Developing A Learning Environment on Realistic Mathematics Education for Indonesian Student Teachers

By: ZulkardiTwente University, The Netherlands2002

SUMMARY

The CASCADE-IMEI study was started to explore the role of a learning environment (LE) in assisting mathematics student teachers learning Realistic Mathematics Education (RME) as a new instructional approach in mathematics education in Indonesia. The LE for this study has been developed and evaluated using a development research approach.

The main problems facing mathematics education in Indonesia – especially in the arena of secondary schools – are the low achievement of pupils in mathematics and their poor attitude toward mathematics. Research cites various potential causes, including inaccurate learning materials, inadequate mechanistic teaching methods, and poor forms of assessment. The Indonesian government, through the PGSM project intended to improve the quality of mathematics education in Indonesia by introducing realistic mathematics education (RME).

The CASCADE-IMEI study was initiated in order to support that initiative. Contrary to the teaching approach that is commonly employed in Indonesia, RME uses contextual problems or applications as a source and a starting point for mathematics teaching. However, introducing RME as a new instructional approach is a complex task. As described in Chapter 1, there are at least three issues that should be considered. First, RME curriculum materials are not easily designed or adopted by teachers because RME materials focus on middle-level and high-level order thinking instead of the low-level thinking only. Second, teachers need training in how to use RME materials in their classroom. Their new role is more that of a guide than of an instructor. Finally, the implementation of RME is not a short-term program or project, but it needs many years to be institutionalized.

This study used several strategies for introducing RME in Indonesia that try to effectively tackle these challenges. First, by choosing teacher education as a research context and prospective teachers as target users, RME is introduced not only to the student teachers, but also to the pupils in the schools through the teaching practice that is conducted by these student teachers. Second, by using the Internet in combination with classroom-based activities, the student teachers are supported in terms of resources and tools for communication and collaboration about the complexities they encounter while they are learning and teaching RME in the schools. Finally, the computer is viewed as a tool that may support teachers and curriculum developers dealing with curriculum development (exemplary lesson materials) and implementation activities.

The aim of this study was to explore the role of a learning environment in assisting Indonesian mathematics student teachers in UPI Bandung learning and teaching RME as a new instructional approach. At the beginning of the study and throughout the development of the LE, guidance was sought from literature relating to RME, curriculum development and implementation, teacher learning in teacher education, exemplary lesson materials, and web-based performance support. Insight from these sources helped to shape the structure of the study as well as the LE itself.

1

The CASCADE-IMEI study used a development research approach aiming at exploring the potential impact of a valid and practical learning environment on Indonesian student teachers' learning of the RME approach. In so doing, the study was guided by the following main research question:What role can the CASCADE-IMEI learning environment play in assisting Indonesian mathematics student teachers with their learning of the RME approach?

Using the development research approach, this study deals with designing and evaluating a learning environment as an intervention. In this study, the intervention is the learning environment, composed of three main components: 1) Internet support, 2) a course and 3) exemplary lesson materials. Also, the study deals with the extracting of design principles of the learning environment through the development and evaluation processes.

The LE has been developed and evaluated in three main phases: 1) preliminary investigation stage, 2) prototyping stage and 3) assessment stage. Throughout the first two phases, five prototypes of the LE were designed and formatively evaluated in both the Netherlands and Indonesia. At the end, the final version of the LE was field-tested in Indonesia.

The development activities of the LE prototypes were conducted at the University of Twente, the Netherlands, while the evaluation activities were conducted either at the University of Twente or at the University of Indonesian Education (UPI) Bandung, Indonesia. In the Netherlands, the evaluation activities were centered on expert appraisals, in which six experts from various disciplines (RME, computer support curriculum development, web design, course development and teacher learning in teacher education) were mainly involved. In Indonesia, the evaluation activities were focused on cooperative evaluations and try-outs. A total of 34 mathematics student teachers (pre-service and in-service) at the department of mathematics education in UPI Bandung were mainly involved. In addition, 6 mathematics teacher educators from UPI Bandung and about 800 pupils from 12 junior secondary schools in Bandung, were peripherally involved. While the teacher educators were involved either as supervisors of student teachers or as observers, the pupils were involved as learners when student teachers conducted their teaching practice.

During the assessment stage, the final version of the LE was field-tested in Indonesia in order to find out about its potential impact in assisting the professional development of student teachers learning and teaching RME as a new approach in mathematics education. In order to study the potential effects of the LE, the fiveeffectiveness levels of a professional development program were used (Guskey, 2000). The fifth level was not directly addressed, however, since the main target users of the study consisted of student teachers in teacher education. At this level, the researcher only explored indications of the potential effects of the LE on pupils. As a result, four sub-research questions were posed:1. What is the perception of student teachers with regard to the LE?2. What knowledge, skills and attitudes did the student teachers acquire as a resultof following the LE?3. What effects did the CASCADE-IMEI LE have on the organization at thedepartment of mathematics education in UPI Bandung?4. To what extent do student teachers apply their RME knowledge and skills in theclassroom?

The results of the formative evaluation and field test in Indonesia are described inChapters 3 and 4, respectively. Ultimately, the study resulted in a final version of theLE.

2

Design PrinciplesVan den Akker (1999, 2002) suggests that knowledge gained from development research can be presented in the form of 'design principles', which are usually heuristic statements of a format such as:

'If you want to design intervention X [for the purpose/function Y in context Z], then you are best advised to give that intervention the characteristics C1, C2,...,Cn [substantive emphasis], and to do that via procedures P1, P2,...,Pn [procedural emphasis], because of (theoretical/empirical) arguments A1, A2,...,An.

Relating to the CASCADE-IMEI study, an overarching design principle can be represented as follows.If you want to design a valid, practical, and effective learning environment for supporting Indonesian mathematics student teachers with a new instructional approach (like RME) then you are best advised to give the learning environment the following characteristics:

- use exemplary lesson materials as the main component of content (consisting of learner materials, assessment materials, and teacher guides) with clear guidelines on how to use the materials;

- use a face-to-face course (which is integrated in one of the method courses in the teacher education) with several activities related to learning the new theory, materials, teaching method and assessment strategy based on the new approach (e.g. RME); and

- use Internet-based support for the course participants that provides additional information, training facilities, advice and communication tools for collaborationand reflection;

and to do the above via the following procedures:- adapt and develop the lesson materials from the available (RME) lesson

materialsthat were derived from both theory and practice into the context of Indonesian teacher education and Indonesian schools;

- design and develop the learning environment (course and web support) by initiating subsequent prototyping cycles until a satisfying LE has been reached; and

- improve the quality of the content, support and structure of the learning environment by conducting formative evaluation activities (expert appraisal, cooperative evaluation, and try-out) with the involvement of experts and target users and make revisions at the end of each prototyping cycle; and

- assess the potential effects of the LE by conducting a summative evaluation via a field test in real practice;

because of the following arguments:- by using lesson materials, student teachers can decide to use the materials as the

learning resources during their course of study, as examples when they learn how to redesign lesson materials, and as additional materials when they conduct teaching practice;

- by following the course, student teachers can overcome the obstacles of being novice teachers and they may learn not only the content and pedagogical knowledge of the new approach, but also how to redesign materials and teach with their

3

materials using the new approach, either in a peer teaching situation orwith the pupils in the schools;

- by using Internet support, student teachers (and other Internet users as well) can learn the new approach (e.g. RME) whenever and wherever they want, either during the course (short-term) or afterwards (in the long term);

- by using a prototyping approach with several cycles, the intervention can evolve towards a valid and practical final version;

- by conducting formative evaluation activities with experts and target users, the validity and practicality of the intervention can be improved and the content can be enriched; and

- by conducting a summative evaluation via a try-out, the potential effects of the intervention can be empirically assessed. All characteristics of the LE, the procedures and the arguments in this design principle will be elaborated in the next sections.

