# >> Behind Good Math Learning – Part 4 : Accompanying Prospective Teachers (by Endang Mulyana)

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In addition to accompanying teachers, I also guide the research of students who use the DDR method. At that time I directed students to design learning based on learning trajectory analysis. However, the obstacle is the low mastery of students about the concepts and principles and structure of the material. I also asked them to read the reference as I did when analyzing the line equation.

One student chose geometry learning problems because he suspected textbooks at school had not considered geometry thinking levels. According to van Hiele, geometry thinking consists of five levels namely, introduction/visual, analysis, informal grouping/deduction, deductiti and rigor. After being checked through instruments developed by Usiskin, it turns out that some grade VII students master the first level, namely visualization. More than that, many students also do not meet the first level. Whereas students of the same age in other countries at least reach the level of analysis. They are the same age but different experiences.

In order to have a conception of the level of thinking, I asked students to analyze the related learning videos: how the structure of the assignment developed, how the student responded, how the pattern of interaction between students, students and teachers, how effectively the use of media, and most importantly, were all students encouraged to think?

I think learning it is thinking so teaching it invites students to think. If students are present in class but are not involved in thinking, just hearing or recording without understanding, that means imitating what the teacher thinks. The student has not fully learned. The characteristic of students who learn (think) is that they will answer when asked and will ask if they do not understand. If they don’t do one of them, that means the student doesn’t learn.

The second task is the deepening of the concept, and the facts/principles and structures related to teaching materials, the accuracy of meaning and language (context). The analysis was conducted to introduce the process of repersonalization and mapping learning trajectory. I also did a question and answer with students about the triangle and its properties. When asked about what a triangle is, students answer it as a combination of three lines (segments). I also asked again if every three lines can always form a triangle? To answer that question to the student was given 9 pieces of lidi measuring 1 cm, 2 cm, and … 9 cm. He was asked to create a triangle that was 15 cm around and found how many triangles (the size of the sides could not be the same size) that could be formed.

There are actually three different triangles that have a circumference of 15 cm, namely triangles measuring 4 cm, 5, cm, and 6 cm, then 2 cm, 6 cm, and 7 cm and triangles measuring 3 cm, 5 cm, and 7 cm. Not just any of these lidi whose size is 15 cm can be triangular, for example 1 cm, 6 cm and 8 cm or 1 cm 5 cm and 9 cm. Students are beginning to understand that not every combination of three numbers can always be the size of the sides of a triangle. I also ended the discussion by asking one key question: how can the three numbers be the size of the sides of a triangle? In order to lead to the discovery of learning trajectory, I also added a few more questions: What are the properties of a triangle?; What is an equal-sided triangle?; Mention the properties of the same side?; What is an equal triangle of legs?; Mention the same properties of the foot?; Based on its properties, how is the relationship between a set of triangles, a triangle of equal legs and a set of equal sides? Draw in a Venn diagram!; What is an elbow triangle?; What are the properties of the right triangle?; What is a blunt triangle?; What are the properties of blunt triangles?; What is a taper triangle?; How are the properties of the taper triangle?; Based on its properties, how does the set of triangles, square triangles, blunt triangles and sets of taper triangles? Draw it in a Venn diagram!

Such exploration helps students find an idea of what they will teach. At a later meeting, we began discussing how to formulate a learning goal on this triangular topic: students can classify the types of triangles depicted in a Venn diagram. I asked again, how many meetings would students have in order to achieve that goal? Students mentioned three meetings with the following details. The first meeting discusses the triangle and the nature of the triangular side size, the second meeting discusses the concept of the same triangle, the same side, the elbows, the taper and the blunt triangle, as well as the properties of the triangles of the same side and the triangle of the same foot, and describes in a Venn diagram the triangular set, the triangle of the same side, and the same set of triangles of the foot. While the third meeting explores the properties of the taper triangle, blunt and elbows, and describes in a Venn diagram the four sets of triangles. Next explore the combination of triangle types based on the size of the sides depicted in a Venn digram into the five sets of triangles.

After getting such a detailed picture, the next question is how will the task structure be presented from each meeting, and how do the tasks sequence? To find the facts about the size of the sides of the triangle, i.e. the requirement of the size of the line to form a triangle, students have the following ideas: 1) Create various triangles of various sizes so that students come to the conclusion that not every three lidi form a triangle; 2) By observing the three lidi that make up the triangle, students can guess the requirements of the size of the lidi-lidi in order to create a triangle; 3) Based on the allegations, the students proved it through a variety of different sets of three lidi; Dan 4) When given three different numbers (think of the same unit), without using the lidi again, students can determine which sets (of three numbers) can form triangles.

In the next meeting we discussed how to create an initial learning situation to make the structure of the task interesting and challenging for students? In fact, the structure already qualifies didactic situations, ranging from action siatuation, formulation, and validation, as well as providing concrete and abstract thinking experiences. However, conceptualizing it familiarly to students is not easy. I tried to help him with a magic square in the form of a puzzle to fill the square square 3×3 to fill by the numbers 1, 2, …, 9, so that the number of columns, rows and diagonals is the same, but there should be no box containing the same number.

The next task is to predict the student’s response to the assignment given, and what help to give to students when students get into trouble. Students again ask for time to think and until the next meeting. Based on experience, show the following facts. First, if all students do not manage to fill the magic square correctly, the teacher instructs the student to fill in the number in the middle box, by asking, “when numbers 1 through 9 are sorted, which number is located in the middle?” Through that question, it is expected to fill the box in the middle with the number 5, and the other box is filled in by means of trial and error. Second, if a student succeeds, to fill the box in the middle, the teacher asks the student to create a puzzle that he answered number 5.]

The above activities are new related to the design of the material, not yet scenario packaging related to: 1) how to manage student interaction with teachers, interaction between students, so that students are always focused and enthusiastic thinking about solving problems given during learning; 2) how to allocate time for each stage; Dan 3) how to package it in terms of the selection of sentences in giving questions or briefings. I realized the student didn’t have enough flying hours. Therefore, I ask students to write down what sentences to use, what actions to take as well as an overview of students’ feelings. I also as a mentor do not expect much for the perfection of the learning process. At least the student produced the original lesson plan so that the video recording of the learning process can be the study material of other students, teachers and lecturers. (continued)