How to Uncover The Psychological Phenomena

Traumatic is a phenomena which is unknown its beginning. This behavior appears due to communication problems. This mistake happens because there are competence differences among people and the differences of their jobs. Traumatic commonly can be identified with strange behavior and the hidden reason appearance from what they have done. It can appear with any explanation or not. This attitude happens to everyone when there is something influencing him/her to do it. For example, because his friend betrayed him, Joel felt disappointed. Then it make him don’t believe with other easily. But, not all traumatic gives bad impacts. There are also positive impacts for some people. We have an example. Every week, we get some assignments in our lecture. We have to collect them in the previous day before the lecture. Initially, it stressed us. Then we get positive influence from that activity. We study more diligent than before. We must have strategy to divide our time to do all of our tasks.

In political view, traumatic is known with the term authorities. This word can’t be explained more detail again. But, traumatic is called accident in philosophical view. An accident is an impact from something which exists. Truly those three words have the same meaning, however their applications are in different zones. Traumatic can be solved by communication. The communication will make the misunderstanding doesn’t occur again.

In communication, there are three elements. They are sender, receiver and constraint. Those elements are in contact to appear giving-receiving activity. In hermeneutics, communication is called spiral dynamic. Communication is flexible and contextual. There is a reality appearing at this communication. Then it causes a readiness to face the challenges. This readiness make someone has an awareness. Beside that, it also makes apperceptions happen. Apperceptions are started by the sensation, a process when the stimuli enter to human sensors. After apperceptions process, there is perception, a process to translate the stimuli which enter to our sensors. Then, it will cause a concept about something appearing in our mind.

Concepts are divided into empiric concept and apriority concept. Empiric concept is a concept which appears after there are some valid proofs. On the other hand, apriority concept usually appears naturally without any stimulus. We have understood it, so it hasn’t to be defined again. For example, we don’t need to define the meaning of chair because we know exactly what the chair is. We don’t need to define the word “is” because this word was structured in our mind as the word used to accompany a word which is defined.

In mathematics learning, communication takes an important role. Recently, some mathematics learning still use one direction communication. Students are placed as object. They are like empty glasses which are ready to accept all information from the teachers. If these phenomena happen continuously, there will be unbalancing in learning process. The common mistake happens when the teacher teaches pure mathematics to the students. Whereas, they should be taught using school mathematics which is started from realistic activity. Therefore, as the next teachers, we should have a capability to change mathematics learning methods. We must use an interesting method to make the students aren’t trauma with math. In math, concept is begun from definition. Then, we get axiom. And we can find the theorem from the axiom. To build a concept, we have to think the quantity, quality, category and relation. These four consist of singular, partial and universal. Those all also are on our mind. So, we should develop them.

The conclusion from the discussion above is the world is traumatic. If you want to live, you have to face the traumatic and feel it as strongly as you can with your limit of tolerance. You have to feel what traumatic is like to finding the meaning of life perfectly.

Mathematical Thinking and Scientific Work

As mathematicians, we should understand about scientific work, mathematical thinking and their relationship. Scientific work is an activity done to find something new and different. We also can define that scientific work is products which are produced by scientific researchers. The example of scientific work are scientific journal, scientific paper, research etc. the specific example of scientific work in mathematics is mathematic research. It also has many characteristics.

First, scientific work is impersonal. It is not related to our mood and feeling. If we do scientific work, we should be professional. Although our feeling is not good enough, but we can’t show that in our work. Second, scientific work has standard. SW can be local, national or international work. SW also has writing criteria. And it depends on what kind of the writing. Third, SW must be objective work. And the last, SW has a regulation that SW is free from plagiarism.

What is mathematics research like? It is like the lectures do research related to mathematics. Or maybe we, as mathematics students do research. Mathematical research is very important because with a lot of research, we can find new formulas, methods, renovations and new concepts in Mathematics. To be a good researcher, we must have good mathematical thinking.

Mathematical thinking is a thinking needed to build our curiosity and ability in mathematics. Mathematics is consistent thinking. It means based on the earlier agreement like the definitions, concepts and procedures. Mathematics consists of pattern and relationship, problem solving, communication and investigation. As we know, mathematics can be divided into Pure Mathematics and Applied Mathematics. Now, we are near with pure mathematics. Pure mathematics is more formal mathematics because it has more axiomatic math. Pure mathematics has a ground of a system. Many mathematicians also have different opinions about mathematics. One expert said mathematics is deductive. The order is from concept, axiom, theorem, proof, then we can find the procedure to prove the theorem. The other expert also said mathematics is a system because mathematics has assumption.