Reflections on Main Characteristics of the LearningEnvironment on RME

In this study, the intervention is a learning environment for student teachers who are learning RME as a new instructional theory in mathematics education. The LE consists of both an Internet-based and a classroom-based component. The main content of these two components consists of RME exemplary lesson materials. This section summarizes the main characteristics of the web site, the course and the exemplary lesson materials, and provides several reflective notes.

RME web supportThe use of Internet pages on the World Wide Web ('web site') to provide support in teacher education appeared to be a promising tool for an expansive country made up of many islands, such as Indonesia. As a result, a web site has been designed and evaluated as a part of the learning environment (LE). This section discusses the main characteristics of this web site with its three main components: content, support and user interface.

The content of the web site refers to the philosophy and characteristics of RME. During the development of the web site, the validity and practicality of the content evolved greatly. Based on synthesizing the theory in the literature study, as well as feedback collected from the evaluations, the following aspects of RME are emphasized in the web site:

- the background information of RME, as well as its characteristics, in order to give users a general view of what mathematics is, how pupils learn mathematics, and how mathematics should be taught from the RME perspective;

- flexible ready-made exemplary learner and teacher materials in various topics, in the sense that the student teachers can adapt the materials to their needs, or use them directly in the classroom;

- a number of computer simulation and game programs in which the users can visualize and simulate mathematical phenomena (e.g. function graphics, geometry concepts) by changing the variables or manipulating the figure by using the mouse;

- examples of students' products or solutions to the lesson materials, so the student teachers or other users can learn from, and gain insight into the learning process of pupils in solving mathematics problems;

4

- guidelines for developing lesson materials, for teaching in an RME classroom and for assessing pupils; and

- examples of assessment problems at all school levels, so the users can learn from them, or use the problems in the classroom.

In the web site, various types of support are provided. Some of them are inherent to the medium of the Internet, such as tools for communication (e-mail facilities and the mailing list) and links to other resources about RME and mathematics education in general. Others were specifically designed and developed by the developer, including:

- a tool for designing lesson plans or templates;- a number of Java applet simulation programs on several mathematics topics, and a

number of mathematical games;- a tutor component; and- a number of video clips showing examples of how to teach using the

realisticapproach.

The user interface of the web site was designed in order to accommodate users so they can easily navigate and access information. The main characteristics of the interface are listed below:

- the menu and submenus are consistently located in the same area on the screen throughout the web site;

- buttons and text, which use two languages (English and Indonesian) are easy to read;- graphics, text and background colors are clear and simple;- some screens of simulation and game programs are interactive, which means that

users can simulate the phenomena by changing the variables;- some documents and information are provided in Word document files to makeit

easy to retrieve or print them from the web site.

The programming language that was used was mainly HyperText Markup Language (HTML). In addition, some mathematical parts were programmed using Java applet and Java script programming languages.

Based on the results of the prototyping and assessment stages, the web site was evaluated both by experts (on web design, computer support and RME) and by target users as being valid and practical.

RME courseIn this study, teacher education in Indonesia is seen as one of the places to start the innovation. Through teaching practice, the innovation may also reach the pupils in the schools.

The content of the course and the support provided during the course also refer to the philosophy and characteristics of RME. Based on synthesizing the theory as well as feedback from several rounds of evaluation, the following aspects of RME (contentand support) are emphasized in the course.

5

- Theoretical background. This aspect focuses on the background information of RME as well as its five characteristics. For instance, the role of context in learning and teaching mathematics and the way the context can be used in the lesson materials.

- Doing mathematics. This aspect focuses on the learning process with an RME approach. Here, student teachers assume the role of pupils. They are not only learning mathematics (content knowledge), but also the way pupils in the schools come to understand mathematics (pedagogical knowledge).

- Support in lesson redesign. This aspect refers to the process of applying the characteristics of RME when (re)designing RME lesson materials. Ideally, all of the RME characteristics are used. As this is not an easy task, student teachers are supported in learning how to adapt the available materials for other relevant contexts and how to use the materials in their teaching practice. As a result, they mainly focus on two tenets of RME: the use of context and the intertwining among strands of mathematics.

- Teaching method. This aspect refers to the use of interactivity (one of the RME tenets) in the teaching process in the classroom. Several critical moments of teaching are presented and discussed. These critical moments consist of interaction, individual work, group work, classroom discussion, student presentation and teacher presentation. Moreover, an RME teaching video is shown and discussed. Finally, student teachers are invited to apply their knowledge and skills in their own teaching practice in the school.

- Assessment strategy. In RME, the assessment should get attention not only at the endof the instruction (summative evaluation), but also during the instructional process (formative evaluation). While the former focuses on the pupils' achievements in the form of scores, the latter focuses on the improvement of pupils' learning. Inthis course, assessment problems are included in the exemplary lesson materials. During the 'doing mathematics session', for instance, a sequence of assessment problems is given to the student teachers. These problems are designed in such a way that they may guide and enhance the student teachers in presenting a mathematics concept. In RME, assessment problems usually cover all three levels of students' thinking: low, middle, and high-level thinking. Therefore, in some cases, students will not directly get the formal solutions; instead, they have to discuss their informal solutions in pairs or in a small group.

The course was organized based on the main activities such as doing mathematics, redesigning lesson materials, teaching practice, and reflection. In the final version of the course, the meetings required six weeks (each meeting about two-hour or 100 minutes courses) and two weeks for teaching practice in the school. Results from the prototyping and assessment stages show that the RME course was perceived to be valid, practical and having several potential effects.

RME exemplary lesson materialsExemplary lesson materials play an important role in this study. The materials include learner materials, assessment materials and a teacher guide. The teacher guide contains procedural specifications, very clear guidelines and specific directions in using the materials and focuses on the essential parts of the innovation – in this case the RME approach. It provides recommendations on how to deal with RME lessons (start, manage and close) and

6

the assessment in RME. Five mathematics topics have been developed and adapted to the Indonesian context: linear equation system, symmetry, matrices, side seeing and statistics. These materials were used in the 'doing mathematics' session and used as a guide for student teachers in developing or adapting their own materials. The materials were provided and used in the course, in the schools and on the web site. Findings from the expert appraisals and try-outs during the prototyping and assessment stages show that the RME exemplary lesson materials were perceived to be valid, practical and having some potential effects.

ResultsIt can be concluded that the LE developed in this study could play a role in assisting Indonesian mathematics student teachers in UPI Bandung in learning and teaching RME as a new approach in mathematics education in Indonesia. The roles of the LE are outlined below.

1. The LE could enhance the student teachers' satisfaction with the content (RME), support, interface and the organization of the LE. This seems to be mainly due to the consistency of the LE and the fact that the LE matches with their needs.

2. The LE could assist the student teachers in learning the mathematical, didacticand practical part of the RME course. As a result, the LE could promote the student teachers' understanding about RME. The LE could also support student teachers in learning how to redesign lesson materials for the classroom level. In addition, student teachers also use the Internet in learning RME. This may stimulate learning how to use e-mail facilities, for instance for reflecting on their experiences. This way of teachers' learning could enhance their understanding of RME and change their beliefs toward mathematics as well as towards their jobs.

3. The LE could bring about changes in the organization and staff of the department of mathematics education as well as on the practicing mathematics teachers at the various secondary schools where student teachers were active. For instance, the LE could promote change in that organization with respect tothe activities of teacher educators and the process of Seminar and final project courses.

4. The LE could have positive impact on developing teaching performance of the student teachers. Because of the LE, they were able to perform as real teachers in the school classroom using the RME approach.

