**Mathematical Models through Mathematical Thinking**

Public school systems need to be more sensitive to the children. The way of teaching a subject is become an important point of view for teacher’s ability. As not all children in one classroom have the same ability even if their teachers give an equal treatment. Thus, in the classroom, the teachers play a major role for developing the children’s thinking. In relation to that statement, I think teachers have bigger responsibility than parents which have merely met their children in home.

Since the children need guidance in learning, school can be a medium for students to get the guidance in learning through teachers. I have been through the situation that made me strongly felt like or dislike; understand or not understand of one subject. Usually, it depends on the ways of teacher’s explanation. Therefore, to be a good teacher, he or she should have a good understanding of the subject and also have some strategies in teaching the children. The strategies that can make children like or become interested in the subject make a meaningful learning in the classroom also.

In mathematics subject, mathematical thinking is an important part of learning mathematics. It is required some activities that can support learning process in the classroom in order to bring the children to think mathematically. From the activity, children are expected to be able to make mental maps in generalizing their ideas, strategies, and representations from the problem across given contexts – models. It means that children will construct their mathematical thinking while organizing their activity, solving the problems, or exploring relationship.

Through the context, mathematical models are existed from *model of* thinking to *model for *thinking in which sustain the process of children’s thinking. Children will move from modeled context situation to modeled focus on mathematical relations. For instance, there is an example of developing model about the bus model in which the main idea is that children move from *model of *to *model for.* Once, Kris a teacher of first-grade* *in Dutch classroom told her class a story about a bus trip. She started asking them whether the children know about taking buses. It seems that Kris want to check the bus context up if it can be use or not. The results prove that the bus context is useful because children are familiar with it.

Kris takes the conversation to another level. She knows that children often watch people entering and leaving a bus. So she asked children to involve in finding the solution of driver’s problem. She said that the driver wanted to know how many people were on the bus. Everytime he came into the bus stop, he counted the number of people who entered and left the bus. However, people are always upset because his counting takes too much time. So that Kris asked children to help the driver to figure out how many people are on the bus at each stop. This strategy can be applied to children who want to learn additive and subtractive operation through the real situation.

Before we enter to further explanation, I want to inform why I use bus as a context. Bus is one of public transportation which is common for many people. Children usually think that bus is used as transportation to go to school or to visit some places. Because of that, Kris took bus as a context in which familiar for children. Even though different places have different rules of bus system, Kris used this context based on the information that children know in general. This condition supports mathematical model that the bus has a certain number of passengers and in another bus stop the number of passengers can be added or reduced. It depends on the passenger who entered or left the bus in which makes mathematical operation happened.

Started by asking some children to involve in the activity, one of the children becomes the driver. Charlotte has been pointed among children who raised their hand because they wanted to be the driver. It is good when Kris chose Charlotte as a driver, she said to other children that Charlotte as the first driver. Even though it was doubtful if there was still time to apply this context by using another driver, she let the children go on and off the bus in each bus stop. After that, she asked Charlotte how many passengers in every bus stop. At first, Charlotte was confused because she only focused on driving. The activity was repeated until Charlotte got an experience with the situation and the others could overcome the bus driver’s problem in mathematizing. In this phase, Kris made a pressure to Charlotte in order that Charlotte knew exactly the passenger who will enter the bus by focusing on the addition and who leave the bus by focusing on the subtraction.

Discussion was happened among children while the activity of bus as a context was still running. Many children are still using the strategy of counting on, because they still cannot explicitly add or subtract the passenger. When the children succeeded in creating situation, they came up with their own models, or representations of the situation. Kris and the children are developing a representation of the situation – a model – through discussion within community. It does not mean that Kris prescribes this model; her intention is to help the children only.

The first model is still representations of action and situations with the list of names in which is closely related to their experience in class. But slowly, they start to not write down the names because they realize that the names are not important. In this case, children try other models such as the arrow model as a *model of* the situation. They serve various models with all specifics which are imbedded. After that, by helping children generalize the model to additive and subtractive situations using open number line as a *model for* of the situation – a model that can eventually be a tool to think with.

The solution from the bus context to the open number line does not happen at once, it needs a process from the bus context until children arrive on their realization. In this case, they work on addition and subtraction so that they can put the model of bus context in their head and work only with open number line.

By Ekasatya Aldila Afriansyah

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I think you’ve made several gramatical mistakes

ohh,can you tell me the mistake so i can look it again and i’ll repaire it. Thanks before, i’ll always wait any comment and suggestion. 🙂