INTERNATIONAL PERSPECTIVE ON DEVELOPING METHOD IN UNCOVER PSYCHOLOGICAL PHENOMENA OF LEARNING MATHEMATICS

By Dian Permatasari

In this world, there are many countries. Because of it, many phenomena of psychology happen. It is needed to be uncovered. To uncover the psychological phenomena especially in mathematics education, it is needed to develop the method. Many countries in this world, so it also have many perspective about developing the suitable method for each county.

In Australia, Peter Gould is written in ‘Fraction Notation And Textbooks In Australia’ that tell us about how to teach the students about fraction. Fraction is one of difficulty aspects of learning mathematics. It represent by one whole number written above another whole number,  where a and b are integers and b ≠ 0. Kieren argued that from the point of view of curriculum is needed 4 sub constructs of rational number, that is, part whole, quotient, ration number, operator, and measure. In this case, there are two types of fraction, that is, partitioned fraction and quantity fraction. Partitioned fraction is fraction that from by partitioning objects in to b equal parts and selecting a out of b parts to arrive at the partitioned fraction a/b. Quantity fractions express fractional quantities and refer to a universal measurement unit, similar to the way that meters can operate as a standard measurement unit. In Australia, fraction is learned through the contexts such as sharing food or use a model. The shapes of it can be circle or squares. The textbook in Australia use regional model to introduce fraction, some students attend to the discrete, countable features of the area models. Further, the underpinning idea of area as a quantifiable attribute is frequently not taught before students are expected to make area comparisons through the interpretation of regional models. This is one of the method to uncover the psychological phenomena that happen when the students learn about fraction.

In the other case, Mr. Isoda Masami define about uncover the nature of mathematics with use the characteristics of the mathematics like +, :, x, - , and etc. Mathematics must approach the real situation or a context problem. Its means that the students must organize, identify the mathematical aspects of the problem, and discover answer and relations. After that, the students can develop the concepts. Following figure illustrates the process of develop basic concepts of mathematics or formal mathematical language.

 Guided Reinvention model

Reflection is the also the other method to uncover the psychological phenomena. Reflection starts from contextual problems. By implementing activities such as solving, comparing and discussing, the student deals with representing a relation in a formula, proving regularities, refining and adjusting models, using different models, combining and integrating models, formulating a mathematical model, generalizing and ends up with the mathematical solution. Then, the student interprets the solution as well as the strategy which was used to another contextual problem. The teacher examine it and give the reason why it is wrong and why it is correct. The mathematics lessons that will be designed in Realistic Approach should represents the characteristics of how the students do matematisation. The way of embed these characteristics into the lesson plan components can be seen in the following diagram:

Mr. Isoda about iceberg. Iceberg is one method to learn about fraction.

The iceberg is modified by Mr. Marsigit and become like in above :

Beside it, in the other side of the world, visual discrimination is found as alternative method in learning. Visual Discrimination is the ability to determine differences between objects or symbols by sight or the ability to discriminate items from a background. The example of visual discrimination skills is Henry is playing in the BLOCKS area with two cars. The cars are exactly the same model but are different colors. You approach Henry and ask, “Henry, how are those two cars the same?” Henry begins to explain that they each have two doors, a roof, a steering wheel, and four tires. Next, you ask, “Henry, how are the cars different?” Henry notes that they are different colors – one is red and one is

blue. In this situation, Henry uses his visual discrimination skills to determine the likenesses and differences between the two cars. Visual determination skill can provide the strong foundation dor future success in learning.