5. The results indicate that the LE could have impact in changing the pupils' beliefs, or in increasing the positive attitude of pupils in the secondary schools toward mathematics.

7

Applying Realistic Mathematics Education (RME) in Teaching Geometry in Indonesian Primary SchoolsBy: Ahmad FauzanTwente University, The Netherlands, 2002

SUMMARYSimilar to other countries (see for example Niss, 1996; NCTM, 2000), the mathematics curriculum for primary schools in Indonesia pays much attention to several important aspects such as developing pupils' reasoning, activity, creativityand attitude, and providing pupils with mathematics skills so that they can handlereal-world problems mathematically. These aspects are crystallised in the goals ofthe mathematics curriculum for Indonesian primary schools as follows:

1. Preparing the pupils to be able to deal with the dynamic world situation effectively and efficiently through practical works based on logical reasoning, rational and critical thinking, caution and honesty.

2. Preparing pupils to be able to use mathematics and mathematical reasoning in their everyday life and in studying other sciences.

Despite its lofty goals, the curriculum appears to have fallen short of its aims, giving rise to the following questions: Why is the quality of mathematics education in Indonesian primary schools still poor? Why do most pupils hate to learn mathematics? (see Marpaung, 1995, 2001), and Why pupils' achievements in mathematics are poor from year to year? (see www.depdiknas.co.id).

These questions indicate that there are some problems in mathematics education in Indonesia, especially regarding the curriculum and the learning and teaching processin primary school.

In the last three decades, the curriculum in Indonesia has been changed four times (Curriculum 1975, 1984, 1994 and 2002). Each curriculum was based on a different approach and each one was presented as an ideal curriculum (see Goodlad, 1984). However the changes from one curriculum to another did not result in a significant improvement for several reasons. Firstly, the changes of the curriculum always followed a Top-Down model (see Noor, 2000), while the need for changes, especially at the school level, was never thoroughly investigated.

Secondly, each curriculum that has been implemented has lacked an implementation strategy. The in-service training provided to support teachers in implementing each revised curriculum seems not to have been effective (see Somerset, 1997; Hadi, 2002). Most teachers who went through the training frequently 'got lost' when they tried to put the new ideas into practice in their schools. Because there was no adequate supervision and evaluation after the training (Somerset, 1997), the teachers preferred to teach in the ways they had used before.

Thirdly, the implementation of the curriculum was never carefully evaluated. The only criteria used by the government to measure the success of the curriculum implementation was pupils' achievements. Meanwhile, data about the process of curriculum implementation

8

such as how the learning and teaching process was conducted in classrooms, how the pupils learned, or the difficulties that teachers faced in implementing the curriculum, remain unknown.

There are also some weaknesses regarding the content of the mathematics curriculum in the primary school. Firstly, the content of the curriculum is burdensome because there are too many topics that have to be taught (see Soedjadi, 2000). Teachers complain about the numbers of topics that they have to teach in a limited amount of time. Pupils complain about having too many exercises and too much homework to complete, while parents frequently become confused when they are helping their children with their homework.

The second weakness is the lack of connection between the topics in the curriculum. As a result, teachers perceive the curriculum as a set of disconnected topics that they have to teach, while pupils experience the curriculum as a number of separate topics that they have to learn.

Thirdly, the curriculum lacks examples of practical applications. Referring to the goals mentioned earlier, the content of the curriculum is supposed to be very rich with practical work and meaningful applications. In fact, the content is dominated by an approach that focuses on introducing and memorising abstract concepts, applying formulas and practising computational skills.

The learning and teaching process in Indonesian primary schools is mostly organized in the traditional way. Teachers become the center of almost all activities in the classrooms ( Fauzan, 1999; Fauzan, Slettenhaar & Plomp, 2002, 2002a; Marsigit, 1999) in which the pupils are treated as an 'empty box' that needs to be filled. In general, the climate in Indonesian classrooms (see Fauzan, 2001; Sommerset, 1997) is similar to that in several African countries, summarized by de Feiter & Akker (1995) and Ottevanger (2001) as follows:

1. pupils are passive throughout the lesson;2. 'chalk and talk' is the preferred teaching style;3. the emphasis is on factual knowledge;4. questions require only single words, often provided in chorus;5. lack of learning questioning;6. only correct answers are accepted and acted upon;7. whole-class activities of writing/there is no practical work carried out.

The impact of these classroom characteristics is that most pupils are not learning the mathematics they need. They also do not have the opportunity to learn significant mathematics. For most pupils, learning mathematics is an endless sequence of memorising and forgetting facts and procedures that make little sense to them, while for most teachers, teaching mathematics has become a routine task in which the same topics are taught or re-taught year after year (see also Battista 1999).

A number of attempts were made by the Indonesian government to overcome the

9

problems. However, most of these attempts were relatively ineffective. Until recently mathematics curriculum and textbooks still did not give the pupils adequate opportunity to learn mathematics, but only the opportunity to memorizing mathematics. Meanwhile, teachers proved reluctant to depart from their traditional methods, and a significant proportion of pupils tended to develop distaste for learning mathematics.

Based on the explanation above, we can summarise some fundamental problems inmathematics education in Indonesia:

1. The content of the mathematics curriculum is burdensome. This leads tosituations in which the learning and teaching process concentrates only on learning objectives and learning outcomes, while the process that leads to these learning outcomes remains a black box. In addition, most of the learning objectives only focus on memorising facts and concepts, and computationalprocedures (i.e. applying formulas).

2. The approach to teaching mathematics is very mechanistic and conventional.3. The changes and innovations in mathematics education have never addressed the

previous two problems because those changes and innovations lacked an implementation strategy.

The rationale for this study emerged from a general dissatisfaction with mathematics education in Indonesia, especially at the primary level, and aimed to contribute to solving the fundamental problems outlined above. This idea was developed by developing and implementing a piece of curriculum material namely Indonesian Realistic Mathematics Education (IRME) curriculum for learning and teaching the topic Area and Perimeter at Grade 4 in Indonesian primary schools. The term curriculum referred to an operational plan for instruction including what mathematics pupils need to know, how pupils are to achieve the identified goals, what teachers are to do to stimulate pupils to develop their mathematical knowledge, and the context in which learning and teaching occur (see NCTM, 1989). In this study the operational plan was crystallized in the form of a teacher guide and a student book.

The IRME curriculum was developed and implemented based on Realistic Mathematics Education (RME) approach through a development research. The results of the literature study suggested that RME was a very promising approach to overcome the fundamental problems. In RME pupils learn mathematics based on activities they experience in their daily life; and they are provided with ample opportunity to reinvent mathematical concepts and to construct their knowledge by themselves (see Gravemeijer, 1994, 1997). Instruction in RME calls for work to be done in-groups, where investigation, experimentation, discussion and reflection form the core of the teaching learning process (de Moor,1991). The development research was applied in this study because it provided sufficient and useful support for the development and implementation of the IRME curriculum. (Note: the term implementation is used here to indicate the process of the classroom experimentation using the IRME curriculum to teach the pupils in Indonesian primary schools). The study followed two "schools of thought" in development research. The first one is mentioned by van den Akker (1999), van den Akker & Plomp (1993), and Richey & Nelson (1996) and the second one proposed by Freudenthal (1991) and Gravemeijer (1994, 1994a, 1999).

10

The aim of the study was to develop and implement a valid, practical and effective IRME curriculum for learning and teaching the topic Area and Perimeter at Grade 4 in Indonesian primary schools. The terms valid, practical and effective refer to the classifications created by Nieveen (1997, 1999), Kirkpatrick (1999) and Guskey(2000). This aim of the study was elaborated further as follows:

1. The development of a valid RME-based curriculum refers to the developmentof 'local instructional theory' (see Gravemeijer, 1999) and to methodological guidelines for further development of RME materials in Indonesia.

