported Situational Motivation Scale (SIMS)), teachers’ mathematical beliefs and individual mathematics performances were also given before and after the activities.

According to the situation and condition, the activities were conducted in different settings. In general, the activities were held during school hours, but some schools’ activities were also organized on a Sunday (holiday) or after school hours. On average, each activity required two to three hours to complete. Four schools carried out activities outside the school (including in some city parks and near landmarks in the city of Semarang), while five schools conducted activities in the school area. These choices were made based on the conditions and situations (such as safety and the availability of teaching and learning time) that were unique for each school.

 

 

The results of the field empirical study revealed that, in general, the activities ran smoothly, the mobile app worked well, and the rules and goals of the programme were comprehensible. Nearly the entire group attempted to work in their own way (without using hints). Most students reported no problems in participating in these activities, including the use of the app. All groups discovered the locations of the tasks, and a total of 73 tasks were completed by all groups in this activity; 14 tasks were not answered or were completed incorrectly by some groups.

The answer to the first research question: the students’ motivation to engage in mathematics The findings indicate that it was easy to involve the students in the activities. Most of the students were actively engaged in the activities, expressed positive feelings and had no problems in carrying out the activities. Students’ engagement was associated with their motivation. The SIMS results show that the students’ motivations to engage in the activity were diverse; however, all had positive Self- Determined Index (SDI) scores (ranging from 2.00 to 16.25 and M ± SD = 7.3180 ± 2.92629), which indicates that their motivations were more often the self-determined or internalized forms of motivation, namely intrinsic motivation (IM) and identified regulation (IR), than external regulation (ER) and amotivation (AM). Students perceived the activity to be interesting or enjoyable (an indicator of IM) and meaningful or valuable (an indicator

 

 

of IR). They were engaged in the activities for their own sake/pleasure/satisfaction, and their engagement was considered to be a self-choice.

Open-ended follow-up questions revealed that students enjoyed the outdoor mathematical activities (30%) and were interested in the use of advanced mobile technology to learn mathematics (23%). They also reported that learning and applying mathematics in the real world were valuable and meaningful for them (18%). Other students mentioned different reasons or did not give a reason, and slightly negative feelings were also mentioned. The SIMS results also show that there was a significant difference in the reported orientation of the students’ motivation before and after they were involved in this programme. There was a change in the orientation from being extrinsically oriented toward being more intrinsically oriented. We also found that there has not been a decrease in the intrinsic motivation of students from the first year to the second year of implementation.

Generally, the project was successful in designing and offering a programme that can promote students’ intrinsic motivation in mathematics; however, this improvement of students’ motivation is only as a stimulus for students to promote their own engagement in mathematics. Their involvement in the learning process is expected to positively affect their performance in mathematics.

 

 

The answer to the second research question: the students’ performance in mathematics As a serious mathematics educational programme, the main focus of this activity is to help students in learning mathematics and constructing their own mathematical knowledge. We found that, by engaging in this programme, the students got opportunities to learn, experience and practise solving real problems by following the stages of mathematizing. These stages are: understanding a problem situated in reality; organizing real-world problems according to mathematical concepts and identifying relevant mathematics; transforming real-world problems into a mathematical problem that represents the problem situation; solving mathematical problems; and interpreting mathematical solutions in terms of the real situation.

Through this programme, students learned and experienced using what they knew to solve problems related to mathematics. The students practised extracting information from the problems they faced and using the mathematical concepts that they knew to solve those problems. They experienced mathematics in the environment and constructed their mathematical knowledge by applying mathematics in solving the problems encountered along the path.

To determine the effect of the intervention on students’ performance in mathematics, a statistical comparison of the students’ scores between the two groups was carried out.

 

 

The results show that the mean gain in performance score of the 272 students in the experimental group (16.2684) was significantly higher (Cohen’s d = 0.31) than that of the 248 students in the control group (4.2540); in other words, the programme enhanced students’ performance in mathematics, especially their ability and skills to understand and apply mathematics to solve problems. Besides the learning arrangement, the teacher was also an important factor in these outcomes.

The answer to the third research question: the teachers’ mathematical beliefs Teachers play an important role in organizing and running the strategy of the learning process. Teaching and learning processes are linked with teachers’ attitudes; as such, this study was also concerned with the influence of the programme on the teachers’ mathematical beliefs in relation to their strategies for fostering students’ motivation to engage in mathematics. The influence was given by holding focus group discussions, teacher training and a workshop for designing tasks, and involving teachers in the processes of observing and evaluating the activity.

