Errors in Problem Solving Story

The errors of students in solving problems closely related stories difficulties experienced by students in solving story problems. Difficulties in resolving the matter of the story students, according to Ahmad Syafri in Rahardjo and Astuti (2011: 14) can be broadly classified as follows.

• Difficulty in understanding the issues (problems), the difficulty in determining what is known and what is being asked in the matter. 
• Difficulties in the settlement plan, which is about the difficulty in translating the story into the model (phrase) mathematics. 
• Difficulty in completing the plan, the difficulty in resolving models (phrase) mathematics.
• Difficulty in seeing (check) returns results that have been obtained. 
• The difficulty in interpreting the answers to the situation of the problem contained in the matter.

Sudjono in Supin (2001:8) describes some of the difficulties in solving the problem in terms of a psychological story is:

• Students do not understand what is read, due to the lack of knowledge of some terms that do not know, whether from foreign languages ​​or from traditional languages.
• Students are not able to live up to what is described in the matter, due to the events in question are not in accordance with the student experience. 
• Students are not able to choose the mathematical concepts that are relevant to the demands of matter in developing a mathematical model. 
• Students are not capable of computing (the math). In this case also means that the students already know conceptually counting regulations, but many can count finished insufficient servings specified time, or there are many wrong answers students because they lack accuracy.

Based on the difficulties of solving the problem raised by the story and Soedjono Syafri Ahmad, the general errors that may occur in solving story problems are (1) an error in determining the things that are known to the matter, (2) difficulty in determining things that are asked by the question, (3) errors in the mathematical model, (4) errors in computing, and (5) errors in interpreting the results obtained.
According to Allan L.White (2005:17) errors in solving math problems include:

a.       Reading Errrors (R)

Errors will be classified as a reading if students can not read key words or symbols written on the issue. This discourages students perform the next procedure in the proper groove problem solving.

b.      Comprehension Errors (C)

Students have been able to read all the words in question, but do not understand the whole meaning of the words, so that students are not able to move further along the grooves solving the right problem.

c.       Transformation Errors (T)

Students have been able to understand what the question is to be found but was unable to identify the operation or sequence of operations, which are necessary to solve the problem.

d.      Process Skills Errors (P)

Students recognize the corresponding operation or sequence of operations but did not know the procedures necessary to carry out operations accurately.

e.      Encoding Errors (E)

Students correctly solve the solution of a problem, but the solution cannot be expressed in the form of proper notation.

From the description of the types of errors completing a math problem above in relation to the type of errors in solving math story problems. Types of mistakes made by students in solving math story problems can be determined based on step-by-step story about the settlement. Reading and comprehension errors are misunderstanding about the story. In the aspect of the student should be able to read the entire sentence and look for important information when reading and understanding about the story. In the aspect of mathematical modeling, students can perform transformation errors, it is because students are not able to change the sentence into a sentence about math. Process skill errors are errors in the process of completing math story problems. In this case, students make the mistake of computing the student made a mistake in arithmetic operations or procedural errors. While encoding errors is error infer that students cannot write about the conclusions in the context of the question.

The Word Problem Of Mathematics

About the story of math is math related to daily life to search for the solution using a mathematical phrase that contains numbers, arithmetic operations (+, -, ×, :), and relations (=, <,>, ≤, ≥) (Rahardjo and Astuti, 2011:8). Tapilouw in Jasti (2008:8) states that "mathematics is a form of a story about a math problem that is expressed in a sentence that needs to be translated into mathematical notation sentence". While Saragih in Jasti (2008:8) states that math story problems is a matter that is presented in the form of short stories that are meaningful, so that a student's understanding of the story is not just a matter of computing factor alone but more than that students must first understand the meaning of a sentence by sentence of the matter, then make a mathematical model, calculating (computing), and then interpret the results obtained in the original question.

Based on the various opinions, it is a matter of math story problems presented in the form of short stories meaningful and related to everyday life that needs to be translated into mathematical notation sentence.

To finish the story about the student must learn to translate a problem situation into a sentence or mathematical notation. Sentence or mathematical notation is a sentence or a complete mathematical notation indicates a problem situation and indicated operations and what is required. It is based on the statement Troutman and Betty (1982:266) that "to solve word problems, children must learn to translate a problem situation into a number sentence or number expression. It is the number sentence or expression that compactly represents the problem situation and indicates roomates operations are to be performed and in what order ".