2. A practical RME-based curriculum refers to the question of whether the RME approach could be utilised in Indonesian primary schools or not.

3. An effective RME-based curriculum refers to the extent to which the RME-approach could address some of the problems in mathematics education in Indonesian primary schools, more specifically in the geometry instruction.

In line with the aim of the study the main research question was formulated as follows:

What are the characteristics of a valid, practical and effective IRME curriculum for learning and teaching the topic Area and Perimeter at Grade 4 in Indonesian primary schools?

This research question was broken down into sub-research questions, and these were investigated in the three stages of the study. The first stage was the front-end analysis, in which the current situation of Indonesian education was analyzed, and literature on RME and research trends in mathematics education was reviewed . The literature review on RMEtheory resulted in the first draft of the IRME curriculum and of the conjecture learning trajectory for learning the topic Area and Perimeter.

The second stage of the study was called the prototyping stage. This stage consisted of the development and implementation of Prototype 1 and Prototype 2 of the IRME curriculum and formative evaluation of each prototype. While evaluation activities in the prototyping stage were focused primarily on the validity and practicality of the IRME curriculum, some aspects of the effectiveness were also evaluated in this stage. The last stage of the study was the assessment stage. In this stage the final version of RME-based curriculum was developed and implemented, followed by a summative evaluation activity. The assessment stage of the study was designed to gain further insights about the practicality and effectiveness of the IRME curriculum.

The evaluation activities that were conducted in this study included interviews and discussions with the Dutch RME-experts, Indonesian subject matter experts, an inspector, principals of two Indonesian primary schools, teachers and pupils, as well as classroom observations, analysing pupils' portfolios, assessments, pre-tests and post-tests. The instruments used for the evaluation were the interview guidelines, observation scheme, questionnaire, and the assessment and test materials. The schools for the classroom experiments in this study were chosen with the main consideration being the willingness of the schools, especially the teachers, to collaborate or to participate in the study. Prototype 1 of the IRME curriculum was implemented by the author in two primary schools in Surabaya, Indonesia, during Fieldwork I (September 1999 - February 2001). Two teachers and one Dutch RME expert observed the classroom experiments. The formative evaluation

11

in this fieldwork was conducted in a rather informal way. The results of the evaluation on the development and implementation of Prototype 1 can be summarised as follows:

1. There were several problems found at the beginning of the classroom experiments such as:- Dependent attitude of the pupils;- Pupils were not used to working on the contextual problems;- Pupils' tendency to get the result without paying attention to the process;- Pupils were not used to working in-groups;- Pupils' lack of motivation, activity, creativity and reasoning.After some efforts by the author (as the teacher) in overcoming the problems and after the pupils became familiarised with the RME approach, some improvement was noted in pupils' attitudes, motivation, activity, creativity, and reasoning. The teachers from the two schools also observed the changes. The contextual problems in the student book and the teaching method as applied by the author played very important roles in these changes.

2. The content and construct of Prototype 1 of the IRME curriculum were considered to be valid after the prototype was evaluated by two Dutch RME experts and reviewed by four Indonesian subject matter experts and two primary school teachers. The findings from the classroom experiments also showed that in general the learning trajectory for learning and teaching the topic Area and Perimeter worked as intended. However, problems regarding the pupils' attitudes found at the beginning of the classroom experiments and also the findings from the pupils' portfolios lead to some changes being made to the contextual problems in the student book.

3. In terms of the practicality, the results from the interviews with the teachers and pupils and the classroom observations showed that the student book was easy to use. The pupils were able to learn as intended according to the RME perspective, after the problems mentioned before were solved. Nevertheless, it was found that the pace planned for some lessons was insufficient because most pupils needed more time to solve the contextual problems than was expected, and also because of the above mentioned problems regarding the pupils' attitudes.

4. It took some time for the pupils as well as the author to adapt to the RME approach. The presence of the RME Dutch expert and the observers during the classroom experiments helped the author to get used to the new teaching style and also to overcome the problems that occurred in the classrooms.

The implementation of Prototype 2 of the IRME curriculum was performed by the author in two primary schools: one school (2 classes) in Surabaya, East Java, and another school (2 classes) in Padang, West Sumatera, during Fieldwork II in Indonesia that took place from August 2000 until March 2001. One teacher from each school was scheduled to teach the IRME curriculum in one class, but after two classroom experiments, both teachers withdrew. The teacher in Surabaya was away from the school because of family business, while the teacher in Padang felt that she was not yet capable of teaching using the RME approach because of inadequate preparation. In the assessment stage of the study it was considered to be important that proper training should be provided for the teachers before they conducted the classroom experiments.

12

The following summarizes the results from the evaluation of the development and implementation of Prototype 2 of the IRME curriculum:

1. The same problems as those experienced in Fieldwork I were also faced at the beginning of the classroom experiments during Fieldwork II. Learning from the experience of the previous fieldwork, the author (as the teacher) could overcome the problems more effectively. The experience gained from Fieldwork I also meant that the author felt more comfortable using the RME approach to teach during Fieldwork II. The benefit was not only in how to handle the problems that occurred in the classrooms, but also in how to react to the pupils' answers or contributions and how to guide and stimulate the pupils in solving the contextual problems.

2. The results of the experts' validation, which involved three Dutch RME-experts, four Indonesian subject matter experts, and one teacher, showed that the IRME curriculum material reached the criteria of the content and construct validity. It was also found that the pupils could learn the topic Area and Perimeter according to the conjecture learning trajectory designed in the IRME curriculum.

3. The experts, an inspector and the principal, all agreed that the IRME curriculum has the potential to develop pupils' understanding, reasoning, activity, creativity and motivation. They also agreed that the IRME curriculum would be usable and useful for learning and teaching the topic Area and Perimeter. The results from the interviews with the pupils and the classroom observations indicated that the student book was easy to use, and the pupils could learn as intended according to the RME point of view. Based on these data it was concluded that the IRME curriculum fulfilled the criteria of practicality. One teacher in Padang, who initially doubted the practicality of the IRME curriculum, also appreciated it as she herself observed some progress in her pupils by the end of the classroom experiments.

4. The investigation on four levels of the effectiveness: pupils' reactions, pupils learning, pupils' use of new knowledge and skills, and pupils' learning outcomes lead to the following conclusions:- The pupils liked the IRME curriculum and believed that it had helped them to

develop self-confidence and reasoning skills.- Most pupils had acquired the intended RME knowledge. They construeseveral

geometry concepts by themselves after performing the activities designed in the IRME curriculum and also found various strategies for solving the contextual problems.

- Most pupils demonstrated that they were able to use the new knowledge and skills that they had gained from an earlier lesson in subsequent lessons. This was not so for a few pupils who lacked knowledge of fundamentalmathematics concepts.

- The pupils' learning outcomes showed that the IRME curriculum had a positive impact on the pupils' confidence as learners and also their understanding, reasoning, activity, creativity and motivation.

- The pupils' achievements on the post-tests were significantly higher than their achievements in the pre-test, and their average achievement on the assessments was more than 8 on a scale of 1 to 10.

The final version of the IRME curriculum was implemented through Fieldwork III (August 2001 –February 20002) in Padang, West Sumatera and Surabaya, East Java. The classroom

13

experiments in Padang were conducted by the author in three primary schools. Three teachers and four student teachers took the role of observers. The author decided to implement the final version of the IRME curriculum himself in order to validate the results gained from Fieldwork I and II, especially regarding the impact of the IRME curriculum on the pupils' learning outcomes.

