International Perspektif on Developing Method to Uncover Psychological Phenomena of Learning Mathematics

By Reni Dwi Astuti

Each country has unique culture that influence the style of mathematics teacher. In this case, the style of mathematics teachers are about the way or the method of mathematics teacher teach students. They use method of teaching that suitable with social interaction and surrondings. For example, teacher in Indonesia that teach mathematics in elementary class give example of the addition problems is about how many the books that on the table and on the chair. The teacher use anything that are near of the student.

Although each country has different way of teaching mathematics, one thing that is not different is the concept of mathematics. There are many concept in mathematics like multiplication, addition, subtraction, division, etc. It means that mathematics is absolute. The truth of mathemtics is not change wherever and whenever mathematics used.

The student in school think the simple of mathematics. They only apply the formula that given by the teacher. University students think more complicated than student in school. They not only think about the application of the formula but also think about the axiom. They must prove the theory or the axioms that given by the lecture. Based on David Tall theory, there are three world of the student in the transition process from mathematics school to mathematics university.  They are conceptual-embodied, proceptual-symbolic and axiomatic-formal.

In mathematics school the conceptual embodied world can be done with some activity. The students learn about concrete object of mathematics. It starts from elementary school, the students learn mathematics with recognition pattern in shape and numbers. The recognition has relationship with geometry and graphs. They think about the concrete objects. The students also do repetition when they learning mathematics in the school. They do routinise of sequence action when they build the concept of arithmetic and algebra. After the students know about object of mathematics, they can make description of them. The students know the characteristics or properties of the object. They also can diferentiate one object with the others, for example the students can mention the different properties between triangle and square. From the properties that have been mention the student can make definition of the object in mathematics. In this case, the definition of the student is base on their knowledge that have been got before.

Proceptual-symbolic world is the growth of conceptual-embodied of the student world. This world is built by student by doing action of thinking concept. It is relationship between process and concept. In the other word, this world can be called procepts. In this world, the student learn about aplication of the formula or the concept in the process to get knowledge. After student know about the symbol in mathematics, they will apllied it to solve the given problems.

The axiomatic-formal world (based on formal definitions and proof), which reverses the sequence of construction of meaning from definitions based on known objects to formal concepts based on settheoretic definitions (David Tall 2008). The subject of this world is adult students that have capablility in math. The university students learn about pure mathematics. They prove that the axioms are right or false by using the given definition. The object of mathematics university is abstract. It’s sophisticated enough for the student. The example of this world, the mathematics student in university must be prove that two line make a plane or not if they intersect each other. In this problem, the student must know the definition of the plane. After they know the definition, they can prove  the given problem. Picture below is a scheme of cognitive development that have been explained.

The explanation before is about the mathematics student when learning mathematics. The next is about mathematics teacher when they teach the student in the class. The mathematics teacher must know three kind of mathematical thinking. This theory is given by Katagiri. Here, teacher not only know about mathematical thinking but also know how to teach mathematics to the students. There are mathematics attitude, mathematics method and mathematics contents. Mathematics attitude is about the attitude of student that related to mathematics, like students account the pocket money from their parents, remember their shoes size, sum their friend, divide their friend the small groups when discussion etc. All of that need mathematics, without it they can’t do that. The next is mathematics method. Method is used by teacher to teach the student. Commonly, Mathematics teacher in Indonesia do explanatory method when they teach. It’s old method, that have been done. The last is mathematics contents. Contents of mathematics is about the propesties of mathematics. It is not only about calculation like division, multiplication, subtraction and addition but also about all of the material like fraction, geometry, differential, integral, etc.

The teachers also must know the role of their self to develop teaching material. The mathematics teacher must develop their ability in some scheme. Base on Mr. Marsigit explanation, the scheme are about lesson plan, apperception, small group discussion, various method, various interaction, various media, student reflection, scheme of acheiving, student’s condition and student work sheet. In this article will be axplain them one by one.

Lesson plan is a plan that used by teacher to arrange the material and the time when teaching. In the othe word, lesson plans use to communicate instructional activities regarding specific subject-matter. Almost all lesson plans developed by teachers contain student learning objectives, instructional procedures, the required materials, and some written description of how the students will be evaluated. The teacher must develop their skill of making lesson plan. Lesson plan must be effective when applied.

Apperception is readiness of the student and the teacher to do teaching learning in the class. The teacher must be ready to give the material to the student. The teacher must prepare all of the material and the mehod that will be use. The students also must be ready to get knowledge from the teacher. They prepare it with study of the topic that will be given by teacher in home. If the students and the teacher ready to have teaching learning, so the process can pass easily.

Small group discussion is done by the students in goup, the member of group consist of two students until five students. The effective group is the group that have three members. Using small groups of students to work on activities, problems and assignments can increase student mathematics achievement. Teacher as fasilitator must arrange the the students into small group and give the rule. Then the teacher give freedom to the group to think about the given problems.

The teacher must use various method  when teaching mathematics. Many method that can be used in teaching learning process. The taecher choose the method base on the condition of the students, the material that will be given, the capability of the student, student backgound etc. For example, there are traditional method like axpository or explanation method, discussion method, investigation methods, etc.

Teacher use various interaction when they teach in the class. There are some interaction that be done by teacher when teaching math. For example, interaction between teacher and the student in whole group. It’s mean that the teacher must manage the big group and touch their heart to learn about math with spech in front of the class. The teacher imposible ask the student one by one in the large class. Another example is interaction between teacher and the student in small group. It had axplained before.   

Various media is used by teacher to explain any topic in math. The purpose is that help the student to understand the topic. The teachers must be capable in this skill. They can prepare before they teach to use any media. The media can make the teaching learning process more interest. The students also don’t fell bored.

Student reflection is form of student participation. The reflection can be done when teaching learning process going on. Reflection is like retell the material that the teacher give. The teacher must give opportunities to the student to give their reflection of the material. The teacher also give the same opportunities to all of the student. So, the student will spech their reflection without any discrimination.

Scheme of acheiving used to know what the students and the teacher want from the teaching-learning process. They must make a scheme of acheiving.

Student’s condition is the other aspect that must be paid attention by the teacher. The condition of the student and the others are different. They come from different family. They have different backgroud and different ability. The teacher can’t consider that all of the student have same ability. So, the teacher can’t apply only one method when teach the students in the class.

The teacher must have ability to make student work sheet. Student work sheet is used to help the techer and the student in teaching-learning process. The content of student work sheet are the explanation of the topic, the assigment that must be done by students, and the exercise for the students. The student can construct their world of math by doing exercise and the othe activity in the student work sheet. The role of the teacher is that make suitable student work sheet. Only the teacher can make the student work sheet that suitable with condition of the students and the school.

After the teachers know about their role when teach their student, we hope they will improve their skill to make teaching-learning be better than before.

Refferences:
www.merga.net.au/.../MERJ_20_2_Tall.pdf
Douglas A. Grouws and Kristin J. Cebulla.Improving student achievement in mathematics. From www.iaoed.org/files/prac04e.pdf -
Dr. Bob Kizlik. 2011.Lesson Planning, Lesson Plan Formats and Lesson Plan Ideas.From http://www.adprima.com/lesson.htm.