Previous : Part 2 : Repersonalization of Mathematics

**Accompanying The Teacher**

I became involved in DDR with Pa Didi and Pa Tatang when they collaborated with elementary school math teacher Gagas Ceria in 2012. The three of us discussed what factors underlie the changing teaching culture so as to make teachers have an independent way of thinking. At that time there was an idea about building a teacher’s belief system: How teachers viewed mathematics, how to learn it and teach it. As previously thought, actually gagasceria teachers and principals have a certain conception of all three things. However, they do not yet believe in what should be understood and done. We also have dialogue guided by fundamental questions such as what mathematics is, what is learning and what is teaching.

Next discussion we try to design the volume learning of a wake. At that time I was the one who guided the discussion by digging into the conception of what volume means, how do students think about having that concept, how to compare the volume of one wake to another wake, how do students get to the volume unit, how do students get to the standard unit of the volume, how do students associate the volume number with the sized numbers (length, width, height) of a beam?

Based on that critical design we are trying to implement it. His observers came from other schools, including LESSON Study activists from UPI. While there are some valuable lessons, such as the importance of interpreting essential concepts and predicting and anticipating student responses, there are some things that require further attention. The planning discussion has not yet reached discussing what is known as a didactic situation or teaching stage. The didactic situation consists of several stages, namely action situations, formulations and validations. In fact, students have difficulty formulating and validating concepts and links between numbers associated with volume and numbers that become sizes (length, width, and height) of blocks.

The analysis became a discussion in revising the learning of the next year’s beam volume. We focus on associating the number of block volumes with the size numbers of the length, width and height of the beam. While the concept of beam volume was presented by the teacher concerned at the meeting the day before. The assignments to be given to students as well as the order in which they are presented, the media to be used are elaborated in detail. For example, students are assigned to look for different pairs of three numbers that result in 8. Number 8 is the number chosen by the model teacher with which the number is not very large so that all students are able to register all possible pairs of the three numbers, namely, 1, 1, 8; 1, 2, 4; 2, 2, 2. I would argue the number 8 only has a prime factor, which is 2. Is it not better to choose a number that has a prime factor of more than 1, for example 2 and 3, such as the number 12?

Once agreed to use the number 12, then how to present it? Simply written on the board, or in the form of paper that can be affixed to the board? How big is it that it is visible to all students but does it waste a place on the board? How do students write answers on the board? Is it what students want or need to set up through the media? We also try out the paper size either that will be pasted directly or written by students. At the event, the student’s response is expected to be to find all possible combinations of the original three numbers resulting from 12.

The second task in groups of students discusses at least two ways of calculating the cubes of units contained in transparent blocks. Originally, the designed beam measured 2 × 4 × 8. But based on the discussion the team is set to × 2 × 5. The main reason is that the learning process in the classroom will be divided into six groups of students, to make six blocks size 2 × 4 × 8 required 384 cube units, while cube units are available in schools of about 200 pieces. Finding a unit cube whose precision is the same as the size of an existing unit cube is quite difficult, if buying a new unit cube is a waste, the existing unit cube becomes useless. Another reason, again related to the coposition of prime numbers, numbers 2, 4, and 8 has only a prime factor of 2, so 64 also only has a prime factor. We want composite numbers that contain more prime factor but not too large. The number 30 has three prime factors namely, 2, 3, and 5 and is not very large. Expected student responses include crossing one-on-one cubes ranging from 1, 2, …, to 30; crossed the rectangular surface of 2 × 3 (6 cubes of units) and found 5 rectangular layers containing 6 cubes of units so that the number of cubes was 6 × 5 = 2×3×5; or find a rectangle measuring 2×5 with 3 layers, or a 3×5 rectangle there are 2 layers. Whichever way the results are used is the same, which is 30. From there the teacher will direct that the number of cubes of the unit is called the volume of the beam. If one unit cube is measuring (1 cm × 1 cm × 1 cm in volume 1 cm^{3),}then the volume of the beam is 30 cm^{3}.

On the third assignment students were given a picture of blocks equipped with a 3×4×5 grid. Students are asked to calculate the volume of the beam. The expected response is that students can calculate the volume by imagining the props used when spelling out the second task so that it obtained the volume of the beam (3×4×5) cm^{3} = 60 cm^{3}. Furthermore, students were given images of blocks without grids but available the size of the ribs, 6 cm, 4 cm, 3 cm. Students are asked to calculate the volume of the beam. The expected response is the volume of the beam is the multiplication of three numbers of each rib size i.e. (6 × 4×3) cm^{3}= 72 cm^{3}.

The fourth last assignment, the student at this meeting, the student is given an L-shaped room wake image and given a grid as seen in Figure 1. Students were asked to calculate the volume of the space’s build in at least two ways. Each way is related to partitioning the build into two block builds as seen in Figure 2 and Figure 3, or into three block builds as seen in Figure 4. To calculate the volume of the build is to hang each beam and then sum it up. Another way of calculating the volume of such blocks is to calculate 6 ́2 ́6 minus the volume of the beam sized 3 ́2 ́4 (Figure 5).

**(continued)**