The implementation of the final version of the IRME curriculum in Surabaya took place in two primary schools (four classes). Two teachers (one teacher from each school) and two Ph.D. students conducted the classroom experiments. Nine observers (four Ph.D. students, one master student, two lectures, and two teachers) took the role of observers during the classroom experiments. In every lesson, at least two observers viewed the classroom activities. One observer focused on the teacher's activities and the other focused on the pupils' activities. The observers and the teachers were trained before the classroom experiments. The results from the assessment stage of the study are summarized as follows:

1. In general it was concluded that the learning trajectory for learning and teaching the topic Area and Perimeter could work as intended for most pupils.

2. The pupils could use the student book without any difficulties and they could learn the topic Area and Perimeter as intended according the RME approach.

3. The teacher guide was useful for the teachers in implementing the IRME curriculum. Three teachers said that the teacher guide was easy to use, while one teacher suggested that the teacher guide should provide more detail.

4. The evaluation on the aspects of the effectiveness: pupils' reactions, pupils' learning, pupils' use of new knowledge and skills, and pupils' learning outcomes (performance and achievement) resulted in the same findings as those in Fieldwork II.

5. The results of the evaluation indicated that the teachers felt positive about the IRME curriculum. In general the teacher could implement the IRME curriculum as intended, although sometimes they still used traditional ways of teaching. The author also observed that on some occasions the teachers could not fully apply the RME knowledge and skills that they gained from the training probably because they were not yet used to the RME approach.

6. The pupils' achievements on the post-tests were significantly higher than those in the pre-tests. The pupils' achievements in the experimental classrooms were also significantly higher than the achievements of the pupils in Grade 4 and 5 who had been taught the topic Area and Perimeter using traditional methods.

7. A significant difference was found between the motivation of the pupils before and after they had been taught the IRME curriculum, especially in terms of selfconcept.

Based on the results from the two stages of this part of the study, it has been concluded that:

1. The IRME curriculum developed and implemented for pupils at Grade 4 in Indonesian primary schools met the criteria of the content and construct validity. It suggests that the learning trajectory designed in the IRME curriculum can be used as a local instructional theory for learning and teaching the topic Area and Perimeter. The way the IRME curriculum was designed can also be used as a reference to design other RME materials. The characteristics of the valid IRME curriculum can be described as follows:

14

- The content of the IRME curriculum included the subjects that were supposed to be taught for learning the topic Area and Perimeter based on the RME point of view. In this case pupils' understanding of the concepts of Area and Perimeter was built by relating the concepts to other magnitudes such as costs, weight, and to irregular shapes. The reason for this is that in reality pupils mostly deal with the concepts of Area and Perimeter in regard to these contexts.

- The content of the IRME curriculum reflected the RME's key principles. When learning the topic Area and Perimeter using the IRME curriculum, the pupils had the opportunity to find out the concepts involved in the topic by themselves. They learned the topic Area and Perimeter based on the phenomena that they were familiar with, so that they could build an understanding of the topic using their informal knowledge. They also had the opportunity to use their own ideas in solving the contextual problems in the IRME curriculum.

- The IRME curriculum reflected the RME's teaching and learning principle- The RME curriculum included some important aspects of realistic geometry,

especially measuring and calculating, and spatial reasoning.- The content of the IRME was sequenced properly, so that the learning trajectory

for learning the topic Area and Perimeter could guide the pupils to learn as intended.

- The goals for each lesson in the IRME curriculum were clearly stated, and the content designed for each lesson was well chosen to meet the goals.

- The relevance and the importance of the units in the IRME curriculum were explicit

2. The IRME curriculum met the criteria of practicality. This condition is characterised as follows:- The IRME curriculum could stimulate pupils' understanding, reasoning, activity,

creativity and motivation in learning the topic Area and Perimeter.- The teaching learning process using the IRME curriculum created pupilscentered

learning.- The pupils could use the student book without any difficulties, and they could

also learn the topic Area and Perimeter (using the student book) as intended according to the RME point of view.

- The teacher guide was useful and easy to use by the teachers. The time setfor each lesson in the teacher guide was adequate.

3. The IRME curriculum met the criteria of the effectiveness as it resulted in some positive impacts on the pupils at Grade 4 in Indonesian primary schools. The positive impacts of the IRME curriculum on the pupils are characterised as follows:- The pupils reported that they liked the IRME curriculum. They said that the IRME

curriculum was useful and gave them more confidence as learners.- Most pupils acquired the intended RME knowledge in which they found out

several concepts involved in the IRME curriculum by themselves. They also developed various strategies in solving the contextual problems. Moreover, they could use the new knowledge and skills that they had acquired in one lesson in the following lessons.

- The pupils developed more positive attitudes towards learning mathematics. They became more independent and engaged actively in the learning process.

15

They also became more motivated and were stimulated to find different strategies in solving the contextual problems. Although their mathematical reasoning had been very weak initially, the pupils demonstrated that by the end of the classroom experiments they were able to reason mathematically.

- The pupils' achievements on the post-test were improved significantly compared to their achievements in the pre-test. The achievement of the pupils in the classroom experiments was significantly higher than the achievement of the pupils who had been taught using traditional methods. The pupils' achievements on the assessments were also satisfactory.

4. The results outlined above indicate that the RME approach could be utilised in Indonesian primary schools. Further, the RME approach could address some problems mentioned earlier in this chapter, especially in changing the classroom climate and providing guidelines in how to develop and implement a good quality curriculum material for teaching mathematics.

16

Design Research on Addition Up to 100 using Mental Arithmetic Strategies on an Empty Number Line

Puspita Sari (3103080)

Freudenthal Institute, Utrecht University, The Netherlands, 2008

Mental Arithmetic Strategies for children in Primary School has been suggested as an alternative to overcome children’s inability to handle numbers and numerical information properly. However, mental arithmetic strategies must be introduced thoughtfully with rich contextual situations, in which children have freedom to develop their understanding under the guidance of the teacher. In addition to this, an empty number line was found to be a powerful model to do mental arithmetic strategies flexibly and to foster the development of more sophisticated strategies.

In this design research, a sequence of learning activities was designed to support children’s development in constructing mental arithmetic strategies to solve addition problems up to 100. Next, the designed activities together with the hypotheses of children’s mental activities while engaging in the activities were put to the test in a teaching experiment. Then the data obtained from video recordings of classroom activities, interviews with children and the teacher, children’s worksheets, and notes taken during the teaching experiment were analyzed in the third phase of this design research.

The result shows that children develop their understanding by means of the ‘measuring situation’ as the contextual situation and the ‘empty number line’ as the proposed model in solving addition problems up to 100 using mental arithmetic strategies. However, the role of the teacher, the classroom culture, and children’s pre-knowledge before the teaching experiment certainly affect children’s performance both in the classroom activities and in their work sheets. On the one hand, some children solve addition problems on an empty number line without a great deal of understanding. On the other hand, they develop their flexibility in solving addition problems using mental arithmetic strategies on an empty number line.

17

Computational Estimation in Grade Four and Five: Design Research in Indonesia

By: Al JupriFreudenthal Institute, Utrecht University, The Netherlands, 2008.

Summary

Introduction and research questions

One calculation form that is used most in our daily life is computational estimation. For instance,

when we are in the supermarket, we often use estimation to know how much money will be spent

before going to the supermarket’s cashier, and after leaving the supermarket we frequently check

whether the calculation in the receipt is reasonable or not. This is an example that mathematics is

actually part of our life. That is why Freudenthal (1991) said that mathematics should be seen as a

human activity.