The steps to solve problems presented by mathematics stories Soedjadi in Muncarno (2008:2), namely:
(1) read about the story carefully to capture the meaning of each sentence, (2) separate and express, in terms of what is known, what is known about, workmanship count what is required, (3) developing a mathematical model of the problem, (4 ) completed in accordance with the rules of mathematical models that get an answer from that question, (5) returns the answer to the answers to the original models.

Skemp in Rahardjo and Astuti (2011: 13) also suggested steps to resolve the matter of the story is: "(1) understanding the problem, (2) mathematical modeling, (3) manipulation of the mathematical model, (4) to interpret the the original problem."

Density of Decimals

Density of decimals : One of the features distinguishing decimals from whole numbers is the density of decimals. Hiebert et al., (1991) found improving the continuity aspects of decimals’ density was particularly difficult. Working with problems involving continuous models in the written tests and the interviews, such as marking a representation of a decimal number on a number line, or finding a number in between two given decimals such as 0.3 and 0.4 were found to be more challenging than working with discreterepresentation task utilizing MAB models. Analysis for this finding suggested that an extra step in finding the unit of the continuous models explained the lower performance on continuous-representation tasks. In Table 2.1, A2 (money thinking) describes students thinking like this.

Likewise, Merenlouto (2003) found that only a small portion of Finnish students aged 16-17 years old in her study changed their concept of density. She attributed difficulties with grasping density to students’ reference to natural numbers and difficulties in extending their frame of reference to rational or real numbers. Some students relied on the possibility to add decimals in their explanations for recognizing density of decimals. Furthermore, she contended that this kind of explanation was based on an abstraction from natural numbers properties rather than a radical conceptual change from natural to real numbers.

Difficulties with density were also evident in studies involving pre-service teachers. Menon (2004) found only 59% of 142 pre-service teachers recognized the density of decimals. A similar trend was noted by Tsao (2005) who found that of 12 pre-service teachers involved in her study, only the six high ability students demonstrated an understanding of density.

The nature of incorrect responses with regard to the density of decimals is reflected in common misconceptions drawing on analogies between decimals and whole numbers. Clearly density will not make sense to students holding misconceptions identified in Table 2.1 such as money thinking, denominator focussed thinking, reciprocal thinking, and place value number line thinking. In general incorrect answers in recognizing the density of decimals could be classified in two categories. The first category of incorrect answers is identifying no decimal existing in between pairs of decimals. Fuglestad (1996) found that most students in her study of Norwegian students claimed there were no decimals in between two given decimals such as between 3.9 and 4 or between 0.63 and 0.64. Similarly, Bana, Farrell, and McIntosh (1997) reported that the majority of 12 year olds and 14 year olds from Australia, US, Taiwan and Sweden displayed the same problem. Only 62% of 14 year olds from Australia and 78% of 14 year olds from Taiwan showed understanding of decimal density. This evidence reflected incorrect extension of whole number knowledge that there is no whole number in between two consecutive whole numbers such as 63 and 64. Note that students holding money thinking (allocated to A2 code in Table 2.1) also will have difficulty in grasping the density notion of decimals and identify no decimals in between decimals such as 0.63 and 0.64. However, these students might identify 9 decimals in between 0.6 and 0.7 for instance, if they interpret decimals only as a number system for dollar and cents.

The second category of incorrect answer translates knowledge of multiplicative relations between subsequent decimal fractions. For instance, Hart (1981) reported that 22 to 39% students age 12 to 15 year-old thought there were 8, 9, or 10 decimals in between 0.41 and 0.42. Similarly, Tsao (2005) observed the same phenomenon in her study with pre-service teachers. She found that three pre-service teachers from a low ability group believed there were nine decimals in between 1.42 and 1.43 by sequencing only the thousandths: 1.421, 1.422,…, and 1.429. Along with most of the students in L and S groups (see Table 2.1), some students holding A thinking, such as A2 thinking with reference to metric measures (m, cm, mm) might possibly respond in this way.

Examples of Mathematics Media and Their Apllication

In relation to instructional media, teaching aids as mathematical manipulative are one of the examples used especially in learning mathematics. It is a concrete object which is designed so that a student can learn some mathematical concepts by manipulating them. The use of manipulatives provides a way for children to learn concepts in developmentally appropriate, hands-on ways.

