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

Learning mathematics is not only just shared and learns about numbers, symbols, and formulas, but also it can be a media to know the students psychology. It can be done if in mathematics learning there are concrete, scheme, and modeling, not only formal learning. Teachers know the students psychology using the characteristics of mathematics like addition, subtraction, division, and multiplication.

Two important things in mathematics, they are mathematics close to daily life (realistic approach that refers not just to the connection with the real-world, but also refers to problem situations which real in students' mind) and mathematics as a process of guided reinvention (where students can experience a similar process compared to the process by which mathematics was invented). Lesson plan components in mathematics in realistic approach consist of

Based on figure above, mathematics learning start from introduction, continued creation/develop symbol model, reason and explanation, and the last is closing/application in daily life. Each of the steps, there must reflection. It because reflection is the most effective method in mathematics learning to know student understanding and students psychology. 

Mathematics trough realistic approach can be done by:

a.       Contributed

b.      Model

c.       Simultaneously

d.      Interaction

e.       Strands of mathematics

In guided reinvention model, learning process start from contextual problem, and then student will describe and solving mathematics. Doing that, they use formal mathematical language that consist of mathematical language and continued to make a solve algorithm.

Learning mathematics is not only by student themselves, but teachers as a facilitators have very important roles. Mathematics teachers play role to improved teaching material. Besides that, teacher is also applying teaching learning scheme in mathematics learning. It consists of:

1.      Lesson Plane

2.      Apperception

3.      Small group discussion

4.      Various method

5.      Various interaction

6.      Various media

7.      Student Reflection

8.      Scheme of achieving competence

9.      Student conclusion

10.  Student worksheet

Shigeo Katagiri said that mathematical thinking can be divided into the following three categories:

1.     Mathematical Attitudes

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

b.      Attempting to take logical actions

c.       Attempting to express matters clearly and succinctly

d.      Attempting to seek better things

2.      Mathematical Thinking Related to Mathematical Methods

a.       Inductive thinking

b.      Analogical thinking

c.       Deductive thinking

d.      Integrative thinking (including expansive thinking)

e.       Developmental thinking

f.   Abstract thinking (thinking that abstracts, concretizes, idealizes, and thinking that clarifies conditions)

g.       Thinking that simplifies

h.      Mathematical Thinking Related to Mathematical Contents

i.        Thinking that generalizes

j.        Thinking that specializes

k.      Thinking that symbolize

l.        Thinking that express with numbers, quantifies, and figures

3.      Mathematical Thinking Related to Mathematical Contents

a.   Clarifying sets of objects for consideration and objects excluded from sets, and clarifying conditions for inclusion (Idea of sets)

b.      Focusing on constituent elements (units) and their sizes and relationships (Idea of units)

c.       Attempting to think based on the fundamental principles of expressions (Idea of expression)

d.   Clarifying and extending the meaning of things and operations, and attempting to think based on this (Idea of operation)

e.     Attempting to formalize operation methods (Idea of algorithm)

f.       Attempting to grasp the big picture of objects and operations, and using the result of this 

understanding (Idea of approximation)

g.       Focusing on basic rules and properties (Idea of fundamental properties)

h.   Attempting to focus on what is determined by one’s decisions, finding rules of relationships between variables, and to use the same (Functional Thinking)

i.        Attempting to express propositions and relationships as formulas, and to read their meaning (Idea of formulas)

Beside explanation about mathematical thinking by Shigeo Katagiri, David Tall was explain how student learning mathematics. He used cognitive development scheme:

 

From figure above, perception as the starting steps to learn mathematics, and continued to perceived prototypes, freehand drawing. After that, described real world object like geometric construction. Next, student imagined platonic object, Euclidean geometry. And the last is formally defined object deductive proof.

Although we had explained above, we must know well about the characteristics of mathematics itself. There are characteristic of mathematics:

1.                        1. Self (the student itself)

2.                        2. Collaboration (teacher and other student)

3.                        3. Motivation (attitude)

4.                        4. Context (mathematics character)

Based on all of explanation above, it can conclude that mathematics learning is not only media to give knowledge (mathematics) but also with mathematics learning, we can know psychology of our students. It is helped by some explanation that explained by Shigeo Katagiri and David Hall. As long as mathematics learning is doing, there are some psychology aspects that we can investigate.

References:

Katagiri Shigeo. 2004. Mathematical Thinking and How to Teach It. University of Tsukuba.

Marsigit dkk. 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.