Many mathematics educators said that estimation is a very important basic skill that should be

mastered by students (Reys, Rybolt, Bestgen, & Wyatt, 1982; Rubenstein, 1985) because this is

useful either for solving mathematical problems at school or daily life. Moreover, according to Van

den Heuvel-Panhuizen (2001), estimation has a didactical function for learning, for instance, doing

estimation beforehand can help to master mental calculation strategies in arithmetic.

However, at schools, estimation has only a small place in mathematics curriculum even over the

world (Reys, Bestgen, Rybolt, & Wyatt, 1982; Reys, Reys, & Penafiel, 1991). In addition, most of

students seem uncomfortable with estimation (Trafton, 1986). When students are given estimation

problems, they frequently solve the problems by an exact calculation.

Based on the above issues we conducted research on computational estimation with the aims: (1) to

investigate students’ strategies in solving estimation problems; and (2) to gain insight into how

students can be stimulated to use estimation strategies instead of using exact calculation in solving

estimation problems. In the light of these aims, we conducted design research with the research

questions: (1) What strategies do students use to solve estimation problems? (2) What are students’

difficulties in solving estimation problems? (3) What kind of problems invite students to use

estimation? and (4) What kind of learning-teaching situations invite students to use estimation?

18

Theoretical framework

Computational estimation is the process of simplifying an arithmetic problem to find a satisfactory

answer, without actually calculating it exactly. Regarding the learning-teaching of computational

estimation, Van den Heuvel-Panhuizen (2001) distinguished three types of questions that are the

driving force behind learning to estimate, namely: (1) Are there enough? (2) Could this be correct?

and (3) Approximately how much is it? With regard to completeness of data from the problems,

there are two kinds of estimation problems: estimation problems with complete and incomplete or

unavailable data. In general, there are three cognitive processes that can be used to solve estimation

problems, namely reformulation, translation, and compensation (Reys et al., 1982; Reys et al., 1991).

Each cognitive process includes different estimation strategies. Reformulation includes, for instance,

rounding, front-end, and substitution strategies. Translation includes, for instance, changing

operations, and making equivalents strategies. And compensation includes intermediate and final

compensation.

In our research in grade 4 and 5 we focused on: (1) an investigation of strategies used by students to

solve estimation problems; (2) an understanding of students’ difficulties in solving estimation

problems; (3) looking for problems that invite students to use estimation; and (4) in particular for

grade 5, the research is also focused on a creation of learning-teaching situations to encourage

students in the use of estimation. To do these we use the theory of realistic mathematics education

(RME) because it offers pedagogical and didactical both on mathematical learning and instructional

materials (Treffers, 1987; Gravemeijer, 1994; Bakker, 2004).

RME is a theory of mathematics education which has been developed in the Netherlands since the

1970s and it has been extended there and also in other countries (De Lange, 1996). RME is shaped

by Freudenthal’s view on mathematics (Freudenthal, 1991), namely: mathematics should always be

meaningful to students and should be seen as a human activity. There are five tenets of RME

according to Treffers (1987) and Bakker (2004). Besides that, there are also principles which are

offered by RME to design learning-teaching environments such as: guided reinvention, and didactical

phenomenology (Gravemeijer, 1994). Based on the tenets and principles of RME, we designed

research instruments and a learning-teaching environment for learning estimation.

19

Research methodology

In this research, design of research was a crucial part of the research. Therefore we used design

research as the research methodology. The core of this kind of research is formed by classroom

teaching experiments (Gravemeijer, 2004). In our case, the purpose of this kind of research is to

answer the research questions about students’ thinking processes and to design an instructional

environment that supports students in learning estimation.

Design research encompasses three phases: a preliminary design, a teaching experiment, and a

retrospective analysis (Gravemeijer, 2004; Bakker, 2004). A design and research instrument that

proved useful during all phases of design research is called hypothetical learning trajectory (HLT),

where it includes: learning goals, learning activities, and hypothetical learning process (Bakker, 2004;

Simon, 1995).

In the preliminary design, we made HLT 1. This was used in the first research period: May-June 2008,

for grade four (10-11 years old) of a PMRI class. The purpose of this research period was to answer

the first three research questions. In addition, the research results of this period would be used to

revise HLT 1 to HLT 2 that would be used in the second research period: July-August 2008. Here

students were asked to solve estimation problems individually. After each lesson, at least three

students were interviewed based on their answers on the worksheets.

From the first research period we found: (1) estimation strategies that used by students consists

only of rounding and front-end strategies to solve computational estimation problems; and (2)

students found difficulties in the use of estimation to solve: problems which numbers involved are

‘too’ far from the nearest tens, hundreds, thousands, or other easy numbers; inkblot multiplication

problems (a kind of incomplete data problem); unavailable data problems; problems which numbers

involved are decimals or fractions; and problems that need more than one step solution.

In the teaching experiment, HLT 2 was used for primary school students of the first semester of

grade five (10-11 years old) of a non-PMRI class. The purposes of this research period were to get

better answer to the first three research questions than in the first research period and to answer

the fourth research question. Here, the students would be asked to solve estimation problems under

the teacher guidance in learning-teaching situations. During the teaching experiment we would use a

video camera and the researcher would also be available in the class to taking notes, pictures, and

help the teacher. In addition, after each lesson, at least three students would be interviewed to

20

Easy

Difficult

know their thinking processes based on their answers on the worksheets. Therefore, we would get

video, audio interview, students’ worksheets, and field notes data.

In the retrospective analysis, all data during the research would be analyzed to answer the research

questions. Here the HLT was compared to students’ actual learning.

First hypothetical learning trajectory and the retrospective analysis

In this part we describe the first hypothetical learning trajectory (HLT 1) which was used during the

first research period and the analysis of the results of this research period. We first describe HLT 1

that was used for primary school students of the second semester of grade four—10–11 years old.

Second, we analyze the results of this research period. And third, we describe the revision of HLT 1

to HLT 2 which would be used in the second research period.

First hypothetical learning trajectory (HLT 1)

In general, we expected that students would increasingly use estimation strategies. This means

during the lessons we predicted that there would be students who solve estimation problems by

estimation strategies and there would also be other students who solve problems by an exact

calculation strategy. We expected number of the latter kind of students would decrease from lesson

to lesson. Briefly, HLT 1 is described in Table S.1 below.

Table S.1: An overview of HLT 1 (used in the first research period: May-June 2008)

Problems Type of numbers Operations Expected

Difficulty

1 – 5

(a/b versions)

Integers

Examples: 50,000;

10,000; 5; etc.

Addition, subtraction,

multiplication

6 – 9

(a/b versions)

Integers, decimals,

simple fractions

Examples: 1,675; 1.5;

1/2; etc.

Addition, subtraction,

multiplication,

division, and a

combination of these

10 – 15

Larger Integers,

decimals, fractions

Examples: 69,999;

9,998; 3/4; etc.

Addition, subtraction,

multiplication,

division, and a

combination of these

(also operations with

fractions and decimals)

21

Overall Percentages of Students Using Estimation, May-June 2008

0

10

2030

4050

60

70

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Problems

Perc

enta

ges

Problems a

Problems b

Retrospective analysis

From students’ answers, as expected in HLT 1, we found two kinds of strategies used by students to

solve estimation problems, i.e., estimation strategies and exact calculation strategy. Students’

answers that used only words (without mathematical reasons) or no answers at all are classified as

unclear reasons. Estimation strategies which were used by students can be classified as rounding

and front-end strategy. This means, in this case, the cognitive processes used by students belong to

reformulation, but none of students used other cognitive processes: translation or compensation.

