To know how long, how big, and how weight something, we need to measure it. Measurement has been a part of human life since centuries ago, when some old civilization used their body to measure the length of object (Wijaya A, 2008). Without measurement we do not know how long, how big and how weight something, or in other word without measurement there is no control. Because of the importance of measurement, measurement has been taught since at elementary school in many countries. In measurement domain, there are three subjects, they are; (1) measuring time, (2) measuring length, and (3) measuring weight. In this paper we focus to measuring time because we think that students have difficulties in learning time. Time is something intangible for students because they cannot see, feel or touch it (Van den Houfel, M. 2004). This domain also challenge us because time measurement have not much literature about that and we also hope by knowing the time, the students will improve student’s disciplines attitude toward time, because by learning time, they will know about time and became aware of time, so they will be discipline in time, for example they will be shy to come late in school. In Indonesia, students learn about time since they are in grade 1. In grade one the focus of time measurement is to develop sense of time. In grade two the focus of time measurement is reading the clock in full hour and half hour. Based on our observation of Indonesian student, when we ask some students in grade 3 to see the clock and ask them “what time it is” most of them have no idea about it. In fact, the time was 8.30 when we ask to them. It was strange because they already have studied about how to tell time in full hour and half hour in grade 2. This situation motivates us to design a learning trajectory in time measurement in grade 2, to help the students more understand about the clock and are able to read the clock. In this paper our goal is to design a learning trajectory that can help the teacher to teach the students in measuring time so the students are able to read the time in full hour and half hour.
b. Mathematical Goal
In our research the main goal is that students will learn about time. Therefore students have to construct their idea about how to tell time, but also about the
strategies how to tell the time. When students will start they already have learned number up to hundred. It is important, because in learning to tell time they need that knowledge. For example to know that 1 minute is equal to 60 seconds, the students need to count the movement of the second hand. In this research we have four goals, they are; (1) Developing student’s sense of time. (2) Students know the units of time; second, minute and hour. (3) Students are able to read the clock (analogue clock) in full hour and half hour, and (4) Students are able to determine duration of certain activities Where in those four goals have each mathematical moment. Talking mathematical moment, it means that we talk the activity for the students, because mathematical moment happens in the activity that the teacher or designer design for the students. To reach the goal, developing sense of time, we design two activities, they are ;(1) We ask students what they can do in one minutes by using stopwatch to measure what they are doing, In the class the teacher could ask the students “what can you do in one minute?” The students may have question “how long one minute is”. This can be a conflict for them, how they can measure one minute. After that the teacher can provoke the students by asking “who can say that we are already one minute doing the activity?” in this situation the teacher provides the stopwatch for the students. In this activity we want the students have experience with one minute, and they know how long one minute is. At the end of the lesson we hope that the students have vocabularies short time because they cannot finish his/her job in one minute and long time because they finish his/her activity less than one minutes and they can develop their sense of time. (2) We ask students to order daily activity based on the time. In this activity, we give pictures to the students where in the pictures, we provide certain activities that they usually do in certain time and certain hour in analogue clock.
For goal number (2), students know the unit of time. We design the activity observing an analogue clock. The purposes to do this activity are to make the students know the parts of an analogue clock, for example; the second hand, minute hand, and hour hand; to make the students know the function of second hand, minute hand and hour hand; and to make they know the movement of those three hand (clockwise). After they know the parts of an analogue clock, we do the
activity observing the movement of second hand and make the relationship with minute. The purpose to do this activity is that we want the students know that one minute is equal to 60 seconds. For the relation between minutes and hour, we give the students project to do in their house. At the end, we hope the students know that 1 minute is equal to 60 second and 1 hour is equal to 60 minute Actually, in learning the relationship among the second, minute and hour, there is a concept of unitizing. Unitizing is using numbers to count groups and the objects in the group simultaneously, make the group a new unit (Dolk and Fosnot, 2001). Unitizing is a process where in that process the students became aware what the unit is and how it works. In this situation, the first unit is second. After the second moves 60 times and the new unit come, one minute that showing by one step movement of minute hand, and after the minute hand moves 60 times and the new unit come, one hour that showing by one step movement of hour hand. For goal number (3), students are able to read the clock (analogue clock) in full hour and half hour; we design the activity reading the clock in full hours and half hour. In learning full hour, we focus that the students are aware about the characteristic of full hour, o’clock, where the hour hand is really in line with the number in the clock and the minute hand is in line with the twelve in analogue clock. In learning half hour, we focus that the students are aware about the characteristic of it, where the hour hand is in the middle of two numbers in analogue clock, and the minute hand is in line with the six, in analogue clock. For goal number (4), students are able to determine duration of certain activities; we design the activity to determine the school hour in the school. Actually Indonesian students from grade one to grade three start studying at 7 o’clock and finish studying at 10 o’clock, but in certain days, the teacher has meeting and they can go home half hour earlier. In this situation, we want the students to determine how long they study at that day.
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