The most important in mathematics is we must have object. Where does it come from? Yeah, it comes from our ideas. It comes from our mind. Because mathematics’ object is abstract. Sometimes, we have difficulties to find the object. So, how do we get mathematics’ object?

The mathematics’ objects are made from the possibility existence and the existence. There are two ways to find mathematics’ objects from concrete objects.

1. Idealization

We assume the object is perfect because we need absolutely plane in math. Just assume that the line is straight, the line is absolutely flat. There is nothing which is perfectly formed.

2. Abstraction

We just learn many characteristics which are should learned. If we find an object, we can explain million characteristics of that object. We can have a lot of perceptions abut that thing. But in mathematics, we just talk about the shape and the size, the value of that thing. Because those all which we learn in mathematics.

The characteristic of mathematical thinking is logic. It comes from our daily life. We can differ the smaller and the bigger ones. There are many things which show mathematical thinking is logic. The first is mathematical thinking based on order. Then, relationship is very important. Every element in mathematics has a relationship with the others although the relationship is disjoint. The third is MT related to Arithmetic operation. MT use operator like addition, subtraction etc. MT also contains If…Then… statement. For example if N is odd number and M is even number, then N times M is even number. Then, how to get conclusion of statement is one of proof that MT is logic. We can use minor premise and mayor premise. MT use thesis, antithesis and hypothesis to conclude a statement.

Mathematical thinking and Scientific work are very important for mathematics world. With those, we can find newest mathematics information. We can do scientific work if we have an idea. And this idea comes from our mathematical thinking. On the other side, scientific work can be the ways to publish and introduce mathematical thinking. So, both of them are complete each other.

If we want to be success mathematicians, we should make a development in our world. One way is increase our mathematical thinking. We can have new ideas then realize the ideas on scientific work.

EXERCISES AND WRITING

1.

1. We will prove that the square root of 2 is irrational number.

Answer: Two is not a perfect square number. So, the result of square root of two of course is irrational number.

2. We will show that the sum angles of triangle is equal to 180 degree.

Answer:

· Create a triangle. Then, make a line which is parallel with the triangle’s base on the top of triangle.

· Assume the triangle is ABC triangle. Then the angle’s name is A,B and C. And the angle C is on the triangle’s top. The angle A on the right corner and the angle B on the left corner.

· There is an angle between the line and the right side of triangle which has the same size with angle B because they are intern passing angles. We call it angle B’. there is also an angle between the line and the left side of triangle which has the same size with angle A. Then, we call it angle A’.

· So, we can see that A’ plus B’ plus C equals to one hundred and eighty degrees because it does a complete rotation from one side of the straight line to the other.

· Because A equals A’ and B equals B’. so, A plus B plus C equals one hundred and eighty degrees.

· It is proven that the sum angles of the triangle is one hundred and eighty degrees.

3. We will show how to get phi.

· First, we wrap a thread around a cylinder. Then, we measure the thread’s length.

· To have good accuracy of measurement, we can do it several times.

· After we find the length of the thread, we can assume that the length is the perimeter of a circle.

· We know that the perimeter of a circle equals two times phi times the radius. With the same radius, we can count that the value of phi approximately equals to three times one four.

4. We will show how to find out the area of region bounded by the graph of y equals x square and y equals x plus one

Answer:

· First, we must sketch the graph. But before that we must find the intersect points between y equals x square and y equals x plus two. Substitute y equals x square to y equals x plus two.

Then, we will find x square equals x plus two. Then, add both sides with negative x minus two. So, we will find x square minus x minus two equals zero.

After that, we can find the factor of the equation above. We find x minus two in bracket times x plus one in bracket equals zero. So, we find x equals two and x equals negative one. Then, we can find the value of y. If x equals two, so y equals four and if x equals negative one, so y equals one.

The intersect points are (2,4) and (-1,1).

· The area of the region bounded by the equations has lower boundary x equals negative one and upper boundary x equals two.

