Behind Good Mathematics Learning – Part 1 : Conception of Learning (by Endang Mulyana)

There’s one question that’s been raised since I became a student three decades ago: What does teaching good math look like? There are indeed many people who talk about the various criteria of good math teachers. However, until now no one has ever shown such a teacher figure.

How do we understand the conception of good teaching? Before answering it, do we first define how good math learning is? More than that, the root of the problem is how we view mathematics. Is mathematics procedural or relational knowledge? Both conceptions will determine whether the learning of mathematics is instrumentalist, structuralist, constructivist or other forms.

I often interact with master teachers who are considered role models for other teachers. It’s actually not surprising that I finally concluded that teachers are more likely to be mechanistic. They teach mathematics as a use of the calculation formula, not teaching thinking. This makes it difficult for prospective teachers to get a good picture of learning conception around them. When they were in school, that’s the kind of picture they captured. I find it difficult for them to explore and develop the meanings of the mathematical problem-solving process. It seems that they prefer to listen to explanations of concepts, theories, examples of questions and work on the questions given by lecturers. As a result, when they are invited to discuss lecture material, they do not master it, do not be confident and avoid with reason to have forgotten. It seems that math learning has taught students to forget!

Conception of Learning

Participation in Lesson Study (LS) activities in 2008 paved the way. Indeed since then it is still difficult to find an Indonesian teacher who has a good conception. But from there I gained inspiration about what and how good learning is. Little by little the conception began to awaken especially after analyzing the math learning video for grade 2 elementary school in Japan.

The first impression of the learning situation is the attitude of students who are orderly, polite and enthusiastic to learn. When the teacher asks, the student hands down without saying an answer. They are waiting to be chosen by the teacher. When someone is elected, he will stand up and answer. Choosing a student to answer is a teacher’s prerogative but is based on reasons related to the overall learning flow.

In the video the teacher shows 5 cards the size of A6 paper without the slightest talking. The male teacher toyed with the card like a magician while waiting for some students. Some students put their hands up hoping to be chosen by the teacher to be given a card. Then the teacher chose a man and a woman to come forward. Male students are asked to select a card, view and remember its contents and hide it so that other students do not know the number written on the card. Then the male student was asked to make a puzzle. “The number on my card is two plus two”. Almost all students put their hands up. The teacher pointed to a student mentioning his guess. “Number four,” the student said. Then the teacher asks the student holding the card to show that the student’s guess is correct. Eventually the teacher put the paper on the board. The same thing was done to female students. The student gave me the puzzle “The number on my card is shaped like glasses”. As with the previous situation, almost all students put their hands up and the designated student guessed it as 8.

What did the teacher do that activity for? Opening lessons should be packed with interesting and challenging. It shows that learning math is fun because students are invited to think. Students who hold cards think spontaneously to create puzzles. In front of the class this student was not given space to imitate or ask a friend’s opinion. He independently created puzzles for other students.

Next the teacher invites students to look to the board. There are numbers 4 and 8 as follows.

4 (     ) 8 = (       )

The teacher asks students to think about the signs of surgery on (    ) between the numbers 4 and 8, and fill in (    ) to the right of “=” as a result. A designated student selects the “add” operation and the result is 12. Then the teacher asks the student’s opinion, is the result 4 + 8 equal to 8 + 4? Most students think the results are the same, but some students say they are not the same. He was also asked to explain and prove it with props. Through these discussions he finally understood that 4+8 = 8+4 as a true fact.

When entering the core activities, the main task is to process the data presented in a carton about the types of activities students help their mother at home. There are 32 images representing 6 different activities, namely; (1) sweep, (2) mop, (3) wash dishes, (4) wash clothes, (5) iron, and (6) tidy the bedroom. Students were asked to look for images that appeared 6 times and how to find them.

What’s interesting is the way the teacher presents the task. Initially the teacher took 6 different pictures (a size A5 and the picture was hidden against the student). The teacher asked “What are you used to do when helping mom work at home?” One student replied that he helped mom wash dishes. The teacher attached a picture of washing the dishes he held on the board. Next appoint another student. When the student replied to wash the clothes, the teacher stuck to the picture about it. Thus, up to the six images are affixed to the board.

Why did the teacher decide to prepare six pictures of the activity? Why does the teacher ask the student first before pasting it? It looks like the teacher had predicted it; maybe before he secretly surveyed it. He intends to surprise and show attention to their activities in their respective homes. Students get more enthusiastic when the teacher slowly opens the roll of paper (size A0).

When the data is presented in its entirety, the teacher explains the task that the student must do. He also made sure all students understood the task, namely looking for the image that appeared exactly 6 times, and how to find it. Each student is given paper (A3 size) as a duplicate of the image (data) affixed to the board and they are asked to work on it for 3 minutes only. When the alarm goes off the stopwatch goes off, most students are not finished yet.

After being given an additional 2 minutes, the teacher asks a student to explain and demonstrate how to get the answer. When the student came forward, the other student was asked to pay attention. It turned out that the student’s answer was not precise because the image allegedly appeared 6 times when it was crossed and has been relatively unmarked. The teacher told the other students to come forward. The student created a table consisting of six columns. Each column represents each image. The table is unusual, it looks like an original creation of its own. The column title below is represented by the image, and the number of images that appear is at the top. He was also asked to explain how to fill the table: cross-marking the copied image in the table. After that, all students are asked to create a table and fill it in, while the student finishes his or her work on the board. In the last session, the teacher explained about the benefits of the table and how to fill it out. With the table students can determine which images appear the most, and how many times each one appears accurately.

That learning situation provides an important picture. Teachers provide learning experiences, not information. From that experience students find ideas complementary and well managed by teachers. Students can present data from the shape of the image scattered into the form of a table along with how to fill the table (perform tally),systematically. The students’ ideas were well presented on the board. Even the teacher never deleted it. At the end of the lesson students write down their learning experiences.

The video mentioned above is one of the references to good math learning. There are indeed many references to learning theory. However, we will still have a hard time imagining it let alone designing it if it does not engage observation and reflection on a learning, either directly or indirectly (video). Without it we would never have the desired conception of learning. If that happens then the teacher will not be able to develop a lesson plan. (continued)