The manipulative materials should relate to the students' real world. Mathematical manipulatives are used in the first step of teaching mathematical concepts, to make the abstract mathematical concepts more concrete to students. Through concrete objects, students can be directed to make representations of mathematical concepts and finally come to concept abstraction. Here is an example of manipulative teaching aid for addition and substraction operation by using cards (Widyantini, 2010).

Mathematics teaching aid for Integer Addition and Subtraction operation.
This media consist of some cards marked positive cards for the cards ("+") and marked negative cards for the card ("-“)

Most of the students at 4th grade still have many difficulties in understanding the arithmetic operations on addition and subtraction of integers. Therefore, teachers always try to develop ways that can facilitate students in understanding addition and subtraction arithmetic operations such integers. One way is to use flah cards marked positive (+) and negative (-). Teaching aid is used to help the students‟ to understand about the concept of integer addition and subtraction operations. Some of the provisions that must be agreed in demonstrating addition and subtraction operations on integers teaching aid are:

1. Providing cards marked "+" and cards marked "-", at minimum of 20 (twenty) cards each. Cards marked "+" are used to represent positive integers and cards marked "-" are used to represent negative integers.

2. Addition operation is the process of adding/giving card "+" or card "- ", while the subtraction operation is the process of taking card "+" or  "-".

The key word is:

give  : add; plus

take  : subtract; minus

c. If the card marked positive meets (in pairs) card marked negative, the result obtained is 0.

Example of the use of the card in addition operation

1). Demonstration to determine the outcome of 2 + 3 = ... Put two "+" cards to represent the number 2 in the first term, and add three "+" cards to represent the number 3 in the second term, the result is five "+" cards. The demonstration shows that 2 plus 3 makes 5. For details, consider the following demonstration.

So, we get 2 + 3 = 5.

2). Demonstration to determine the outcome of 2 + (-3) =…

As in the example no.1

Demonstration to determine the outcome of 2 + (3)

So, we get 2 + (-3) = -1.

Examples of the use of the card in subtraction operation

1). Demonstration to determine the outcome of the 3 + (-2) Put the card marked "+" as many as three to represent the number 3 in the first term. Furthermore, take two cards marked "+" to indicate the minus 2 and show the rest of the cards. From this demonstration, 1 is obtained from the subtractions 3 + (-2). For more details, notice the following demonstration.

So, we get 3 +(-2) = 1

2). Demonstration to determine the outcome of the 2 +(-3)
The process is identical to example 1.

Consider the following demonstration

So, we get 2 +(-3) = -1

Beside those manipulative teaching aid, we also have to know the use of electronic media such as Computer Assisted Instruction (CAI) as mentioned above. Some mathematics softwares are available in the market or published in the website. We might be free to download or buy them. Some examples of mathematics softwares for primary school are Fraction Munchers (MECC), Math Blaster for drill and practice programs. Math Shop (Sholastic Inc), The Island of Dr. Barin (Sierra Discovery Series) - a combination of mathematics and science – are mathematics software for simulation programs (Sharp, 1996). . Drill and practice math program help students become more proficient in their math skills and concepts. The program gives students the needed practice in a highly motivating format and assists the less academically adept child in mastering concepts. Program such as Fraction Munchers (MECC), which focuses on fraction types, equivalent fraction, comparison of fraction, and fraction expression, uses the Pacman arcade format to motivate the learners to improve their skills. The users direct the fraction munchers to “eat” numbers or fractional expressions that match a phrase that appears at the top of the screen. If the users eat an incorrect fraction or are caught by a “troggle” creature, he or she will lose a muncher. The game automatically advances to the next level after the student successfully clears the screen of all the target values.

The Relationship Between Media and Education

A. Definition of Media.
The study of media in education implicitly assumes that each medium entails some particular attributes that matter in learning depending on the symbol system it involves (Salomon, 1981). Media are our cultural device for selecting, gathering, storing, and passing knowledge on in representational forms. Representation, as differentiated from direct experience, is always coded within a symbol system. If one attempted to remove picture from film, cartography from maps, or language from texts, what would be left? Media, without symbol systems, are as inconceivable as mathematics without numbers.

According to the cognitive theories of learning, all cognition and learning are based on internal symbolic representation. If symbol systems are central to media of communication and thinking, then the interactions and interdependence between the two systems cannot be disregarded. For example, it is possible that symbolically different presentations of information differ as to the mental skills of processing that they require. It is also likely that the major symbol systems of the media cultivate mental skills differentially and that one learns to use media‟s symbolic forms for purposes of internal representation. The symbol used in media and thought is quite striking.