The findings indicated that the project changed the teachers’ mathematical beliefs in relation to their strategies for fostering students’ motivation in mathematics. The comparison test shows that there was a change in the teachers’ mathematical beliefs from traditionally oriented

 

 

beliefs toward more inquiry-oriented beliefs, i.e. after being involved in this programme, the teachers were more likely to believe that: maths is a tool for thought; the students’ goal is to understand; students should have some autonomy; maths ability is prone to change; and students will want to engage in maths if the tasks are interesting and challenging.

The regression analysis results revealed that there was a significant effect of teachers’ mathematical beliefs on students’ motivation (y = 2,239 + 0,492x) with the contribution at a rate of 77.7%. The more teachers implement inquiry-oriented mathematical beliefs in their teaching strategies and practice, the more their students have intrinsic motivation to learn mathematics. This shows that although this project aimed to improve students’ motivation by designing a learning environment and a supporting tool, it also affected teachers’ mathematical beliefs in relation to their teaching strategies and practices for fostering student motivation in mathematics.

Remarks from the answers to the three research questions Finally, the project was successful in designing and offering a programme that can intrinsically motivate students to engage in mathematics. The design and arrangement of the math trail and the mobile app, as well as a combination of both, created a pleasant situation and attractive environment that offered valuable knowledge and experience in mathematics. They embody the aspects

 

 

of enjoying or being interested in the activity and the use of advanced mobile technology for learning mathematics (an indicator of intrinsic motivation), and of the value and meaningfulness of the mathematical tasks and the activity (an indicator of identified regulation), which were generated through the implementation of this project.

The students were interested in the use of advanced mobile technology for learning mathematics and enjoyed the outdoor mathematical activity. The students also believed that the mathematical tasks and the activity were valuable and meaningful for them. The intervention with the mobile app-supported math trail programme also enhanced students’ performance in mathematics, especially their ability to understand and apply mathematics to solve problems. The change in teachers’ mathematical beliefs, from being traditionally oriented to being more inquiry- oriented, which occurred as an effect of the project implementation, also contributed to this improvement in students’ intrinsic motivation to engage in mathematics.

Some criticism of the MathCityMap project However, some feedback on the implementation of the project needs to be considered, such as: the need for the optimization of mobile phone use to motivate students; aspects of the learning mathematics outcomes; and the need for a system that helps the teachers to prepare and run the activities. Firstly, we will criticize the students’

 

 

motivation to engage in this activity. Although the SDI scores of students affected by the use of the mobile phone are the highest, the data show that the majority of students were interested in these activities because they were sited outdoors. This is important because one of the goals of this project was to develop the concept of a math trail by taking advantage of mobile technology. Therefore, the potential of mobile technology to support outdoor mathematics learning needs to be further exploited.

Currently, the use of the mobile phone is directed toward supporting the improvement of students’ intrinsic motivation. However, based on the sociocultural viewpoint, learners need encouragement to initiate and engage in a learning process, despite the fact that the encouragement of external factors needs to be reduced gradually so that in the end students must learn for its own sake, as suggested by the cognitive view.

Here, the use of a mobile phone can play a role. The possibilities of this role, for example, include the use of these tools for outdoor games or giving scores in recognition of what has been achieved by learners. These are factors that extrinsically motivate students to engage in an activity. These rewards, however, may differ from the basic concept of the math trail, but these findings provide benefits for the development of the math trail and its innovations by taking advantage of the use of mobile technology, so that the programmes developed through this

 

 

project can be optimized to increase students’ motivation to engage in mathematics.

Secondly, it is about mathematical skills. The results of the study show that once the students were involved in the activities, they had the opportunity to learn mathematics, construct mathematical knowledge and improve their performance in mathematics. The topic and subject matter used in designing tasks for this programme were adapted to the learning objectives and standards of competence expected by the school mathematics curriculum.

However, the expected learning mathematics outcomes, or more specifically the aspects that are examined in the final maths examination, are not always related to real problem- solving abilities and do not always require the problem- solving skills that are trained through the math trail activity. Even some (multiple-choice) test items can be solved by using “the quick solution”.