· After we know the region, we can count the area. Assume A is the area of the region. Make a partition on the graph, then we will find delta A equals x plus two minus x square in bracket times delta x (delta A is found from the subtraction of upper graph y equals x plus two and the lower graph y equals x square in bracket times delta x).

· A equals define integral of x plus two minus x square from lower boundary x equals negative one to upper boundary x equals two.

· A equals a half times x square plus two times x minus one third times x cubeb in bracket from x equals negative one to x equals two.

· A equals a half times four plus two times two minus one third times eight in bracket minus open bracket a half times one plus two times negative one minus one third times negative one close bracket.

· A equals two plus four minus eight third minus a half plus two minus one third.

· A equals four and a half.

· So, the area of the region bounded by y equals x square and y equals x plus two is four and a half.

5. We will show how to determine the intersection point between the circle x square plus y square equals twenty and y equals x plus one.

Answer:

· Substitute y equals x plus one to x square plus y square equals twenty.

· We will find x square plus x plus one in bracket square equals twenty.

· X square plus x square plus two times x plus one equals twenty.

· Add both sides by negative twenty. So, we find x square plus two times x minus nineteen equals zero.

· Then, to find the value of x, we can use ABC formula

First x or second x equals negative two plus minus the square root of open bracket four minus four times two times negative nineteen close bracket over four.

· First x or second x equals negative two plus minus the square root of one hundred and fifty six in bracket over four.

· First x equals negative two plus two times the square root of thirty nine in bracket over four. So, x equals negative one plus the square root of thirty nine in bracket over two. So, the value of y equals negative one plus the square root of thirty nine in bracket over two plus one. Y equals one plus the square root of thirty nine in bracket over two.

· Second x equals negative two minus two times the square root of thirty nine in bracket over four. So, x equals negative one minus the square root of thirty nine in bracket over two. So, the value of y equals x equals negative one minus the square root of thirty nine in bracket over two plus one. Y equals one minus the square root of thirty nine in bracket over two.

· The intersection points are x equals negative one minus the square root of thirty nine in bracket over two, y equals one plus the square root of thirty nine in bracket over two. And x equals negative one minus the square root of thirty nine in bracket over two, y equals one minus the square root of thirty nine in bracket over two.

Differential Mean Value Theorem

Theorem

If function f continue on the closed interval [a,b] and is differentiable at the points on the interval (a,b) that there will be at least a number c on the interval (a,b) with differential of function f with x is c equal to function f with x is b minus function f with x is a in bracket over b minus a on the bracket (f’(c)=f(b)-f(a)/b-a).

How can we find this theorem?

There is a proof to prove that the theorem is right.

For example, there is an equation y equal to function f. That function is on the interval [a,b], so, we will find that the corner points are (a,f(a)) and (b,f(b)). There is a line lies an those two points with the equation y equal to function g and has a gradient equal to f(b) minus f(a) over open bracket b minus a close bracket (m=f(b)-f(a)/b-a).

The line equation of y equal to function g through the point (a,f(a)) is

· y-y₁=m(x-x₁) (y minus y one equal to gradient times open bracket x minus x one close bracket)

· g(x)-f(a)=f(b)-f(a) over b-a in bracket times (x-a) (function g minus function f with x is a equal to function f with x is b minus function f with x is a over b minus a in bracket times open bracket x minus a close bracket)

· g(x) equal to f(b) minus f(a) over b minus a in bracket times open bracket x minus a close bracket plus f(a) (g(x)=(f(b)-f(a)/b-a) times (x-a) +f(a))

There is a function S. function s is found from f(x) minus g(x).

Substitute g(x)=(f(b)-f(a)/b-a)times (x-a)+f(a) to s(x)=f(x)-g(x). So, we find

S(x)=f(x)-(f(b)-f(a)/b-a)(x-a)-f(a) (s(x) equal to f(x) minus open bracket f(b)minus f(a) over b minus a close bracket times open bracket x minus a close bracket minus f(x).

S’(x)= f’(x)-(f(b)-f(a)/b-a) (differential of function s equal to differential of function f minus f(b) minus f(a) over b minus a in bracket)

S(x) continue at (a,b) because s(x)=f(x)-g(x) which function f and function g continue on the interval [a,b]

We will show that s’(c)=0 which c is element of(a,b).