Bruner (1964) considers “that the development of human intellectual functioning from infancy to such perfection as it may reach its shape by a series of technological advances in the use of mind. The growth depends upon the mastery of techniques and cannot be understood without reference to such mastery. These techniques are not, in the main, invention of the individuals who are “growing up”; they are rather skills transmitted with varying efficiency and success by the culture-language being a prime example. Cognitive growth, then, is in a major way from the outside in as well as from the inside out.”

It is difficult to ignore that the possible role of media‟s symbol systems played in the cultivation of mental skills is not just as a carrier of information about the skills or as a carrier of skill-models, but rather as the mental-skills-to-be. As Bruner argues (1964; 2) that internal representation of the environment depends on learning “precisely the techniques that serve to amplify our acts, perceptions, and our ratiocinative activities”. Media, to which we all are heavily exposed, must surely be included among these techniques. Our era, the twentyfirst century, can be characterized as the age of media and technology.

As channel for information and entertainment, mass media surround us day and night. Vygotsky´s theories of social interactionism inform us that learning takes place through engagement with contextualised and situationalised socio-cultural environments and thorough ´contact with a culture of material and social resources that everywhere support cognitive activity´ (Crook, 1994: 32).

B. Types of Media
The above types of media are the most complete audio and visual motion (there are pictures, sounds, and motion). But, even the most comprehensive nature of truth is relative; in this case the TV media are still incomplete when compared with the combination of a videointeractive and computer program. TV programs do not "interact" with students actively, while interactive video does.

Media with sole ability are of course visual media only or audio-only media. Meanwhile, such real media or model displaying a visual threedimensional shape (such as statues) are not included in Bretz classification. Schramm (1977) divides media according to the number of students (audience) that they serve, namely bulk (lots spread over large areas), classical (quite small and concentrated in one place), and individual. Distribution, according to Schramm, appears in the following table:

In addition, media can also be divided according to which learning objectives can be achieved. Certain media are good to be applied to reach “visual identification” (kind of visual silence goals), but these types of media are less well when they are used to teach things such aspsychomotor skills or attitudes (affective).

C. Media and Student Development
The use of media will always be associated with the presented material, the chosen strategy and the level of students' progress. To be able to deliver good teaching, teacher needs to understand the child's cognitive development. According to Piaget, children cognitive development is divided into four stages (Wadsworth, 1994), namely:
1. Sensory motor stage (0 - 2 years old)
2. Pre-operational stage (2 - 7 years old)
3. Concrete operational stage (7 - 11 years old)
4. Formal stage (more than 11 years old)

Piaget's theory shows that in the beginning children learn through concrete or tangible things in the sense that it can be observed using the senses of children. To understand abstract mathematical concepts, children need concrete objects. In addition, Bruner (in Sukayati, 2009) divides the process of student learning into three phases: enactive, iconic and symbolic.

a. Enactive phase
In this stage, students are required to learn knowledge by using concrete objects or real situations.

b. Iconic phase
After learning knowledge with real objects or concrete objects, the next stage is the iconic stage where students study knowledge in the forms of pictures or diagrams as a manifestation of the activities that use concrete or real objects.

c. Symbolic phase
In addition to the above two stages there is one more stage, the symbolic stage, in which students create knowledge in the form of an abstract symbol. In other words, students must undergo an abstraction process.  Based on Bruner‟s opinion, learning should be started using real objects
first. Therefore, the process of mathematics learning should take place using models or real objects to certain topics that can help student‟s understanding. Thus, it is clear that the demonstration in mathematics is essential.

Based on the above explanation, the transition from primary school students to junior high school is concrete operational stage for the formal stage of learning. Hence, in learning mathematics students still need demonstration to promote their understanding and fascination of meaningful mathematics. Some functions of teaching aids in mathematics learning are as follows:

1. To make easier to understand a concept in mathematics. Example: flash cards, plane shapes teaching aid, solid shapes teaching aid, signs of "+" and "-”.
2. To strengthen or practice more on the concept that has been given. Example: card game of addition and subtraction operation of integers, card game of the value of fractions,   card game of decimal fractions, and card game of algebra,
3. To motivate or arouse students' interest in a concept. Example: the logic of power, base two, demonstration of Al- Khwarizmi, and the limit of rows.
4. As learning resources. Example: how to use a tool as a learning resource.