Therefore, this programme cannot be used as a single strategy in the process of teaching and learning because the students need to be equipped with mathematical skills in other aspects. The programme is also not the best strategy when used alone, but it can become one of several strategies used to complement each other. This programme can take a role in improving the mathematical literacy of students.

 

 

Although based on research results, teachers involved in this project experienced a change in their beliefs from traditional orientations toward being more inquiry- oriented, but they are also required to prepare students for the examination and teach the subject matter in accordance with the requirements in the curriculum, i.e. there was a conflict between their beliefs and their needs. Therefore, time efficiency and the effectiveness of the strategy in achieving the objectives need to be managed carefully. Although this programme provides many advantages, organizing the students to do outdoor activities requires extra time and attention.

Before implementing the programme, the teacher or educator also takes time for preparation, such as designing tasks and uploading them into the web portal system. In designing math trail tasks, although the environments present many ideas for mathematical tasks, safety, comfort and public order factors are also constraints. Discussion, evaluation and validation of the tasks are also required to produce good math trail tasks in accordance with the concept. These require the development of a system that helps teachers to prepare math trail tasks and activities for their students.

However, overall, it can be concluded that the project was successful in designing and offering an engaging mathematics learning environment. Students easily engaged in the activities and these activities positively

 

 

affected students’ attitudes and motivations toward learning mathematics. Students were intrinsically highly motivated to engage in these interesting and fun mathematics activities. Their involvement encouraged them to experience the process of mathematization and construct their own mathematical knowledge. As a result, their performance in mathematics improved. Teachers have a role in achieving these outcomes. Afterward, the mathematical beliefs of the teachers involved in this project became more inquiry-oriented. This supports efforts to increase the students’ intrinsic motivation in mathematics.

6.2.  RECOMMENDATIONS

Based on the conclusions of this study, some recommendations are given at the end of this report on the study for two reasons. Firstly, the aim of this part was to concentrate the findings into concrete suggestions for implementing the programme and utilizing the results of this study. Secondly, the recommendations were aimed at highlighting the issues that were not addressed in this study and require further research and development.

In this study, it was reported that math trails, supported by the use of a mobile phone app, have many potential advantages. Schools can take advantage of the results of this project by integrating the mobile app-supported math trail programme into the teaching and learning processes. The public can also participate in this programme for

 

 

leisure by walking around the city while learning mathematics. The public can freely use the mobile app and run the math trail activities that have been designed, use the existing math trails that have been designed and tested in this study, and design new math trails and upload them into the portal system by following the specification procedures and technical requirements formulated in this study.

Implementation of this programme can be strengthened through cooperation with several parties, such as city governments, schools and education authorities. Cooperation can also be established with the tourism department, museum managers, travel managers and others. With this cooperation, this educational programme can be synchronized with the programme of the parties, e.g. mathematics city tours. Financial support can also be obtained through sponsorship cooperation with several parties, such as a mobile phone service provider or mobile device manufacturers and sellers.

However, based on the results of this study, some conditions and requirements must be considered. The important role of influencing student motivation and performance in mathematics must be designated when designing the app, tasks and activity. The relevance and value of the task must be clearly identified and linked to the concept and objectives of the project. Most importantly, the students must enjoy and be attracted to the programme,

 

 

both in terms of completing the mathematical tasks and doing activities outdoors, and in using the mobile app.

As a pilot study, the research was conducted in a limited manner in one place. Although this pilot study provided information about the overall project implementation and provided suggestions for models of the programme, further development needs to be carried out, both in the development of the concept of the programme and its supporting tools. Further research needs to be conducted to inform the development of this project in other places/cities with different conditions, which will enrich the information available and improve the model of the programme. It will also be possible to dig deeper and explore more broadly the implementation of the programme and its impact on students’ motivation and their performance in mathematics, teachers’ mathematical beliefs, and other aspects and areas of study.

 

both in terms of completing the mathematical tasks and doing activities outdoors, and in using the mobile app.

As a pilot study, the research was conducted in a limited manner in one place. Although this pilot study provided information about the overall project implementation and provided suggestions for models of the programme, further development needs to be carried out, both in the development of the concept of the programme and its supporting tools. Further research needs to be conducted to inform the development of this project in other places/cities with different conditions, which will enrich the information available and improve the model of the programme. It will also be possible to dig deeper and explore more broadly the implementation of the programme and its impact on students’ motivation and their performance in mathematics, teachers’ mathematical beliefs, and other aspects and areas of study.