# The Development of Children’s Mathematical model in the Bus Problem

Choosing a good context, a situation intended to develop mathematizing, is not an easy task for the teacher. Teacher needs to find a real or imaginable situation that can be used to both engage and help students in grasping mathematical ideas. An example of good contexts is the bus problem where students are asked to check the number of passengers. It is considered as a high quality context since not only experienced real by the students but also has the potential to prompt students to model addition and subtraction. In the light of it, this essay has how children shift from *model of* to *model* *for* in the bus problem as the main issue.

The bus problem as a context is started by telling first-grade students a story about a bus trip and asking their experiences related to it. Once they get engaged to the conversation and have an accurate picture of bus travel, the teacher brings them to the next phase, the introduction of the context. In this occasion, the teacher gives the students a situation to be imagined. There is a bus driver with his small bus that can carry a certain number of passengers, let’s say 12. Since there are only 12 seats, the bus driver wants to know the number of people on the bus in each stop, in order to decide how many more he can allow on. The students are told that it always takes too much time for the driver to count at each stop and, of course, it makes the people on the bus get angry. Hence they are asked to help the bus driver figuring out how many passengers are on the bus at each stop. Working with this context, children are supposed to get engaged and be able to mathematize the situation. They also, at the end, are intended to develop a generalized model that can be referred when discussing adding and removing.

Using students’ ideas or suggestions as a starting point is the best way to start classroom activity. They will feel their ideas are valuable and it can serve as a trigger for next brilliant ideas. In this context, students’ immediate responses for the bus problem that states the driver can keep track of when the passengers get on and off can be used as a starting point before moving to the next phase, experiencing the situation.

Giving chance the students to experience such a situation by playing game or role play makes them enjoy the lesson and get more engaged to it even though there is also possibility students forget the real problem. In this context, the students play bus with several routes and stops. One acts as the driver and the rest are the passengers. Teacher doesn’t need to ask the driver how many people on the bus at each stop. Instead, he just needs to ask after a couple of stops in order to emerge dilemma, the need to keep track with addition and subtraction, in the driver’s mind. To get involved in mathematizing, the students must feel the problem by themselves and take it as their own.

Experiencing the problem by themselves doesn’t automatically ensure students have model or representation of the situation. It is teacher task to help the development of the students’ mathematical model and it can be done by guiding students through the situation, asking questions, and discussing students’ answers to get others’ responses. In addition, by probing and elaborating students’ reasoning, teacher can get sufficient knowledge before implicitly asking students to make their own model of the situation. In the bus problem, for example, after discussing how many passengers will be on the bus if there are 2 people on the bus and 3 waiting on the stop, the students will experience adding and removing people. Even though they have strategy to do it and other similar cases and furthermore discussing their result, they still have not represented the problem. They will start to do so when the teacher succeeds emerging the situation where they will need to come up with their own model of the situation, in this case by asking the student to show what happen at each stop after they experience it for several times.

Most students will start their representation or model of the situation by writing down all the passengers’ names since it is the easiest way to depict the situation. On the other hand, some students think names are not important in this situation due to the heart of the problem is to check the number of passengers. In the light of it, they tend to make picture of a bus with all its stops and all numbers indicate precisely how many passengers at the beginning, how many get off, and how many are waiting at the bus stops. This model shows the actions involved in the real situation and seems more interesting for the students, so after the discussion in the classroom most students, that realize list of names is not important, will use this *model of *that in fact is also developed by the students. Teacher in this situation doesn’t need to prescribe the use of this *model of* because different students will develop their own model that represents the actions and situations. They will decide by themselves which model is the best for the situation after considering many things and what teacher should do is taking strong guidance role in the lessons.

When leaving out the names and changing by drawing a bus, students already make generalization of the model. Their model is not only for their specific situation but also for another similar problem related to the bus. Later on, students realize that drawing a bus and its stops are not necessary and tend to focus to the situations in which people get on and off the bus, so they change the drawing into box and arrow representing the result of the action. In this phase, students succeed to generalize the model, but their model is still a representation of the situation.

The students will reach the next level of the model when successfully developing model for depicting addition and subtraction, the open number line. It does not represent every whole number, but only numbers yielded from particular action. This model does not come up separately but emerges as a generalization of their previous model developed from bus problem. In this phase, the model is not related to the real situation anymore. Instead, it is representation for the mathematical ideas in the situation, in this case addition and subtraction, and can be used by the students as a tool for solving problems and overcoming situation related to these concepts.

As the conclusion, using the bus problem as context related to addition and subtraction can give students chance to do mathematizing. They with teacher’s guidance can develop their own model of the situation in the bus problem and gradually generalize it to the more formal stage, model for addition and subtraction that can serve as a tool for students when dealing with the problems related to these mathematical operations. Their models from the lowest to the highest stage are list of names, bus and stops, boxes and arrows, and empty number line. (ham)