Objects and Characteristics of School Mathematics

Ebbutt and Straker (1995) mention the characteristics of school mathematics as follows:

1. Mathematics as search activity patterns of relationships.
The implications of this view of learning are: (a) giving students the opportunity to conduct discovery and investigation to determine patterns of relationships, (b) providing opportunities for students to experiment with a variety of ways, (c) encouraging students to discover the existence of sequence, difference, comparison, grouping, etc., (d) encouraging students to draw general conclusions, and (e) helping students understand and find the relationships between understanding one another.
 

2. Mathematics as requiring imagination, creativity, intuition and invention.
The implications of this view of learning are: (a) encouraging the initiative and providing an opportunity to think differently, (b) encouraging curiosity, the desire to ask, deny the ability, and the ability estimation, (c) appreciating the unexpected discovery as it is useful to think of it as an error, (d) encouraging students to discover the structure and design of mathematics, (e) encouraging students to appreciate the discovery of other students, (f) encouraging students to think reflexively, and ( g) not recommending just one method only.
 

3. Mathematics as problem solving activities (problem solving)
The implications of this view of learning are: (1) providing an environment that stimulates math learning problems, (2) helping students solve math problems using his own way, (3) helping students learn the necessary information to solve math problems, (4) encouraging students to think logically, consistently, systematically, and to develop documentation systems/records, (6) helping students learn how and when to use various teaching aids/mathematical educational media, such as: terms, calculators, etc..

4. Mathematics as a means of communication
The implications of this view of learning are: (1) encouraging students to recognize the nature of mathematics, (2) encouraging students to make examples of the nature of mathematics, (3) encouraging students to explain the nature of mathematics, (4) encouraging students to justify the need for math activities, (5) encouraging students to discuss math problems, (6) encouraging students to read and write mathematics, (7) respecting for students' native language in discussing mathematics.

Meanwhile, according to Bell (1998), the direct object of the lesson of school mathematics can be classified as follows:

a. Fact
Fact is the agreement or convention made in mathematics, for example terms (names), the notation (symbols), and agreement (convention).
Example:

1. Notation or symbol “ ^" for the word "and" in mathematical logic.
2. An agreement "On the number line, the right of “0” is positive, and the left of “0” is negative.

How to teach the facts can be done with various techniques, among others are memorization, drill, demonstrations, etc.

b. Concept
Concept is an abstract notion or idea that allows someone to classify the object or event and to determine whether an object or event is an example of abstract ideas or not.
Example:

1. The concept of function is described with or without examples.
2. The concept of the natural numbers: 1, 2, 3, 4, ....

Several concepts are fundamental understanding that can be captured naturally, vividly, and without having to be defined. Example: set, dots, etc..

Meanwhile, another concept is explained, defined or given a constraint using the previous concepts. For example, the understanding of prime numbers is explained using the understanding of factors (prime numbers are numbers which have exactly two factors). Understanding the factors is described as part of the multiplication. Understanding multiplication is repeatedly described as a summation. Understanding the sum is described as a merger of two disjoint sets. So, the concepts that form a network concept are also called concept maps.

How to teach the concept:
Start with the belief that the students already have the prerequisite knowledge and deductive approaches, as well as inductive, or it could be the perception, abstraction and generalization.

c. Principle
Principle is a statement which states the entry into force of a relationship between some of the concepts. The statement may declare the properties of a concept, laws, theorems, or propositions true in that concept. Similar to concepts, principles are also tiered.
Example:

1. The sum of the first –n natural numbers is (½)n (n + 1)
2. Rectangle can occupy exactly the frame with 4 ways.

How to teach principles:
Identify the concepts which are already known, then use a process of inquiry, guided discovery, group discussions, problem solving, demonstrations, etc.

d. Operation/procedure
Operation or procedure is the working steps in mathematics, for example, the steps in multiplying compound, procedure, and solving the equation. This procedure is also called algorithms. This procedure is to accelerate progress, but still based on the correct logic. Therefore, this object is also called as a skill.

Various Approaches in Mathematics

The international trends in mathematics education call for mathematics teaching and learning to enable learners in seeing, connecting, and applying mathematics in real-life. This call is advocated in a movement to teach mathematics as a subject matter closely related to other subject matters such as science, commerce and daily life experiences.