Do you know what the role of the teacher? The role of the teacher is to develop the teaching learning scheme. Teaching learning scheme is one of the methods that can be developed by the teachers. Teaching learning scheme comprise with

                    i.            Lesson plan

                  ii.            Apperception

                iii.            Small Group Discussion

                iv.            Varian method

                  v.            Varian interaction

                vi.            Varian media

              vii.            Students reflection

            viii.            Scheme of achieving

                ix.            Student’s conclusion

                  x.            Student’s worksheet

Mathematical thinking is also the method to uncover the psychological phenomena. Shigeo Katagiri defined mathematical thinking as an understanding of the necessity of using knowledge and skills and learning how to learn by oneself, and the attainment of the abilities required for independent learning. There are many type of mathematical thinking that defined by Shigeo Katagiri, that is :

A.     Mathematical Attitudes

1.    Attempting to grasp one’s own problems or objectives or substance clearly, by oneself

1)        Attempting to have questions

2)        Attempting to maintain a problem consciousness

3)        Attempting to discover mathematical problems in phenomena

2.    Attempting to take logical actions

1)        Attempting to take actions that match the objectives

2)        Attempting to establish a perspective

3)   Attempting to think based on the data that can be used, previously learned items, and assumptions

3.    Attempting to express matters clearly and succinctly

1)        Attempting to record and communicate problems and results clearly and succinctly

2)        Attempting to sort and organize objects when expressing them

4.    Attempting to seek better things

1)        Attempting to raise thinking from the concrete level to the abstract level

2)        Attempting to evaluate thinking both objectively and subjectively, and to refine thinking

3)        Attempting to economize thought and effort

B.     Mathematical Thinking Related to Mathematical Methods

      1.      Inductive thinking
2.      Analogical thinking
3.      Deductive thinking
4.      Integrative thinking (including expansive thinking)
5.      Developmental thinking
6.      Abstract thinking (thinking that abstracts, concretizes, idealizes, and thinking that clarifies conditions)
7.      Thinking that simplifies
8.      Thinking that generalizes
9.      Thinking that specializes
10.  Thinking that symbolize
11.  Thinking that express with numbers, quantifies, and figures

C.     Mathematical Thinking Related to Mathematical Contents
1.      Clarifying sets of objects for consideration and objects excluded from sets, and clarifying conditions for inclusion (Idea of sets)
2.      Focusing on constituent elements (units) and their sizes and relationships (Idea of units)
3.  Attempting to think based on the fundamental principles of expressions (Idea of expression)
4.    Clarifying and extending the meaning of things and operations, and attempting to think based on this (Idea of operation)
5.      Attempting to formalize operation methods (Idea of algorithm)
6.      Attempting to grasp the big picture of objects and operations, and using the result of this understanding (Idea of approximation)
7.      Focusing on basic rules and properties (Idea of fundamental properties)
8.  Attempting to focus on what is determined by one’s decisions, finding rules of relationships between variables, and to use the same (Functional Thinking)
9.   Attempting to express propositions and relationships as formulas, and to read their meaning (Idea of formulas)

Ebutt and Straker define the cognitive development as a one of the nature of the student. Cognitive development consist of

      1.      Formal geometries : formally defined object (deductive proof)
2.      Euclidean geometries : imagined platonic object (Euclidean proof)
3.      Practical geometries : described real world object (in hierarchy) (geometry construction)
4.      Shape geometries : perceived the prototypes (can be differences) (perception)

Not only it, but also the teacher must be know the nature of learning mathematics :

      1.      Self : the students learn by themselves.
2.      Collaboration : teacher and student
3.      Motivation : attitude
4.      Context : mathematics characteristics

In above is some method that developed in some country. There many method that be developed in this world. Every country develops their method. On developing the method, each countries has their perspective. They think that the method is suitable if it applies in their country. This method is used to uncover the phenomena that happen in teaching learning process especially in mathematics learning. So, perspective of each country influence on developing method in the other country to make an alternative method that better than before.

References

Peter Gould. Fraction Notation And Textbooks In Australia.pdf. Australia

Marsigit, Atmini Dhoruri, Sugiman, Ali Mahmudi. LESSON STUDY: Promoting Student Thinking on the Concept of  Least Common Multiple (LCM) Through Realistic Approach in the 4^th Grade of Primary Mathematics Teaching.pdf. Indonesia

Dr. Marsigit. Pengembangan Kompetensi Guru Matematika Melalui Model-Model Pembelajaran, Lesson Study dan PTK Melalui Peningkatan Peran MGMP.pdf. Indonesia