Next we present an overview of students’ global performances in the use of estimation during the

first research period (see Figure S.1). In general, we see that for problems 1 to 7 there is an upward

trend in the use of estimation strategies, both for a and b versions. This trend, however, does not

continue. Further, we can also make the following observations: (1) there is only a small difference in

the use of estimation between the a and b versions of problems 1 to 9, except for 4; and (2) there is

a sudden drop in the use of estimation strategies after Problem 7. This was a contradiction to our

expectation in HLT 1. We therefore go on to analyze the data in search of possible explanations.

Figure S.1: Overall percentages of students using estimation in the period May-June 2008

Note: Problems 1 to 9 have a and b versions, whereas Problems 10 to 15 do not have versions.

In Problems 4 (a and b) we find out possible reasons why there is large difference between these

two versions. For the second observation we analyze Problems 8, 9, 12, and 13. From the analysis we

found students’ difficulties in solving these problems, we also find out possible reasons why the

difficulties happened to students. In addition, we can find out characteristic of problems that less

22

invite students to use estimation. Therefore, based on this analysis, we can answer the second and

the third research questions.

Revision of the HLT

Based on the retrospective analysis, we then revised the HLT 1 to HLT 2. In the revision, we reduced

problems, re-ordered problems, and we decided to use the a versions only. The re-arrangement of

problems from the first to second research period can be seen in Table S.2.

Table S.2: Order of problems used in the second research period: July-August 2008

P. May-June 1.a 8.a 3.a 4.a 10 11 14 15 6.a 7.a 1

2

13

P. July-August 1 2 3 4 5 6 7 8 9 10 1

1

12

Note: This table means, for example, Problem 1 (in the second research period) = Problem 1.a (in the first research period); Problem 2 = Problem 8.a; and so forth.

Second hypothetical learning trajectory and the retrospective analysis

In a similar manner to the previous part, we describe: HLT 2 that was used in the second research

period, an analysis of the results of the second research period, and a proposal to revise the HLT 2

based on the analysis. Our analysis, in particular, is focused on answering four research questions.

Second hypothetical learning trajectory (HLT 2)

In general, like in the HLT 1, we expected that students would increasingly use estimation strategies

from lesson to lesson. This means during the lessons we predicted that there would be students who

solve estimation problems by estimation strategies and there would also be other students who

solve problems by an exact calculation strategy. We expected that number of the latter kind of

students would decrease from lesson to lesson except perhaps for new type of problems. Briefly,

HLT 2 is described in Table S.3 below.

23

Easy

Difficult

Table S.3: An overview of HLT 2 (used in the second period: July-August 2008)

Problems Type of problems Type of numbers Operations Expected difficulty

1 – 4

Problems with complete data

IntegersExamples: 50,000; 1,675, etc.

Addition, Multiplication, Combination: division and multiplication

5 – 8

Problems with complete data

Larger integers, decimals and fractions

Examples: 69,999;

3/4; etc.

Combination: addition, multiplication, division of integers; Combination: addition, multiplication with a (simple) fraction; Combination: addition and multiplication with fractions

9– 12

Problems with

incomplete or

unavailable data

IntegersExamples: 9998; 28…; 4…..; etc.

Addition, subtraction, multiplication, Combination:

multiplication, division

Retrospective analysis

The analysis is specifically focused on answering the four research questions. Similar to the results

of the first research period, based on students’ answers, students also used two kinds of strategies

to solve estimation problems, namely: estimation strategies and exact calculation strategy.

Estimation strategies which were also used by students were rounding and front-end strategy. This

means, in this case, the cognitive processes used by students belong to reformulation. However,

none of students used translation or compensation. From analysis of video recordings and field

notes we found that the teacher did not guide the students to use other cognitive processes

(translation or compensation) rather than reformulation to solve estimation problems. The

difference between results in the first and second research period is: in the second research period,

none of the students used any estimation strategy to solve estimation problems with incomplete or

unavailable data.

24

Overall Percentages of Students Using Estimation, July-August 2008

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Problems

Perc

enta

ges

Percentages

Phase 1 Phase 2 Phase 3

Similar to the analysis of the first research period we first present an overview of overall percentages

of students using estimation during the second research period (see Figure S.2).

Figure S.2: Overall percentages of students using estimation in the period July-August 2008

From the graph (in Figure S.2), for problems 1 to 8 except for Problem 5, we might see an upward

trend in the use of estimation strategies as we expected in HLT 2. However, surprisingly after

Problem 8 none of the students used an estimation strategy. In the HLT 2 we classified problems into

three groups: Problems 1 – 4, Problems 5 – 8, and Problems 9 – 12. We in retrospect understand this

classification. From the graph, we can distinguish the graph into three phases (see Figure S.2). In

phase 1, the average percentages are around 21 %, which means that around 21% students solved

problems using estimation strategies. In phase 2, the average percentages are around 59%, which

means that around 59% students solved problems using estimation strategies. However in phase 3,

it is surprising because none of the students used an estimation strategy—this result is absolutely

different from the result of the first research period. Therefore, there are at least two observations

that need more explanation: (1) Problem 5 evoked a sudden high percentage in the use of

estimation; and (2) none of the students solved the inkblot problems (Problems 9 – 11) and an

unavailable data problem (Problem 12) by any estimation strategy.

25

In Problems 5 we find out possible reasons why students are invited to use more estimation

strategies, i.e., context of this problem (namely ice cream) is experientially real for students; the

problem is open such that elicit students to use both different strategies and different answers; the

problem can be solved flexibly by different operations (addition, multiplication, or division); numbers

involved in the problem are easy to round off; the teacher’s role and classroom situations are

important in guiding students to use estimation.

For the second observation we analyzed Problems 8 and 9 because there are two large different

results between the first and second research period. In the first research period these two

problems—previously problems 6.a and 7.a respectively—were relatively successfully done using

estimation strategies, namely 55% and 50% students solved the problems by estimation strategies

respectively, however none of students solved these problems using estimation in the second

period. The reasons can be the following: (1) students did not recognize the problems as estimation

problems; (2) students might not think to use positional system of numbers (considering magnitudes

of numbers) to solve these problems; and (3) the teacher’s role is very influential to students,

namely the teacher told to students how to solve the problems by exact (trial and error) calculation

which are then followed by the students.

Proposal for revision of HLT 2

Based on the retrospective analysis, we revised the HLT 2 in order to invite more students to use

estimation. In revising, we took into account the following factors: the mathematical problems

themselves; design of problems; and classroom cultures. Mathematical problems include, for

instance, operation of numbers, and type of numbers. Problems with addition and subtraction were

given first, problems with multiplication and division afterward. Problems with complete data are

given first, next problems with incomplete or unavailable data.

Design of problems includes, for instance, difficulties, context of problems, selection of numbers,

and types of questions. Problems with more than one step solution are generally more difficult than

problems with one step solution. Numbers that are close to nearest tens, hundreds, thousands, or

other easy numbers can guide students to do rounding off for estimation. And questions which do

not require exact answers can invite student to use estimation strategies.

Classroom cultures include, for instance, students’ own strategies, teacher guidance, group and

classroom discussion (interactivity in learning-teaching situations). In the learning estimation,

26

students should be encouraged to come up with their own strategies under the teacher guidance in

the interactive learning-teaching situations.

Conclusion and discussion

In this part we present answers to the research questions. Next we discuss research findings and

give suggestions for classroom practices and future research.

Answer to the first research question

Strategies used by students to solve estimation problems from both research periods, as predicted in

the HLT, can be classified into two: estimation strategies and exact calculation strategy. Estimation

strategies that were used can be identified as rounding and front-end strategies, where the rounding

strategy is used most. According to Reys et al. (1991) these estimation strategies belong to a

cognitive process which is called reformulation. Other cognitive processes which did not emerge

from students, during the research, are translation and compensation.