Freudenthal (1973; 1983, 1991) promotes the philosophical idea of mathematics teaching and learning as a human activity. Freudenthal‘s notion of ‗mathematics as a human activity’ was developed further in collaboration with his colleagues in Freudenthal Institute. It is now accepted as a well known theory in mathematics education called Realistic Mathematics Education‘ (RME). The basic notion of RME is that mathematics should be undertaken as an activity for students to experience mathematics as a meaningful subject under the guidance of teachers. One of the basic principles in RME is guided reinvention principle, which stresses the importance of learners to experience learning process as a process where in they get to ‗reinvent‘ mathematical properties and notion under the guidance of others. In 1993, Freudenthal noted the link of guided reinvention principle with the aim of developing common sense in learning process:

…I have pointed out that, in invented and reinvented on a manifold of places on earth independently, mathematics, unlike any other science, has been and still is a matter of common sense: in the course of individual histories and that of mankind, gradually refined common sense. So didactically, it seems to be no exaggerated requirement to have this knowledge reinvented by the learner, albeit under guidance. (Freudenthal, 1993, p.72)

Gravemeijer and Doorman (1999, 116) contend that in guided reinvention process, ―the learners come to regard the knowledge they acquire as their own private knowledge, knowledge for which they are themselves responsible‖. In this case, contexts from real-world or a story serve as a starting point where students explore and reinvent mathematical notions in a situation that is ‗experientially real‘ for them (see Gravemeijer and Doorman, 1999).

In a similar vein, Schoenfeld (1994) argues that thinking mathematically is far more important than just building an inventory of mathematical topics. He discriminates between ‗knowing‘ a list of mathematical contents and ‗knowing‘ to do and think mathematically. In his view, knowing mathematics is characterized by his ability to use mathematics in dealing with both novel and familiar situations. Hence, he advocates the use of problem solving as a strategy to enable students in developing mathematical thinking.

In line with RME approaches, a more general approach to teaching and learning, known as Contextual Teaching and Learning (CTL) (see e.g., The cornerstone of tech prep, 1999) also underscores the use of contexts for teaching and learning approaches. This approach advocates the use of context as a tool to help learners in making sense of the content as reflected in the following quotation: 

According to the contextual teaching and learning theory, learning occurs only when students (learners) process new information or knowledge in such a way that it makes sense to them in their own frames of reference (their own inner worlds of memory, experience and response). This approach to teaching and learning assumes the mind naturally seeks the meaning in contexts – that is in relation to the person‘s current environment- and it does so by searching for relationships that make sense and appear useful. (The cornerstone of tech prep, 1999, p.1)

Similarly, NCTM (2000) promotes the use of daily-life contexts that allow students to experience mathematics from an informal setting to a more formal and abstract mathematics. The emphasis on assisting students to see and make connections among various mathematical ideas is written in Chapter 3 of the NCTM Standards (2000). Furthermore, it underscores the role of teaching and learning process in making students aware of the mathematical connections through the use of probing questions:

By emphasizing mathematical connections, teachers can help students build a disposition to use connections in solving mathematical problems, rather than see mathematics as a set of disconnected, isolated concepts and skills. This disposition can be fostered through the guiding questions that teachers ask, for instance, "How our work today with similar triangles is related to the discussion we had last week about scale drawings?" Students need to be made explicitly aware of the mathematical connections.

The notion of mathematical literacy promoted by OECD (2004) also emphasizes on the function of mathematics in daily life. OECD‘s (2004) definition of mathematical literacy depicts a broader spectrum of what constitutes mathematics. This definition goes beyond school mathematics curriculum:

Mathematical literacy is an individual‘s capacity to identify and understand the role that mathematics plays in the world, to make a well-founded judgment, and to engage in mathematics in ways that meet the needs of that individual‘s current and future life as a constructive, concerned and reflective citizen. (p.72)

Mathematical modelling and the process involved in it are considered as one of the most central notions in mathematical literacy (Kaiser & Willander, 2005; Stacey, 2009). The modelling cycle involves a process of translating real-life problem into a mathematical model (mathematisation) and a process of reinterpretation the mathematical solutions back to the real world problems. Going through this cyclic process, it is expected that mathematics could be explored as something that is closely related to our daily life situations. A didactical modelling

process proposed by Kaiser and Blum (in Kaiser & Schwartz, 2006, 197) illustrates this in Figure The use of contextualised tasks in interdisciplinary settings that are meaningful for students is often used in mathematical modelling to promote mathematical literacy (see e.g. Stillman, 2000; Ng & Stillman, 2009).

