**Introduction**

The paper contains a design research aimed to change the paradigm of students in Indonesia, by far they are taught by their teachers without deep understanding. This fact is contradict to Freudenthal’s idea that viewed mathematics as a human activity. Therefore, the writer of the paper intended to make design research.

In the beginning, the writer chosed the number sense as the domain. This research is try to develop agrounded theory for this domain, especially for young learner on age 6 or 7. Based on Indonesia context by developing local instructional theory on guiding the development for number sense with the support of structuring. For those need, this research will aim to:

- Explain children’s thingking process and achievement in exploring structure in relation on how they perceive numbers
- Support children’s number sense growing process by engaging their ability on structuring.

To explain children’s thingking process and achviement, the research will be guided and will answer these following questions.

- What is the role of structure in the relation on how a child perceives numbers?
- How can children at the early development engage the structure to support their growing process of number sense?
- How does the process of symbolizing evolve when children engage structure on their effort to grow the sense?
- What is the role of socio-mathematical practice in motivating an individual’s number sense development?

The questions above are grounded theory on this domain, and will be resulted from the retrospective analysis and the refined hypothetical learning trajectory based on the teaching experiment and for the second aim, supporting children’s number sense growing process by engaging their ability on structuring, the research will try to answer the following research questions:

- What kind of context, means, and instruction that support children’s number sense growing process through structuring?
- What kind of context, means, and instruction that encourage children to evolve the process of symbolizing?
- What kind of activities that stimulate the emergence of socio-mathematical practice that motivate an individual’s number sense development?

**Theoretical Framework**

To build grounded theory, it will be elaborated as some following theories :

- Number Sense

Number sense refers to a person’s general understanding of numbers, their relationship and operations; and the ability to handle daily-life situations that include numbers. Counting is not only component of number sense. Based on the researchers and discussions, it has been generalized the number sense component as following understanding numbers, understanding meaning of operation and how they relate to one another, and compute fluently and make reasonable estimation.

- Structuring and Its Relation with Number Sense

In this research, structure will be occupied in two ways. First, structure is used to build a number image for the children and enables them getting familiar with numbers; second, structure and number image is occupied by the children to ease them on further counting and further formal operation.

- Realistic Mathematics Education

Realistic mathematics education (RME) is a theory of mathematics education that offers a pedagogical and didactical philosophy on mathematical learning and teaching as well as on designing instructional materials for mathematics education. According to Freudenthal, the emphasis should be placed on the activity mathematizing which include generality, certainty, exactness, and brevity.

This research will be done based on this theory. The mathematics will be learned by young children in the realistic way in the sense of experiantially real and also based on prior mathematizing. The process will be progressive, using the imagery of the previous activity to do the next activity.

- Symbolizing and Emergency of Models

Symbol is any situation in which a concrete entity such as a mark on paper, an icon on a computer screen, or an arrangement of physical material is interpreted as standing for signifying something else. In this research symbols and models that will be occured during the process is attempted come from the children as a result of the reflection of their learning process and take place in the individual practice, classroom discussions, and negotiation.

- Social Norms and Socio-Mathematics Norms

In this research we will be focus on the normative aspects of mathematics discussion specific to students’ mathematical activity.

- Indonesian National Curriculum for Grade 1

As the consequence that the research will examine the early development of number sense, the research chooses the student from grade 1 as the experimental subject. In this grade in Indonesia, the number knowledge is taught in two semesters. In the formal condition, in the first semester, the number is limited up to 20, and in the second semester, the number will be learned up to two digits numbers. The basic competences are counting, ordering, addition and substraction, and solving the problems related to addition and substraction.

**Methodology and Subjects **

- Hypothetical Learning Trajectory

The tasks on this research are selected based on hypothesis about learning process; the hypothesis of the learning process is based on the task involved.

- Desing Research Methodology

The goal of preliminary phase of design research of experiment is to formulate a local instruction theory that can be elaborated and refined while conducting the intended design experiment. A design research cycle consists of preparation and design phase, teaching experiment, and retrospective analysis.

- Reliability and Validity

The internal reliability in this research will be improved through discussing the critical episodes that might be happened during the teaching experiment with the supervisors and colleges during the PMRI meeting but the external reliability will be presented by recording the teaching experiment using the video and observation sheet.

The internal validity in this research will be kept by testing the conjecture during the retrospective analysis. The external validity is mostly interpreted as the generalizability of the result.

- Description of Experimental Subjects

The research will be done in the first grade at SD BOPKRI Demangan III Yogyakarta, Indonesia, under supervision of Sanata Dharma University (http://www.pmri.or.id/lptk/lptk2.php?Id=10)

**Mathematical Phenomenology**

Mathematical Phenomenology is the study of mathematical concept in relation to phenomena. It is organized from mathematical point of view. Some Mathematical Phenomenology that deal with the early number sense are cardinality-magnitude, one-to-one correspondence, hierarchical inclusion, and untizing.

- Exploratory Interview and Didactical Phenomenology

Didactical phenomenology is the study of concepts in relation to phenomena with the didactical interest. One purpose of didactical phenomenology is to find problem situation that can be used for the guided reinvention of the concepts for the end goals. But we also need to know about students’ prior knowledge. The task inteviews had been done in May- June 2008. One of the purpose of this interview was aiemed to know what kind of strategies students used to solve the problem related with counting.

Freudenthal said that the constitution of cardinal numbers can be perfomed by:

- Eliminating from structures with the same substratum the structuring component in order to arrive at substratum sets
- Transforming the inclusion realtion into the order realtion (‘less’instead of’contained in’)
- Using isomorphism of structures wit diffeerent substratum to compare different sets
- Using transitivity of equality and order (of numerosity number)

When cardinal of sets is connected by structure, cardinal equivalency of sets is more shown by one-to-one mapping rather than by counting out.

**Conjectured Local Instructional Theory for Developing Number Sense in Grade 1**

Conjectured local instruction theory consists of conjectures about a possible learning process, together with conjectures about possible means of supporting that learning process.

- Learning Goal of Students

The main mathematical goal in this research is the emergence of the number sense.

- Hypothetical Learning Process

The teaching experiment will be held for three weeks, four days in a week, and seventy minutes for every lesson. In this chapter, we will start making the hypotetical learning process for those four weeks. The theme for the first, the second, and the third are respectively butterfly-wings, inventory, ice-cube-tray.

- Cascade of Learning Tools

Imageries |
Tools |
Ideas |

– | Butterfly picture | Constructing context based meaning |

Structure of Butterfly wing | Butterfly – shape paper | Getting familiar with structure |

Structured object | Picture – card of Structured object and number card | Representing the number with Structured object |

Structured object | Classroom stuff | Structuring in five or tens (or other structures) |

Structured and unstructured object | Inventory list | Understanding the place value idea |

Pack (tens) and loose (units) | Ice cube tray | Get more understanding on place value. |

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