Answer to the second research question

We categorize students’ difficulties into three, namely difficulties that were related to: (1) numbers,

(2) degree of complexity of problems, and (3) students’ habits. Difficulties related to numbers

include: difficulties in using numbers that are not clearly factors of other numbers; rounding off

numbers that are “too” far from the nearest tens, hundreds, thousands, or other easy numbers; and

solving estimation problems that have to do with fractions or decimal numbers. Difficulties related

to degree of complexity of problems include: number of steps to solve problems (problem with more

than one step solution are generally more difficult than problems with one step solution); translating

problems with a lot of information and inventing realistic data to solve problems (because students

are used to solve problems with all information is given); connecting various information from the

problems themselves and the outside (knowledge, experience, etc) to find an answer; thinking

reflectively with what students have done (because students are used to solve problems without

looking back to the answers whether reasonable or not); and solving problems that they had never

encountered before. And difficulties related to students’ habits include, for instance, most of

students are not easy to convince to use estimation strategies in solving computational estimation

problems because they need assurance that exact calculation is not necessary.

27

Answer to the third research question

Based on our experience with the students involved in the research period , we speculate that the

following characteristics of computational estimation problems can invite more students to use

estimation strategies: (1) contexts used in problems should be experientially real for students; (2)

the problems should be open; (3) the numbers involved in the problems should initially be close

enough to the nearest tens, hundreds, thousands, or other easy numbers; (4) problems can be

solved using different operations (for instance addition, multiplication, etc); and (5) the questions

used in the problems should not require exact answers.

Answer to the fourth research question

Based on our experience with students involved in the second research period, we found two

aspects that might invite more students to use estimation in solving computational estimation

problems, namely the role of the teacher and the lesson structure in the classroom learning-teaching

situations. Regarding the teachers’ role, we found in our case (Indonesian culture) that the students

would generally follow what have been explained by the teachers. Therefore, a ‘stronger’ teacher, in

guiding to influence students to use estimation strategies, is important. Regarding the lesson

structure of the classroom learning-teaching situations, in particular, we could say that group and

class discussions might motivate students to share opinion, justification, and (estimation) strategies.

Discussion

In this part we discuss: students’ estimation strategies, inkblot problems, bus problem, preparing

teacher(s) for conducting design research in Indonesian cultures, teachers’ role and classroom

cultures’ differences between PMRI and RME, and possible future research.

Students’ estimation strategies. As we mentioned earlier, the estimation strategies used by

students only include rounding and front-end strategy. One reason why only these strategies

emerged in students’ answers is because the problems which were used do not clearly invite

students to use other strategies. Therefore, we need to improve or use other estimation

problems. One possible way in improving problems is by changing the nature of numbers

involved in the problems—so the problems invite students to use other estimation strategies

(e.g. 3750 + 1675 + 2990 + 4990 to be 3950 + 3975 + 4090 + 4019. Therefore, we can solve this

as 4 x 4000, for instance, besides it can also be solved by 4000 + 4000 + 4000 + 4000 =…)

28

Inkblot problems. One possible reason why students find it difficult to solve inkblot problems is

because the students might not think of positional system of numbers (students are used to

work addition, subtraction, or multiplication from the right to the left, whereas an estimation

makes more sense to work from the left to the right, for instance, from hundreds to tens, and to

units). We suggest three ways to improve the inkblot problems in order to make the students

would be aware of positional system of numbers. First we might simplify the problems by

reducing the inkblots. Second, we could change the format of problems without reducing the

inkblots, namely from a column addition to a row addition, from column multiplication to a row

multiplication, etc. And third, we can combine the first and second ways.

We found interesting students’ answers that used a rather different strategy, namely excluding

impossible options to find an answer. In the analysis this strategy is classified as an exact (trial

and error) calculation strategy. However, we think this strategy is different from the common

exact calculation because when we are excluding impossible options, actually we are using

different cognitive processes rather than cognitive processes that happen if we use the exact

calculation strategy. After all, we do not know yet what kind of cognitive processes which is used

in this strategy

Bus problem. We found in the first research period the students were less tempted to solve the

bus problem by estimation strategies than by exact calculation strategy. More surprisingly, none

of students solved the bus problem by estimation strategies in the second research period. One

possible reason is because most of the students have not been trained to think reflectively: they

are only used to doing calculation, without looking back to check the calculation results. To make

students think reflectively we should slightly change the question. Do not ask whether make

sense or not the news but we change it to be could the number of buses bring 9998

supporters?

Another possible reason is because students are not used to combine information from the

problem itself and from outside the problem—in this case real-world knowledge or experience.

This might be caused by students’ view that mathematics (arithmetic) and real-world contexts

are separate systems. Thus, in learning-teaching situations, we think teachers should give

opportunity to students to solve problems that combine information from the problems

themselves and outside the problems. Moreover, giving rich context problems to students

hopefully would change their view: from the view that mathematics and contexts are separate

systems to a new view that mathematics and context can be connected.

29

In the PMRI class, students might have used to solve contextual problems which combine

information from problems themselves and outside the problems—this might explain why there

were students solved the bus problem by estimation strategies. In the non-PMRI class, however,

students are not used to solve problems which combine information from the problems

themselves and outside the problems—this might also explain why none of students use

estimation strategies to solve the bus problem.

Preparing teacher(s) for conducting design research in Indonesian cultures.

One of the three phases in design research is the teaching experiment. We tried to implement

the teaching experiment based on the plan. However, based on field notes and video recordings,

we found difficulties particularly concerning preparation of the teacher as indicated in the

following: (1) The teacher did not always follow the plan in teaching-learning situations, for

example, in introducing lessons, the teacher sometimes used different contextual situations

from the context that used in the problems; (2) We assumed that the teacher understood the

philosophy of RME. However, in our view, the teacher gave too much guidance to students for

instance she told to students how to solve the inkblot problems. This implies students did not

solve problems on their own: they followed the teacher’s strategy.

Since in Indonesian cultures we should give a great respect to the teacher, we then were

reluctant to give suggestions—it is impolite. Moreover, since we are younger than the teacher,

we should very appreciate to the teacher’s decisions. Such cultural issues are important to take

into account when we try to implement design research in educational practice, particularly in

the teaching experiment phase in Indonesia. This also could be a consideration when we will

conduct co-design research in the PMRI project, for example.

Teachers’ role and classroom cultures’ differences between PMRI and RME.

RME is developed in the Netherlands and PMRI is the Indonesian version of RME. Although

between these two countries had a very close connection in the past, they have very different

cultures. This might happen also in educational practice, such as follows: (1) Classroom social

norms that generally established in Indonesian situations are: in general students are not used

to expressing their thinking in front of the class, students are reluctant to ask questions to the

teacher if they do not understand yet, students try to avoid different arguments either with the

teacher or other students that expressed directly in the class. This could be in contrast if we

compare to the Dutch students; and (2) another classroom social norm that established

particularly in the non-PMRI class is that students are dependent on the teacher’s explanation:

students will follow what the teacher told to students. This might be different from the Dutch

30

situations: Dutch students might not follow everything from the teacher’s explanation and they

are more critical and perhaps less polite, where they seen more used to think for themselves.

Therefore, such potential differences are important to take into account when we try to

implement a RME approach in Indonesian classroom situations through PMRI.

Possible future research. In this research we used paper and pencil to find out how students

solve computational estimation problems. However, in daily life, we frequently use estimation

strategies mentally to solve problems that are encountered. Thus, we propose that the research

on estimation might be enhanced with estimation problems without paper and pencils to make

them more experientially real to students and avoid the possibility of exact calculations.

Whether this will work in Indonesian context is an interesting topic for future research.

31