S(x) continue on the interval a and b, so function s reach maximum and minimum point at (a,b).

a. If the value of s(x)maximum and s(x)minimum equal to 0 that s(x) equal to 0 that s(x) equal to 0 which every x is element [a,b]. If s(x) equal to 0 that s’(x) equal to 0 which every x is element of(a,b). It means if c is element (a,b) that s’(x) equal to 0.

b. If the value of s(x)maximum or s(x)minimum is not equal to 0 that s(x)maximum or s(x)minimum reached in the point c on the opened interval [a,b]. It can’t be reached on point a and b because s(a) and s(b) equal 0.

Because s(x) has maximum or minimum value on (a,b), it means s(x) have differential in every point at interval (a,b).

So, if c in (a,b) produces s(c) maximum or s(c) minimum that c is the stationery point from function S.

S’(c) equals 0……………(1)

Then look again the formula

S’(x)= f’(x)-(f(b)-f(a)/b-a)

S’(c)= f’(c)-( f(b)-f(a)/b-a)……………..(2)

From 1 and 2, we can find

f’(c)-( f(b)-f(a)/b-a)=0

f’(c)= f(b)-f(a)/(b-a) The theorem is proven.

The problem related to Differential Mean Value Theorem

F(X)=X square plus X on the interval [-2,2]. Find the value of C!

Solution:

F’(X) equals two times x plus one

f’(c)= f(b)-f(a)/(b-a)

2c+1= f(2)-f(-2)/2-(-2) (two times c plus one equals f(2) minus f(-2) over two minus negative two)

2c+1=4+2-4+2/4 (two times c plus one equals four plus two minus four plus two over four)

2c+1=1 (two times c plus one equals one)

C=0 (c equals zero)

How To Collect Data

To find data which are valid and guaranteed, we must collect them with right ways. There are many methods which are used to collect the data:

1. Interview

Interview is a way to collect data by holding direct interface. The interview must be done with interview’s guide which contains questions list based on the goal.

There are 2 kinds of interview:

a. Stuctured Interview is one kind of interview that the varieties and orders from the questions are arranged before.

b. Unstuctured Interview is an interview which is not secured tightly. It is more flexible because the questions can be increased.

The characteristics of good questions are:

· Based on the goals or the problems of the experiment.

· Clear and undoubtful.

· Fit in the knowledge and the experiences of interviewee.

· The questions can’t be related to someone’s personal.

The positive side from interview is the needed data can be received directly, so the data are more accurate and can be responsibled.

The negative side is it can’t be done in big scale and is not easy to find the personal informations.

2. Quesioner ( ) is a way to collect data by sending or using quesioner which contains many questions. The advantage of using this method is it can be done at big scale and can find personal informations. The negative side is the results are not accurate, not all questions are answered. Even, not all quesioners are not given back.

3. Observation (Supervision) is collecting data’s method by observe the experiment’s object or the events( human, no_living creatures or natural indication). The data can be found to know the habits and behaviours of human, no_living creatures or natural indication. The advantage of the observation is the data can be trusted. But, it can be found false interpretation to the observed events.

4. Test and Objective Scale is a way to find data by giving test to observed objects. This way is used mostly in psychology to measure the characteristics of human personality. There are some examples of objective scale test:

a. Intelligence and Talent Test

b. Personality Test

c. Attitude Test

d. Test about Value

e. Academic Test etc

5. Projective Method is observe or analyze an object by external expression from that object in drawing or writing. This method is used in psychology to know the emotion and personality of someone. The weakness of this method is the same object can be concluded variously by different observers.

Rectangle

1. Definition

Rectangle is a two dimensional shape which has two pairs parallel lines and one of its angles is 90 degrees. It is also defined as parallelogram which has four right angles. Rectangle is not always equilateral. A rectangle with vertices ABCD can be written ABCD.

2. The Characteristics of Rectangle

a. It has the same diagonals.

b. Its diagonals intercept in the middle.

c. Its angle has the sum 360 degree.

d. Each angle is ninety degree.

e. Opposite sides has the same length.

f. Opposite sides are parallel.

3. How to find the perimeter and the area of a rectangle

Assume that the length of a rectangle is L and its width is T

· Its perimeter is from the sum of all sides’ size. So, P(perimeter) equals L plus T plus L plus T. Then, we can find P equals two times open bracket L plus T close bracket [2(L+T)].