Horizontal and Vertical Mathematization

Treffers (1987, 1991) differentiates two types of mathematization, namely vertical and horizontal, which is depicted by Gravemeijer (1994) as reinvention process (Figure 2).

Figure 1 Horizontal and Vertical Mathematization (Gravemeijer, 1994)

In horizontal mathematization, students start from contextual problems. They try to describe the problems using their own language and symbols, and solve the problems. In this process each student could use his/her own strategies that might different from others. In vertical mathematisation we may start from contextual problems, but in the long run students develop certain procedure which can be used to solve similar problem directly without utilize context. Gravemeijer (1994) calls this as mathematisation of mathematical problems to distinguish with horizontal mathematisation which is mathematisation of contextual problems. RME might be distinguished from other theories of mathematics instruction such as mechanistic, empiristic and structuralistic, based on the existence of horizontal and vertical mathematisation components (Treffers, 1991).

Figure 2 Mathematisation and direction (Treffers, 1991)

The two mathematisation components do not exist in mechanistic mathematics education. Mechanistic approach is algorithmic in nature. It tends to use method of telling and drill to exercise mathematical formulas and procedures. In empiristic theory of mathematics education horizontal mathematisation is clearly realized in informal procedure as a basis for learning. However, without support of models, schemas and the like, it is difficult to arrive at formal level. In structuralistic education, operation, mathematical patterns, and the like are concretized by using tools and media of instruction as representation of mathematical ideas and concepts. Vertical mathematisation is developed by using those structured materials. However, application of mathematics could no be reached, except students have understood how to use learned procedures. As consequence students would not be able to develop further their natural and informal procedures.

Outdoor Mathematics

Should our students always learn mathematics inside the classroom? Could they learn it outside? For students to learn mathematics effectively and successfully, they must be interested and love to study mathematics. In addition, they must be involved. Therefore, good teachers should be able to select and use suitable methods. Outdoor mathematics can be defined as mathematics teaching and learning process which is held outside the classroom. That learning process with its activities will help students to connect and apply their knowledge to their world and to other subjects. Bringing learning material inside is better than doing nothing. However, there are still limited approaches for outdoor mathematics, and the students are still trapped inside the classroom.

Why is Outdoor Mathematics important?
Mathematics becomes more powerful and meaningful when we go outside the classroom and begin to use it as a tool for studying other things. Moss (2007) states that if a child is to keep alive his inborn sense of wonder, he needs the companionship of at least one adult with whom he can share it, rediscover with him the joy, excitement, and mystery of the world they live in. Students are naturally curious about the world around them and enjoy using mathematics to help them understand their world; from measuring their height and weight, dividing cookies, and playing games. Therefore, mathematics is a natural part of their world. By means of this natural curiosity, they are able to construct mathematical knowledge through both psychological and sociocultural perspectives of constructivism.

Outdoor mathematics experiences not only help learners to see connections between mathematics and other disciplines, but also help learners feel more connected to their natural world. In addition, if they have been guided by teachers in applying mathematics in real situations outside the classroom, the students are more likely to successfully retain and use parts of their knowledge; and finally they can apply it to their own needs.

As mathematics teachers at primary school, we have experienced that there are many mathematics classes that are not easy to handle in the classroom. Won‟t they be even more difficult outside? Tran Vui (2001) writes that the most important principle, students may feel “difficult” is if they are not interested in what they are doing. When students have already judged that something is useful, then the subject problems largely disappear. Students need to know clearly what is expected from them. This is the teacher‟s normal task to ensure that the class is properly organized, after the discussion in which a real notice has been taken based on the students‟ input.

Another question has been raised by Tran Vui (2001) that is: “Can all of mathematics be taught this way?” It could be said that the outdoor approach will give a chance to acquire various mathematical skills and techniques. There are so many ways of doing mathematics outside the classroom than one can imagine. 

Outdoor Mathematics in Teacher Training
Teachers need to experience mathematics in ways that they will be expected to teach it; they need to experience outdoor mathematics in the natural world. Teachers are more likely to implement outdoor activities in their own classes if they have experienced it in their own learning experiences.