· Its area A equals L times T (A=L.T)

· Its diagonal’s length equals the square root of open bracket L square plus T square close bracket.

4. Kinds of Rectangle

§ Square is rectangle which has the same length in four sides.

§ Oblong is used to call the non square rectangle.

5. Problems Related to Rectangle

· Find the perimeter and the area of a rectangle, if the length equals 12 cm and its diagonal line is 13 cm!

Solution:

First, we must find the width of rectangle. T equals the square root of open bracket D square minus L square close bracket. T equals the square root of open bracket thirteen square minus twelve square close bracket. T equals the square root of twenty five. T equals five. So, the width of the rectangle is 5 cm.

Then, we can find its perimeter and area.

P=2(L+T)

P=2(12+5) (P equals two times open bracket twelve plus five close bracket)

P=34 (P equals thirty four)

So, the perimeter is 34 cm.

A=L.T

A=12.5 (A equals twelve times five)

A=60 (A equals sixty)

So, the area is 60 cm.

Reflection of Videos

Last meeting, Mr. Marsigit showed us many videos. There were 6 videos and 4 them are related to Mathematics. Then, we’re asked to retell the contents of the videos with our own words. it is very interesting. I hope I can develop my English again. Here, I will write my story about the videos.

1. Video I

Be Different

The title of the video is Death Poet Society. In this video, there was a teacher who talk to his students in a class. He talked about what should we do as human. Shakespeare said that someone has many characters. And the characters can make he or she look at something in different point of view. So, he told that we must see something from different side. We must try to be different. Don’t be afraid to find new ground to increase our self.

2. Video II

If we believe others, we will be trusted easily. As human, we must believe that we have ability to build our self. With trust, we can create many things because we believe that we can. No matter where we come from, if we can trust someone sincerely, we will be loved by others. so, start to believe others with believing our self first.

3. Video III

What you know about math? Math is very complicated. There are a lot of kinds of math. Math is known because of calculator. If we see someone use calculator, we assume that he or she are doing something related to math. Some pieces of math are trigonometry, logarithm, exponents, limit, etc. The most popular in math is Mathematics League. Many students are attracted to join math competition. If we want to success in Mathematics, we must do math works with our self. We should not cheat others work.

4. Video IV

In this video, we can learn how to solve problems related to integral and differential.

Problem: We must get y(dependent variable) from the equation dy over dx equal to four times x square.

Solution:

· First, we move dx to right side of the equation. We multiply both sides with dx. So, we find

dx times dy over dx equal to four times x square times dx.

· Then, the equation becomes dy equal to four times x square times dx

· Then integrate the equation above. So we will find integral from dy equal to integral from four times x square times dx.

· The result is y equal to four third times x cube plus arbitrary constant.

5. Video V

A. Problem: Find the value of x from x-5=3(x minus five equal to three)

Solution: We will find the value of x from the equation x-5=3

· Add two sides by 5.

x-5+5=3+5 (x minus five plus five equal to three plus five)

x=8 (x equal to eight)

So the value of x from the equation above is eight.

B. Problem: Let 2/3 times x=8(two third times x equal to eight). Find the value of x

Solution: We will find the value of x from 2/3 times x=8

· Multiply both sides by 3/2

3/2 times 2/3 times x= 8 times 3/2 (three second times two third times x equal to eight times three second)

x=12 (x equal to twelve)

So the value of x is twelve.

C. Problem: Let 5-2x=3x+1(five minus two times x equal to three times x plus one). What is x?

Solution: We will find the value of x from the equation 5-2x=3x+1

· Subtract both sides by 3x

5-2x-3x=3x+1-3x (five minus two times x minus three times x equal to three times x plus one minus three times x)

5-5x=1 (five minus five times x equal to one)

· Then, subtract both sides by 5

5-5x-5=1-5 (five minus five times x equal to one minus five)

-5x=-4 (negative five times x equal to negative four)

· Multiply both sides by -1/5

-1/5. 5x=-4.-1/5 (negative one fifth times five times x equal to negative four times negative one fifth)

x=4/5 (x equal to four fifth)

So, the value of x is four fifth.

D. Problem: Let 3-5(2m-5)=-2. Find the value of m!

Solution: We will find the value of m from 3-5(2m-5)=-2 (3 minus 5 times open bracket 2 times m minus 5 close bracket equal to negative two)

· Multiply 5 by the number in the bracket, 2m-5

3-5(2m-5)=-2

3-10m+25=-2 (three minus ten times m plus twenty five equal to negative two)

28-10m=-2 (twenty eight minus ten times m equal to negative two)

· Then, subtract both sides by 28

28-10m-28=-2-28 (twenty eight minus ten times m minus twenty eight equal to negative two minus twenty eight)

-10m=-30 (negative ten times m equal to negative thirthy)

· Multiply both sides by -1/10

-1/10.-10m=-30.- 1/10 (negative one tenth times negative ten times m equal to negative thirty times negative one tenth)

m=3 (m equal to three)

So, the value of m is 3

E. Problem: Find the value of x from the equation x/2+1/4=x/3+5/4(x times a half plus a quarter equal to x times one third plus five fourth)!

Solution: We will find the value of x from the equation above

· Multiply this equation by the smallest union multiplication of 2,3, and 4. It is 12.

12(x/2+1/4)=12(x/3+5/4) (twelve times open bracket x times a half plus a quarter close bracket equal to twelve times open bracket x times one third plus five fourth close bracket)

6x+3=4x+15 (six times x plus three equal to four times x plus fifteen)

· Then, subtract both sides by 6x

6x+3-6x=4x+15-6x (six times x plus three minus six times x equal to four times x plus fifteen minus six times x)

3=-2x+15 (three equal to negative two times x plus fifteen)

· Multiply both sides by -1/2

-1/2.3=-1/2(-2x+15) (negative a half times three equal to negative a half open bracket negative two times x plus fifteen close bracket)

-3/2=x-15/2 (negative three second equal to x minus fifteen second)

x=-3/2+15/2 (x equal to negative three second plus fifteen second)

x=12/2 (x equal to twelve second)

x=6 (x equal to six)

So, the value of x is six.

F. Problem: Find the value of x from the equation 0,35x-0,2=0,15x+0,1(oh point three five times x minus oh point two equal to oh point one five times x plus oh point one)

Solution: We will find the value of x from the equation above

· Multiply both sides by 100/one hundred

100(0,35x-0,2)=100(0,15x+0,1) (one hundred times open bracket oh point three five times x minus oh point two close bracket equal to one hundred times open bracket oh point one five times x plus oh point one close bracket)

35x-20=15x+10 (thirty five times x minus twenty equal to fifteen times x plus ten)

· Collect the numbers which have x variable in one side

35x-15x=10+20 (thirty five times x minus fifteen times x equal to ten plus twenty)

20x=30 (twenty times x equal to thirty)

· Multiply both sides by 1/20

20x.1/20=30.1/20 (twenty times x times one twentieth equal to thirty times one twentieth)

x=30/20 (x equal to thirty twentieth)

x=3/2 (x equal to three second)

So, The value of x is 3/2 (three second).

6. Video VI

a) If logarithm base x of A equal to B, so x to the power or B equal to A.

Ø If we multiply logarithm base x of A equal to B by C, we will find

C times logarithm base x of A equal to B times C.

Ø Then, we make quadrant the equation x to the power of B equal to A by C, we will find

x to the power of open bracket B times C close bracket equal to A to the power of C.

So, logarithm base x of A to the power of C equal to B times C.

Ø From two equations above, we can conclude that C times logarithm base x of A equal to logarithm base x of A to the power of C.

b) Logarithm base x of A plus logarithm base x of B equal to log base x of A times C.

We will show the truth of that equation.

Ø Log base x of A equal to L. So, x to the power of L equal to A.

Ø Log base x of B equal to M. So, x to the power of M equal to B.

Ø Log base x of open bracket A over B close bracket equal to N, so we find that x to the power of N equal to A over B.

Ø Then, based from the first and second statement, we can conclude that x to the power of N equal to x to the power of L over x to the power of M.

Ø So, x to the power of N equal to x to the power L minus M.

Ø Then we can see that N equal to L minus M.

Ø So, we will find that log base x of A over B equal to L minus M.

Ø And finally log base x of A over B equal to log base x of A minus log base x of B is proven.